Symmetric Homoclinic Orbits at the Periodic ... - World Scientific
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International Journal of Bifurcation and Chaos, Vol. 24, No. 8 (2014) 1440006 (18 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414400069
Symmetric Homoclinic Orbits at the Periodic Hamiltonian Hopf Bifurcation Lev Lerman∗ and Anna Markova Department of Differential Equations and Mathematical Analysis, N. I. Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Ave., Nizhny Novgorod 603950, Russia ∗[email protected] Received February 2, 2014 We prove the existence of symmetric homoclinic orbits to a saddle-focus symmetric periodic orbit that appears in a generic family of reversible three degrees of freedom Hamiltonian system due to periodic Hamiltonian Hopf bifurcation, if some coefficient A of the normal form of the fourth order is positive. If this coefficient is negative, then for the opposite side of the bifurcation parameter value, we prove the existence of symmetric homoclinic orbits to saddle invariant 2-tori. Keywords: Periodic Hamiltonian Hopf bifurcation; homoclinic orbit; reversible system; normal form.
1. Introduction and Set-Up Periodic Hamiltonian Hopf Bifurcation (HHB) has received much attention from different angles [Heggie, 1985; Bridges et al., 1995; Glebsky & Lerman, 1996; Jorba & Olle, 2004; Olle et al., 2008]. This bifurcation occurs in many applied systems, see [Heggie, 1985; Howard et al., 1986; Olle et al., 2004; Kao et al., 2014]. The bifurcation was investigated in detail, mainly from the viewpoint of the existence of invariant tori (or invariant curves for the case of a symplectic map). Another restriction (unnecessary in our opinion) in the papers mentioned (except [Glebsky & Lerman, 1996]) was the assumption of an irrational ratio for the frequencies in the critical case. In our study below, we avoid this assumption of irrationality. We think the KAM type results obtained in the papers mentioned should be supplemented with similar results under the assumption of the absence of strong resonances. This would lead to their applicability for an open set of Hamiltonians. Nonetheless, an important aspect of the problem has remained unsolved. This concerns the appearance of homoclinic orbits to a periodic orbit
of the saddle-focus type that arises as a result of the periodic HHB at the positive sign of some characteristic quantity A (see below), as well as homoclinic orbits to periodic orbits and invariant saddle two-dimensional tori which exist in the system for the opposite sign of this quantity A. It is well known that the existence of such orbits in the system is one of the main indicators for the chaotic behavior of the system [Smale, 1965; Shilnikov, 1967] and therefore of its nonintegrability [Cushman, 1978; Kozlov, 1983; Koltsova & Lerman, 1999]. Especially it is clear when we have in addition the transversal intersection of the related stable and unstable manifolds for the periodic orbit [Smale, 1965; Shilnikov, 1967; Koltsova & Lerman, 1999]. The proof of existence of homoclinic orbits at this bifurcation is rather a delicate problem as is seen in [Gaivao & Gelfreich, 2011]. The point is that the splitting of stable and unstable manifolds for the related bifurcation in a two degrees of freedom analytic Hamiltonian system is exponentially small in perturbation. This follows from [Neishtadt, 1984]. It is clear that the problem is not easier for the case of periodic HHB. Nonetheless, sometimes the existence of homoclinic orbits
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can be proved by geometric methods using a feature of the system that appears rather frequently in application problems. This feature is the reversibility of the related Hamiltonian system. Existence of homoclinic orbits in a reversible system (usually not Hamiltonian) at the related reversible Hopf bifurcation was proved in [Iooss & Perouem´e, 1993] using the Lyapunov–Schmidt type method. Here, we extend this result onto the case of three degrees of freedom using the geometric tool. We consider both possible cases of the direct and inverse bifurcations, which corresponds to the case A < 0 and A > 0 below. Let us formulate the problem that we are investigating more precisely. We consider a Hamiltonian system on a smooth1 six-dimensional symplectic manifold M with a closed nondegenerate skew symmetric 2-form Ω with a smooth Hamiltonian H (as an example, one may think of the coordinate linear space R6 with 2-form dx ∧ dy w.r.t. coordinates (x, y) = (x1 , x2 , x3 , y1 , y2 , y3 )). We assume the system to have a periodic orbit γ of period T and the set of multipliers for this orbit consists (except for the common double unit) of two double ones e±iν = ±1 on the unit circle in the complex plane (it is a so-called Krein collision of multipliers [Arnold et al., 1993]). Then this periodic orbit will belong to a smooth two-dimensional symplectic cylinder filled with other closed periodic orbits. This family of periodic orbits is smoothly parameterized by the value h of the Hamiltonian. In a neighborhood of the critical orbit γ in M , one can construct symplectic coordinates in such a way that one variable θ (mod 2π) is angular along the periodic orbit, its conjugated variable I is some smooth function of the Hamiltonian and the remaining four variables go in the transverse directions to the symplectic cylinder [Bryuno, 1989]. Now, if one fixes the level of Hamiltonian H = H(γ), then near γ one can express I as a function of the remaining four variables, θ and h, take angular variable as a new “time” and then one gets a smooth one-parameter family of 2π-periodic nonautonomous Hamiltonian systems with two degrees of freedom (such systems are frequently called systems with two and a half degrees of freedom). Each system of the family obtained has a 2π-periodic solution at each (small enough) value of the parameter ε (this parameter corresponds to the former value H −H(γ)). Without 1
losing in generality, one can regard this periodic solution (x(t, ε), y(t, ε)) as given (x(t, ε), y(t, ε)) ≡ (0, 0) for all small ε. This situation is further studied later on.
2. Normal Form and Strong Resonances Multipliers for the linearized system at zero 2πperiodic solution of 2π-periodic system depend on ε. Now multipliers are four complex numbers and our assumption means that at ε = 0, we have a periodic solution with two double multipliers on the unit circle e±2πiω , 0 < ω < 1, ω = 1/2, distinct from ±1. We assume in addition that at ε = 0 the monodromy matrix of the linearized system has, as its Jordan form, two two-dimensional boxes. By the Lyapunov theorem (see, for instance, [Ivanov & Sokol’skii, 1980; Bryuno, 1989]) this family of linearized systems is reduced to the family of linear systems with real constant coefficients by a linear real 2π-periodic symplectic change of variables with smooth dependence on ε. Our assumption implies that the reduced linear system has at ε = 0 the matrix for which its Jordan normal form consists of two two-dimensional boxes. Then, as the parameter ε varies near zero, the related linear (autonomous) Hamiltonian systems will have as eigenvalues either a quartet of complex numbers for one side of ε = 0 (a saddle-focus equilibrium for the linear system) or two different pairs of pure imaginary numbers for another side from ε = 0 (an elliptic equilibrium) (the dependence of multipliers on the parameter for the related monodromy matrix of linear 2π-periodic system is plotted in Fig. 1). This change of the equilibria types is standard for the autonomous Hamiltonian Hopf bifurcation [van der Meer, 1985]. To study solutions in a neighborhood of the zero periodic solution, we apply the usual normal form method [Ivanov & Sokol’skii, 1980; Bryuno, 1989]. First, we assume the linear part of 2π-periodic system to have been reduced to the linear normal form with constant complex coefficients. After that the normal form transformations with 2π-periodic coefficients are applied in order to eliminate nonresonant terms of the third and fourth orders in the periodic Hamiltonian. In contrast to the autonomous Hamiltonian Hopf bifurcation, resonances can exist here. In order to kill all third order
If another is not specified, by “smooth” we understand C ∞ -smooth. 1440006-2
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Fig. 1.
Multipliers at the periodic HHB.
terms, one needs to require the absence of the resonances ω = 1/3, 2/3, and to normalize the fourth order terms, one needs to demand in addition the absence of resonances ω = 1/4, 3/4. All these resonances are usually said to be strong. In the absence of strong resonances a generic one-parameter family of normalized real Hamiltonians up to the fourth order terms looks like (1) (see [Ivanov & Sokol’skii, 1980] and recent [Olle et al., 2005]): δ ε H4 = (p21 + p22 ) + ω(p1 q2 − p2 q1 ) + (q 21 + q 22 ) 2 2 + (q 21 + q 22 )[A(q 21 + q 22 ) + B(p1 q2 − p2 q1 ) + C(p21 + p22 )],
(1)
where δ, ω, A, B, C depend on ε, δ(0), ω(0) = 0 and we assume a genericity condition A(0) = 0 to hold, ε is the governing parameter. For further use, it is convenient to scale the system in order δ = 1 for every value of ε near the critical ε = 0. More exactly, the scaling allows one to make δ = ±1, in order to change the sign of δ one needs to reverse time (or to pass H → −H). This is assumed to be done later on. Now, as is easily verified, for the vector field XH4 with ε positive, the equilibrium at the origin is elliptic, and it is saddle-focus as ε < 0. In [Ivanov & Sokol’skii, 1980] the normal form was derived using the Deprit–Hori method (see, for instance, [Markeev, 1978]). The normal form can also be constructed by the customary generating function method. Let us remark that the normal form (1) is an autonomous Hamiltonian, though the system is 2π-periodic. Moreover, when the strong resonances are absent, the truncated normal form of the fourth order coincides with the normal form at the standard Hamiltonian Hopf bifurcation. It implies that all dynamical features found in this
integrable two degrees of freedom system [van der Meer, 1985; Arnold et al., 1993; Glebsky & Lerman, 1995] take place for this truncated normal form (for the case of irrational ω/2π these analogies and features are discussed in [Olle et al., 2005]). If we shall try to normalize the Hamiltonian up to terms of higher orders, we need to demand the absence of resonances of higher orders. Then the truncated Hamiltonian will be also autonomous and the normal form coincides with related standard normal form of the Hamiltonian Hopf bifurcation again and the periodic terms will be shifted to the higher terms. For instance, if ω is irrational and Hamiltonian is analytic or C ∞ -smooth, then resonances of any order are absent and the Hamiltonian can be formally transformed to the autonomous normal form being the same as for autonomous Hamiltonian Hopf bifurcation [Sokol’skii, 1977; van der Meer, 1985; Arnold et al., 1993; Glebsky & Lerman, 1995; Olle et al., 2005]. In particular, this autonomous Hamiltonian is integrable and it implies that all homoclinic phenomena that arise for the full analytic Hamiltonian are of an exponentially small order [Neishtadt, 1984; Gaivao & Gelfreich, 2011]. When Hamiltonian is normalized up to the terms of fourth order, the remainder is 2π-periodic and of fifth order in (p, q). The more advanced study of the normal form for the irrational ω, estimates on the remainders and analytic issues related with these transformations, as well as KAM type results for this case, one can find in [Olle et al., 2008], for the numerical simulations related to two models of coupled Hen´ on type symplectic reversible maps, see [Jorba & Olle, 2004]. For 2π-periodic system, it is also natural to consider its Poincar´e map through period 2π. The system with Hamiltonian H4 is integrable and
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autonomous, therefore its Poincar´e map is an integrable symplectic map being 2π-shift along the orbits of the vector field XH4 . As ε < 0, this map has at the origin a fixed point of the saddle-focus type and for ε > 0 this map has an elliptic fixed point. The Poincar´e map for the full system is a perturbation of this integrable map. In order to apply geometric methods for proving the existence of homoclinic orbits, we assume in addition the initial three degree of freedom system to be reversible with respect to involution L and the system has a symmetric critical periodic orbit γ. This assumption makes the related nonautonomous 2π-periodic vector fields to be reversible and related Poincar´e maps be reversible symplectic diffeomorphisms. Recall some proper definitions [Devaney, 1976; Vanderbauwhede & Fiedler, 1992; Lamb & Roberts, 1998]. Definition 2.1. A nonautonomous vector field v(x, t) on a smooth manifold M , x ∈ M , is said to be reversible w.r.t. an action of a smooth involution L : M → M , L2 = idM , if the following identity holds:
DLx (v(x, t)) = −v(L(x), −t). A diffeomorphism f : M → M of the manifold M is said to be reversible w.r.t. an action of a smooth involution L : M → M , L2 = idM , if the following identity holds: L ◦ f ◦ L = f −1 . This definition implies that if x(t) is a solution of a reversible vector field x˙ = v(x, t), then x1 (t) = Lx(−t) is its solution as well. For a reversible diffeomorphism f the related property reads as follows: if {xn } is an orbit: xn+1 = f (xn ), n ∈ Z, then {yn }, yn = Lx−n is an orbit as well. Since the vector field we study is nonautonomous and 2πperiodic, we consider the graphs of solutions (i.e. integral curves) in the extended phase space M ×S 1 , S 1 = [−π, π] mod 2π, and will consider L as acting in this manifold as follows: L1 (x, t) = (L(x), 2π −t). Then in sections t = 0 and t = π = −π(mod 2π) we have fixed points of involution L1 corresponding to Fix(L) ⊂ M . For the case under study, a nonautonomous Hamiltonian vector field is generated by an autonomous Hamiltonian vector field with three degrees of freedom. We assume in addition this
latter vector field to be reversible w.r.t. a smooth anti-symplectic involution L that has a threedimensional smooth submanifold of fixed points Fix(L). Anti-symplectic means the identity L∗ Ω = −Ω holds. Recall that for the case of anti-symplectic involution the set of its fixed points Fix(L) is a Lagrangian submanifold [Meyer, 1981] being threedimensional for our case. We also assume that Hamiltonian H is invariant w.r.t. L : H ◦ L ≡ H and the vector field XH has a symmetric periodic orbit γ of the type described above. Recall an orbit of a reversible (autonomous) vector field to be symmetric, if it is invariant w.r.t. the action of L. There is a characteristic property of a symmetric periodic orbit of a reversible vector field [Devaney, 1976; Vanderbauwhede & Fiedler, 1992]: such an orbit intersects the manifold Fix(L) at exactly two points which are apart from each other through a half period in time. Moreover, if some flow orbit intersects Fix(L) at two points, then this orbit is periodic and its period is twice that of the time distance between these intersection points. When an orbit intersects Fix(L) at a unique point, then this orbit is symmetric nonperiodic (for instance, it can be a symmetric equilibrium, if the orbit coincides with this point, also it can be a symmetric homoclinic orbit to either a symmetric equilibrium or a symmetric periodic orbit, also it can be a symmetric quasi-periodic orbit). Generically, symmetric periodic orbits of a reversible vector field in case, if the dimension of Fix(L) is equal to the half of dim M , form a oneparameter family, even if the system is a nonHamiltonian one. If these conditions are met and the vector field is in addition a Hamiltonian one, then this family of symmetric periodic orbits locally near a basic symmetric periodic orbit γ coincides with that family of periodic orbits for the Hamiltonian vector field which is obtained by continuing in the value of H (if the family is locally unique). To explain the genericity condition mentioned, let us consider a reversible system on a smooth 2n-dimensional manifold and let L be its reversing involution with n-dimensional submanifold of Fix(L). Let γ be a symmetric periodic orbit which intersects Fix(L) at two points m, m1 . It is always possible to choose two cross-sections N, N1 to the flow through the points m, m1 , respectively, such that both cross-sections would contain a related local piece of Fix(L). Then the smooth transition flow map S : N → N1 is defined near point m,
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S(m) = m1 . Denote R, R1 the pieces of Fix(L) in N, N1 , respectively, and consider the image S(R) ⊂ N1 . One has S(R) ∩ R1 = ∅ (it contains m1 ) and assume this intersection be transverse and hence it occurs along some smooth curve σ1 . Every point on σ1 being close enough to m1 has its preimage in R, they make up a curve σ ⊂ R ⊂ Fix(L). All flow orbits through points on σ intersect Fix(L) also at points on σ1 , therefore they intersect Fix(L) twice and all are periodic orbits. Periodic orbits for which this transversality condition is met were called elementary in [Devaney, 1976], we will also use this term. Now let us consider the case, when M is a smooth symplectic 2n-dimensional manifold, H is a smooth Hamiltonian, L is an anti-symplectic involution and XH is reversible w.r.t. L with Fix(L) being a smooth submanifold in M , H ◦ L ≡ H. Let γ be an elementary symmetric periodic orbit for XH and m, m1 be its intersection points with Fix(L). Choose two cross-sections N, N1 at these points which contain pieces R, R1 of Fix(L). Then we have the smooth curves σ ⊂ R, σ1 ⊂ R1 . Suppose, in addition, that the Poincar´e map Nγ → Nγ , Nγ = N ∩ {H = H(γ)}, does not have a unit eigenvalue of its linearization at m. In this case, the cylinder formed by periodic orbits near γ, being the continuation of γ in H, is unique and transverse to the level H = H(m). It implies this cylinder to coincide with the cylinder formed by symmetric periodic orbits through points of σ near γ and hence the tangent vector to σ at the point m (it belongs to Tm R) is not tangent to submanifold H = H(m). Thus submanifold H = H(m) (of dimension 2n − 1) and Fix(L) intersects at m transversely. So, we have proved the following assertion: Lemma 1. Let XH be a smooth reversible Hamiltonian vector field on a smooth symplectic 2n-dimensional manifold, L be a reversible anti-symplectic involution and γ be an elementary symmetric periodic orbit of XH . If γ does not have any unit multipliers except for always existing double unit, then near points {m, m1 } = γ ∩ Fix(L) submanifolds Fix(L) and H = H(γ) are transverse and all periodic orbits of XH being close to γ are symmetric.
In this case, traces of periodic orbits of the family on the submanifold Fix(L) form two smooth
local curves σ, σ1 near two points m, m1 being the traces of γ. We assume γ elementary, then the lemma implies that five-dimensional smooth levels of the Hamiltonian near γ intersect threedimensional disk Fix(L) transversely and thus along two two-dimensional disks. In view of the invariance of the Hamiltonian levels w.r.t. the autonomous Hamiltonian system, one can choose symplectic coordinates near γ in such a way that in these coordinates (x, y, I, θ) periodic solutions of the family are given as x = y = 0, I = I0 , and after transition to the time θ the related 2π-periodic Hamiltonian vector field will be reversible w.r.t. to the involution L1 with its Fix(L1 ) consisting of two two-dimensional disks that cut the periodic integral curve x = y = 0 at two points in sections θ = 0, π (to get this we use results of [Meyer, 1981] and [Bryuno, 1989]). Now let us write down the vector field obtained in a neighborhood of the solution X = (x, y) = 0 (we shall preserve for the time variable here and below its usual notation t). X˙ = I∇H(X, t, ε),
∇H(0, t, ε) ≡ 0.
Since we work in a small neighborhood of the set {0} × S 1 ⊂ R4 × S 1 , one may assume involution L to be linear, due to results [Bochner, 1945; Meyer, 1981]: Vector-function I∇H obeys the identity LI∇H(X, t, ε) = −I∇H(LX, −t, ε), due to the reversibility. Linearized at the zero solution system has the matrix B(t, ε) = ID 2 H(0, t, ε) which in turn satisfies the identity LB(t) = −B(−t)L. Then the normalized fundamental matrix Φ(t) at t = 0 obeys the identity LΦ(t) = Φ(−t)L. It is also a symplectic matrix for any t. By the Lyapunov theorem the linear transformation with the matrix S(t), S(t) = Φ(t)e−Ct , reduces this periodic system to that with constant coefficients. Matrix C is the logarithm of matrix Φ(2π) divided at the period 2π. Since this latter matrix has no negative eigenvalues, matrix Φ(2π) does have a real logarithm [Gantmacher, 1959]. Thus the linearized system by a linear real symplectic 2π-periodic change of variable is reduced to a linear Hamiltonian system with constant matrix C = ID, D = D. Further, similar to [Haragus & Iooss, 2011], we make a linear symplectic change of variables that transforms simultaneously matrix L to the form L = diag (−1, 1, 1, −1), and a quadratic Hamiltonian to
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the form in coordinates (p1 , p2 , q1 , q2 ) 1 H2 = (p21 + p22 ) + ω(p1 q2 − p2 q1 ) 2 ε + (q 21 + q 22 ). 2 After that, following [Bryuno, 1989] we normalize the terms of higher order with the preservation of reversibility w.r.t. to L. Under the assumption of the absence of strong resonances, we come to the normal form (1) up to the fourth order terms.
3. Case A > 0 The main result of this section is the following theorem which we formulate for the initial Hamiltonian system with three degrees of freedom, though the proof will be done for the related one-parameter family of periodic Hamiltonian systems with two degrees of freedom. Theorem 1. Let H be a smooth Hamiltonian on a smooth six-dimensional symplectic manifold (M, Ω) and vector field XH has a periodic orbit γ whose nontrivial multipliers are two double nonsemisimple complex numbers on the unit circle exp[±iν], ν = 0, π (mod 2π). In this case, γ belongs to a symplectic cylinder filled with periodic orbits of XH close to γ and we suppose, to be definite, that for levels H = h < 0 the related periodic orbits on the cylinder are hyperbolic (saddle-focus type) and for levels H = h > 0 the related periodic orbits on the cylinder are elliptic ones. We assume, in addition, the vector field to be reversible w.r.t. smooth antisymplectic involution L having three-dimensional set of fixed points, H be invariant under the action of L and γ be symmetric. Then if some quantity A (see above) is positive and h < 0, then the saddlefocus periodic orbit in the level H = h is symmetric and has at least four symmetric homoclinic orbits.
It is worth remarking that this theorem gives not only the existence of homoclinic orbits themselves, but also localize their intersection points with Fix(L). This section is devoted to the proof of this theorem. Remark. Homoclinic solutions of this theorem are
so-called one-round or one-hump ones, each of them goes around an unperturbed symmetric homoclinic solution only one time. It is well known [Shilnikov,
1967] that if stable and unstable manifolds of the full 2π-periodic system intersect transversely along this persistent homoclinic solution, then there are other multiround homoclinic orbits and many other orbits described by a symbolic system. The system with Hamiltonian H4 is integrable, its additional integral is K = p1 q2 − p2 q1 . Due to this fact, if A > 0 and ε < 0, stable and unstable manifolds of the saddle-focus equilibrium at the origin merge forming a “homoclinic skirt” [van der Meer, 1985; Arnold et al., 1993; Glebsky & Lerman, 1995]. The same is valid for the related Poincar´e map for the Hamiltonian vector field XH4 considered as nonautonomous 2π-periodic. This map is 2π-shift along the orbits of the Hamiltonian system generated by XH4 . The homoclinic skirt is topologically a sphere with two of its points identified, the point of gluing corresponds to the saddle-focus. For the map, this set is filled with discrete homoclinic orbits to the fixed point. For the system with the Hamiltonian H4 considered as 2π-periodic in the space R4 × S 1 this homoclinic skirt corresponds to the set of merged three-dimensional stable and unstable manifolds of zero periodic solution, thus it is topologically a direct product of the homoclinic skirt and S 1 . For the full system with the Hamiltonian H, because of the presence of periodic terms of higher order, stable and unstable manifolds of the periodic solutions are generically split. Our aim is to discover which homoclinic solutions do survive. The idea of the proof is to find an equation for the homoclinic skirt and to show the transversal intersection of the skirt with Fix(L), after that to apply the transversality theorem to the stable manifold of the full system to prove its intersection with Fix(L) to persist. But the case under consideration is more subtle, since the size of the skirt is of the order |ε| and it shrinks to the fixed point as ε → −0. Therefore, to get a regular intersection of homoclinic skirt and Fix(L) one needs to blow up a neighborhood of the fixed point (or the related periodic solution). To do this, we use coordinates depending on ε in such a way that the size of the skirt would become finite. To this end, we make a transformation √ qi → −εQi , pi → −εPi , i = 1, 2 √ −ε. and use a new small parameter µ = After that, the system in coordinates Q, P with respect to 2-form dP ∧ dQ coincides with the
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Hamiltonian H = ω(P1 Q2 − P2 Q1 ) +
µ 2 (P + P 22 ) 2 1
µ − (Q21 + Q22 ) + (Q21 + Q22 )[µA(Q21 + Q22 ) 2 + µ2 B(P1 Q2 − P2 Q1 ) + µ3 C(P 21 + P 22 )] + ···, (2) here dots mean the terms of order five and higher in (P1 , P2 , Q1 , Q2 ) and they are of at least the second order and higher in µ. Now as an integrable approximation, we choose H0 : H0 = ω(P1 Q2 − P2 Q1 ) +
µ 2 (P + P 22 ) 2 1
Fig. 2.
µ − (Q21 + Q22 ) + µA(Q21 + Q22 )2 2
saddle for which (r, P ) obeys the equation:
µ2
and terms of the order and higher we consider as a perturbation. In order to find stable and unstable manifolds of the saddle-focus equilibrium for the vector field XH0 , we use symplectic polar coordinates similar to [Glebsky & Lerman, 1995]: ϕ , r ϕ P2 = P sin ϕ − k cos , r P1 = P cos ϕ + k sin
Q1 = r cos ϕ, Q2 = r sin ϕ,
Phase portrait of the reduced system for K = k = 0.
(3)
dP1 ∧ dQ1 + dP2 ∧ dQ2
P 2 − r 2 + 2Ar 4 = 0. This homoclinic orbit corresponds to the homoclinic skirt Γ of integrable system with two degrees of freedom. In coordinated (P1 , P2 , Q1 , Q2 ) the skirt is written in a parametric form with parameters (P1 , Q2 ): P1 Q2 Γ : P1 , , Q1 (P1 , Q2 ), Q2 , Q1 (P1 , Q2 ) where Q1 (P1 , Q2 ) (1 − 2AQ22 ) + (1 − 2AQ22 )2 − 8AP 21 . =± 4A
= dP ∧ dr + dϕ ∧ dk. In new coordinates H0 casts µ k2 µ 2 ˆ P + 2 − r 2 + µAr 4 . H0 = ωk + 2 r 2
As was remarked, L acts as follows L(P1 , P2 , Q1 , Q2 ) = (−P1 , P2 , Q1 , −Q2 ).
Then we get two differential equations for variables (r, P ) (the reduced system) which do not depend on other variables but depend on k as a parameter. Two more equations are k˙ = 0 and ϕ˙ = −ω −µk/r 2 . Let us set k = 0, keeping in mind that homoclinic orbits to the equilibrium at the origin belong to this level. The reduced system then has a saddle at the origin, it corresponds to the saddle-focus equilibrium of XH0 and to the zero periodic orbit of the related 2π-periodic system. One more equilibrium, center, corresponds to an elliptic periodic solution of XH0 and belongs to the level H0 = 0. The phase portrait of the reduced system is plotted in Fig. 2. The reduced system has a homoclinic orbit to the
Then the set of fixed points for L is given by the equations: Fix(L) = {P1 = 0, Q2 = 0}. Homoclinic skirt intersects the plane of Fix(L) at two points: 1 (4) 0, 0, ± √ , 0 . 2A An explicit calculation shows that the twodimensional tangent plane to the skirt at intersection points (4) is spanned by two vectors (1, 0, 0, 0) and (0, 0, 0, 1). The tangent plane to Fix(L) is
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formed by vectors (0, 1, 0, 0) and (0, 0, 1, 0). Thus the skirt is transverse to Fix(L) at both points. If we consider this integrable system as being 2πperiodic, one need to regard the skirt as threedimensional in coordinates (P1 , P2 , Q1 , Q2 , t). Also for 2π-periodic integrable system, we consider involution L1 instead of L with Fix(L1 ) being two two-dimensional disks at the sections t = 0 and t = ±π. Since for the autonomous integrable system stable manifold of the saddle-focus equilibrium intersects 2-disk Fix(L) √ transversely at two different points (0, 0, ±1/ 2A, 0), then for the same system considered in R4 × S 1 stable manifold of the zero periodic solution intersects Fix(L √1 ) transversely at four √ different points (0, 0, ±1/ 2A, 0, 0) and (0, 0, ±1/ 2A, 0, π). The tangent vectors to the skirt are (1, 0, 0, 0, 1), (0, 0, 0, 1, 1), (0, 0, 0, 0, 1) and to Fix(L1 ) are (0, 1, 0, 0, 0) and (0, 0, 1, 0, 0).
3.1. Existence of homoclinic orbits To prove the theorem we wish to use the averaging method in its classical form [Bogoliubov & Mitropolski, 1961; Sanders et al., 2007]. If the system is in the standard form of the averaging method (we assume the system to be periodic or uniformly almost periodic in t) x˙ = µF (x, t, µ)
coordinate change (P1 , P2 , Q1 , Q2 ) → (ξ1 , ξ2 , η1 , η2 ), dP1 ∧ dQ1 + dP2 ∧ dQ2 = dξ1 ∧ dη1 + dξ2 ∧ dη2 , using a generating function S(P1 , P2 , η1 , η2 , t) = (P1 η1 + P2 η2 ) cos(ωt) − (P1 η2 − P2 η1 ) sin(ωt). In fact, this change of variables is used because all orbits of the linear Hamiltonian system with the Hamiltonian ωK are 2π/ω-periodic (except for the equilibrium). The transformation we make is nothing else but the linear symplectic 2π/ω-periodic transformation with the fundamental matrix of this linear system. This transformation just removes the term ωK in the Hamiltonian. Hamiltonian is transformed in accordance with the formula: H(P1 , P2 , η1 , η2 , t) +
˜ 1 , P2 , η1 , η2 , t). = H(P Then new Hamiltonian casts: ˜ = µ (ξ 21 + ξ 22 ) − µ (η 21 + η 22 ) + µA(η 21 + η 22 )2 H 2 2 ˜ 1 , ξ2 , η1 , η2 , µ, t). + µ2 h(ξ
and its averaged system x˙ = µF0 (x, µ),
∂S (P1 , P2 , η1 , η2 , t) ∂t
F0 (x, µ) = lim
T →∞ 0
T
F (x, t, µ)dt
for a fixed small µ > 0 has a hyperbolic equilibrium xµ , then the full system has in a O(µ2 )neighborhood of x0 a hyperbolic periodic solution (if F is periodic) or an almost periodic solution with the basic of frequencies as for F satisfying the exponential dichotomy of the same type as the equilibrium x0 . These solutions of the full system have their local stable and unstable manifolds which periodically or almost periodically depend on t. But we need to drag these manifolds till the set Fix(L1 ) and to prove their C 1 -proximity to the related manifolds of the averaged system in order to be able to use the transversality theorem. To reduce the problem to the standard form of the averaging method we need first to remove in the Hamiltonian the term ω(P1 Q2 − P2 Q1 ). To this end, we make a symplectic 2πω-periodic in time
(5)
The Hamiltonian contains terms of periods 2π and 2π/ω. This implies that if ω is irrational, the Hamiltonian and its related system will be quasiperiodic in t with two incommensurate frequencies, but if ω is rational, ω = q/p, q, p ∈ N, 0 < q < p and p, q have not common divisors, then the Hamiltonian and the related system will be 2πq-periodic in t. It is worth remarking that in both cases the averaged Hamiltonian up to the terms of the order µ will be the same integrable Hamiltonian H0 without the rotation term ξ1 η2 − ξ2 η1 . After 2π/ω-coordinate change the twodimensional disks which were the set Fix(L1 ) in the extended phase space R4 × S 1 , now become in R4 × R a countable set of 2-disks at the sections t = 2πm/ω and t = π + 2πm/ω. Their coordinate representations become
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and ξ1 cos((2n + 1)πω) − ξ2 sin((2n + 1)πω) = 0, η1 sin((2n + 1)πω) + η2 cos((2n + 1)πω) = 0. For the averaged autonomous system we proved the transversality of stable manifold of zero equilibrium O and Fix(L). In the extended phase space we get the cylindrical surface being the direct product of the stable manifold of O and R, this surface intersects transversally the countable set of 2-disks uniformly. Now, if we prove sufficient uniform C 1 proximity of the stable manifold for quasi-periodic or 2πp-periodic solution of the full system existing near O × R, then we shall get the countable set of intersection points. Returning to the initial system with the Hamiltonian (2), we come to the existence of four homoclinic solutions for the zero periodic solution. Indeed, let us consider for the system after scaling its 2π-Poincar´e map from the section t = 0 to t = 2π. If we consider its approximation up to O(µ2 ) order, this symplectic diffeomorphism is integrable, has a saddle-focus fixed point at the origin and its stable and unstable manifolds coalesce forming the homoclinic skirt. This skirt intersects transversely the set Fix(L1 ) at the section t = 0 at two points. Below, it will be proved that the stable manifold of the saddle-focus fixed point will be sufficiently close in C 1 topology for positive µ small enough to the stable manifold of the integrable approximation up to its intersection with Fix(L1 ) and hence intersects it transversely at a close point giving a symmetric homoclinic orbit to the fixed point. Thus we get two such points. But the same is true for the Poincar´e map constructed from the section t = −π to t = π. Here, we also get two intersection points of the stable manifold and the other piece of Fix(L1 ). Thus, we obtain four symmetric homoclinic orbits to zero periodic solution of 2π-periodic Hamiltonian system for each negative ε small enough in its modulus. Here, we need specially to prove a sufficient C 1 proximity of invariant manifolds till its intersection with the set of disks is obtained, the standard averaging methods do not pay special attention to this question [Bogoliubov & Mitropolski, 1961; Sanders et al., 2007]. It follows only from the standard averaging theory the existence of a periodic solution
being close to the set “{equilibrium}×S 1 ” in the full system and the smoothness of its stable (unstable) manifolds. We shall prove in Sec. 5 C 1 -proximity of these manifolds on the global piece till their crossing with Fix(L1 ) (in fact, on a bit longer piece) in two steps: first, we prove this locally near the periodic solution and after that in times of the order 1/µ in a neighborhood of a symmetric homoclinic orbit of the unperturbed system up to its intersection with Fix(L). Let us remark that for the Poincar´e map defined by 2π-periodic reversible system on the cross-section t = 0, the transversality of stable manifold of the fixed point and Fix(L) does not imply the symmetric homoclinic orbit be transverse, that is stable and unstable manifolds of the fixed point for the Poincar´e map intersect transversely. Here for the case of four-dimensional reversible symplectic diffeomorphism two possible cases can occur: (1) this homoclinic orbit is indeed transverse, and (2) this homoclinic orbit has a first order tangency of these manifolds.
4. Case A < 0 When A is negative, the types of periodic orbits p = q = 0 for 2π-periodic system, as ε varies, are the same as for the case A > 0 (recall that we scale δ = 1). But an essential difference with the case A > 0 is the local bifurcation that occurs here for values ε > 0 [van der Meer, 1985; Arnold et al., 1993; Glebsky & Lerman, 1995]. It is well known that a one-parameter family of periodic orbits branches out from the equilibrium of the integrable √ vector field XH4 in a neighborhood of order ε in this case.2 The collection of these periodic orbits forms a two-dimensional set being as before a topological sphere with two of its points identified. This set is filled with periodic orbits (except for the equilibrium itself), but these orbits are of three types. Namely, locally in a O(ε)-neighborhood of the equilibrium the orbits are elliptic (on the related level of Hamiltonian H4 such an orbit is enclosed by invariant 2-tori), they lie on the related level of the Hamiltonian in pairs, as one from two different local Lyapunov families. When moving along each of these two Lyapunov families further from the equilibrium, one meets a parabolic periodic orbit, two
2
This phenomenon gave one a reason to call this bifurcation as Hamiltonian Hopf bifurcation [van der Meer, 1985] in analogy with the usual Andronov–Hopf bifurcation [Shilnikov et al., 1998]. Following the historical sequence it would be better to name this bifurcation as Hamiltonian Andronov–Hopf bifurcation. 1440006-9
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parabolic orbits marked cut off from the singular sphere an annulus filled with hyperbolic (saddle) periodic orbits. Stable and unstable manifolds of each such orbit belong to the same level K = k as the periodic orbit itself, and two sheets of stable and unstable manifolds merge forming a homoclinic skirt, two√other sheets leave the neighborhood. As |k| → 1/6 −A, the hyperbolic periodic orbit at the level K = k with its skirt and the elliptic periodic orbit lying inside the skirt approach and coalesce into the related parabolic periodic orbit and disappear for greater |k|. At the level K = 0, there is a unique hyperbolic orbit, two sheets of stable and unstable manifolds coalesce forming a homoclinic skirt for this periodic orbit. The period of this orbit is equal to 2π/ω, so, if ω is rational, we get here a resonance with the period of the system. In this case, for the full system in the related 2πPoincar´e map instead of a closed invariant curve we have on the cylinder a resonance band with periodic points of elliptic and hyperbolic types forming a heteroclinic chain [Kaper & Kovacic, 1996]. When the system under study is reversible, then periodic orbits of the integrable system are symmetric and their skirts intersect transversely the set Fix(L1 ) at two points giving two symmetric homoclinic orbits for each corresponding periodic orbit. All this will be shown below for the reader’s convenience. For the periodic Hamiltonian Hopf bifurcation in the absence of strong resonances, the vector field XH0 describes only the behavior of the averaged system (in the scaled variables). This integrable system has the whole family of saddle periodic orbits for all values k close to zero of integral K. It means that we cannot apply directly the conclusions of the averaging theory, instead we have to use results of the theory of normally hyperbolic invariant sets [Fenichel, 1974; Hirsch et al., 1977]. Our main result in this section is the following theorem which we again formulate for the initial autonomous Hamiltonian system with three degrees of freedom. Theorem 2. Let H be a smooth Hamiltonian on a smooth six-dimensional symplectic manifold (M, Ω) and vector field XH has a periodic orbit γ whose nontrivial multipliers are two double nonsemisimple complex numbers on the unit circle exp[±iν], ν = 0, π (mod 2π). In this case, γ belongs to a symplectic cylinder filled with periodic orbits of XH close to γ and we suppose, to be definite, that for levels H = h < 0 the related periodic orbits on the
cylinder are hyperbolic (saddle-focus type) and for levels H = h > 0 the related periodic orbits on the cylinder are elliptic ones. We assume, in addition, the vector field to be reversible w.r.t. smooth antisymplectic involution L having a three-dimensional set of fixed points, H be invariant under the action of L and γ be symmetric. Then if some quantity A (see above) is negative and h > 0, then near the elliptic periodic orbit at the level H = h, there exists a Cantor set of saddle symmetric invariant twodimensional tori whose stable and unstable manifolds intersect each other along at least four symmetric homoclinic solutions. We outline only the main points of the proof, since it follows along the same lines as above for the case A > 0. After the same √ transformations as before, only with ε > 0, µ = ε, the Hamiltonian casts: µ H = ω(P1 Q2 − P2 Q1 ) + (P 21 + P 22 ) 2 µ + (Q21 + Q22 ) + µA(Q21 + Q22 )2 + O(µ2 ) 2 and we consider the terms of order µ2 and higher as a perturbation. In symplectic polar coordinates (3) we get a Hamiltonian up to O(µ2 ) terms µ k2 2 2 4 ˆ H0 = ωk + P + 2 + r + 2Ar , 2 r which generates a reduced system with one degree of freedom, its phase portrait is as in Fig. 3. This system has a saddle equilibrium at the point P = 0, r = −1/4A whose two separatrices coalesce forming the homoclinic loop. This homoclinic
Fig. 3. Phase portrait of the reduced system for K = k = 0, A < 0.
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loop belongs to the level of the reduced Hamiltonian ˆ 0 = µ [P 2 + r 2 + 2Ar 4 ] = − µ . H 2 16A Using this relation, the equality k = 0, we express P via r and after integration of the equation r˙ = µP we derive expressions for the solutions on the homoclinic skirt to the symmetric periodic orbit of the system with the truncated Hamiltonian of order µ in coordinates (P1 , P2 , Q1 , Q2 ): P1 =
cos(ωt − ϕ0 ) , √ √ 2µt −8A cosh2 2
sin(ωt − ϕ0 ) , √ √ 2µt −8A cosh2 2 √ 2µt 1 cos(ωt − ϕ0 ), Q1 = √ tanh 2 −4A √ 1 2µt tanh sin(ωt − ϕ0 ), Q2 = − √ 2 −4A
Fig. 4. Phase portrait of the reduced system for K = k = 0, A < 0.
P2 = −
from where we deduce that for the integrable system, the homoclinic skirt of this symmetric saddle periodic orbit intersects Fix(L) transversely giving homoclinic orbits with initial points at t = 0 through the points with ϕ0 = π/2, 3π/2. The periodic orbit has the period 2πω, when k = 0, since 2 for the integrable system, √ one has 2ϕ˙ = −ω 4− µk/r with √ r = r(k) = 1/ −4A + αk + O(k ), α = 2A −4A < 0. Here P = 0, r = r(k) are coordinates of the saddle equilibria of the reduced system for a fixed k. Symmetric saddle periodic orbits of the integrable system for k = 0 but small also have homoclinic skirts, since the related saddle equilibria P = 0, r = r(k) of the reduced system have homoclinic orbits (see Fig. 4). Now we consider the 2π-Poincar´e map of the full system being a perturbation of order µ2 for the Poincar´e map of the integrable approximation. Since the integrable system has the normally hyperbolic cylinder filled with periodic orbits, then the Poincar´e map of the full system also has the normally hyperbolic cylinder for k sufficiently close to zero (the smallness of k depends on ω and µ) [Fenichel, 1974]. The cylinder of the unperturbed Poincar´e map is foliated with closed symmetric
invariant curves, each of them intersects Fix(L) at rotation numbers are 2π(ω + √ two points, their −4Aµk + O(k3 )), so vary strictly monotonically in k (k plays the role of the action variable on the cylinder). Due to the Moser theorem [Moser, 1962], the majority of invariant curves on the cylinder, whose rotation numbers are Diophantine, persist. Every such persisting invariant curve is symmetric and possesses smooth two-dimensional stable and unstable manifolds [Fenichel, 1974] which can be drawn till their intersection with Fix(L). Then we get for the Poincar´e map of the full system the existence of symmetric homoclinic orbits to every such persistent invariant curve.
5. C 1 -Proximity of Stable Manifolds of Integrable and Full Systems In this section, for the case A > 0, we prove C 1 proximity for stable manifolds of integrable and full systems till their intersection with the set of fixed points of involution. This will be proved in two steps: first, we prove this locally near the periodic solution and after that in times of order 1/µ in a neighborhood of a symmetric homoclinic orbit of the unperturbed system up to its intersection with Fix(L). The related system for Hamiltonian (5) system after some additional transformations can be written in the following form:
u˙ = µAu + µf (u, v) + µ2 f1 (u, v, µ, t) (6) v˙ = µBv + µg(u, v) + µ2 g1 (u, v, µ, t),
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where u = (u1 , u2 ), v = (v1 , v2 ), A, B are 2 × 2 real matrices such that eigenvalues of A have negative real parts, and those of B have positive real parts, f , g, f1 , g1 are smooth vector-functions of the third order in (u, v). Integrable system corresponds to (6) with omitting terms f1 , g1 . The proof of existence and smoothness of the stable manifold for a periodic solution for system (6) and its C 1 -proximity to the stable manifold of the integrable system follow the lines in [Shilnikov et al., 1998]. In fact, the local part was proved in [Bogoliubov & Mitropolski, 1961] but there the manifolds were proved to be only Lipschitz continuous. Proposition 1. Zero periodic solution of system (6) has local smooth stable manifold which is C 1 -close to the stable manifold of zero periodic solution of the integrable system up to terms of order µ. Proof. Let us consider the following system of
integral equations with parameters t0 , u0 , µ: t µA(t−t ) 0 u0 + eµA(t−τ ) [µf (u(τ ), v(τ )) u(t) = e t 0 2 + µ f1 (u(τ ), v(τ ), µ, τ )]dτ +∞ v(t) = − eµB(t−τ ) [µg(u(τ ), v(τ )) t + µ2 g1 (u(τ ), v(τ ), µ, τ )]dτ. (7) On the stable manifold every solution of the system (6) is also a solution of the system (7) and vice versa. To prove the existence of solutions for this system, one uses the Banach contraction map principle considering the r.h.s. of (7) as an operator acting in the space of bounded continuous vector-functions (u(t), v(t)) defined on the semi-line t ≥ t0 with sufficiently small norm max u , v . This operator depends continuously on the parameter u0 , so its fixed point, if exists, also depends continuously on u0 .
At the first step, it is proved that this operator transforms the space into itself and is contracting. Therefore, there is a unique solution for u0 small enough. Let us notice that the solution obtained (u(t, t0 , u0 ), v(t, t0 , u0 )) at the section t = t0 gives the function v0 = v(t0 , t0 , u0 ). In R × R4 , its graph in the tube around solution (u, v) = (0, 0) is the stable manifold of this solution. One needs to prove its smoothness and uniform C 1 -proximity to the stable manifold of the integrable system considered as nonautonomous. At the next step, we prove that the stable manifold of zero solution for the full system deviates from that for the integrable system by the quantity of order µ locally. Indeed, solutions for the full sysint tem satisfy (7) and a solution (uint ∗ (t), v ∗ (t)) for integrable system with the same initial condition satisfies the same system with omitting the O(µ2 ) terms. Subtracting them on both sides and making similar estimates as before we get ˜ 2) (1 − Cρ
max
t∈[t0 ,+∞)
int
u∗ (t) − uint ∗ (t), v∗ (t) − v ∗ (t)
≤ µC˜1 . Since ρ is small, the solutions of the integrable and full systems are distinct by order µ, so the stable manifold of periodic solution for the full system deviates from that for the integrable system by order µ. Further, we show the derivatives for the stable manifolds for the integrable and full systems are also close in a neighborhood of zero solution. The solution of system (6) depends on three parameters: (u∗ (t; u0 , t0 , µ), v∗ (t; u0 , t0 , µ)). First we shall show that there is a continuous derivative (u∗ , v∗ ) in u0 . To this end, we formally differentiate (7) in u0 and derive for derivatives def
∂u∗ (t; u0 , t0 , µ), ∂u0
def
∂v∗ (t; u0 , t0 , µ). ∂u0
U ≡
V ≡
The approximation scheme is then as follows t ∂f ∂f µA(t−τ ) 2 ∂f1 2 ∂f1 µA(t−t ) 0 Un (τ ) + µ Vn (τ ) dτ (t) = e + e µ Un (τ ) + µ Vn (τ ) + µ U n+1 ∂u ∂v ∂u ∂v t0 +∞ ∂g ∂g µB(t−τ ) 2 ∂g1 2 ∂g1 Un (τ ) + µ Vn (τ ) dτ. e µ Un (τ ) + µ Vn (τ ) + µ Vn+1 (t) = − ∂u ∂v ∂u ∂v t 1440006-12
(8)
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We seek for solutions of the system (8) using the method of successive approximations as well. If we show the existence of a continuous solution for this system w.r.t. variables (U, V ), this will imply the solution for system (6) depends smoothly on u0 . As a first approximate solution, we take U1 (t) = eµA(t−t0 ) ,
V1 (t) ≡ 0.
The estimates on the r.h.s. of (8) allow one to assert that the sequence (Un (t), Vn (t))
converges and a continuous limit exists (U∗ (t), V∗ (t)) for the approximations (Un (t), Vn (t)) as n → +∞. It means that solution (u∗ , v∗ ) for the system (6) depends smoothly on u0 . The same is valid for the integrable system. Next, we show the derivatives in u0 for solutions of the full and integrable systems differ by order µ. Indeed, if we omit perturbation terms f1 , g1 , it is seen that the derivative in u0 of the solution for the integrable system has to satisfy the following system:
t ∂f int ∂f int µA(t−τ ) int µA(t−t ) 0 + e µ U ∗ (τ ) + µ V ∗ (τ ) dτ U ∗ (t) = e ∂u ∂v t0 +∞ ∂g int ∂g int µB(t−τ ) int e µ U ∗ (τ ) + µ V ∗ (τ ) dτ, V ∗ (t) = − t ∂u ∂v where def
U int ∗ ≡
∂uint ∗ (t; u0 , t0 , µ), ∂u0
def
V int ∗ ≡
∂v int ∗ (t; u0 , t0 , µ), ∂u0
∂f ∂g ∂g int int int int related Jacobi matrices ∂f ∂u , ∂v , ∂u , ∂v are taken at (u∗ , v ∗ ) and U ∗ , V ∗ ≤ 2C0 . Again we prove the differences int max max U∗ (t) − U int ∗ (t) , max V∗ (t) − V ∗ (t) t
t
are of the order µ. So, derivatives in u0 for solutions of integrable and full systems are close up to the terms of order µ. In the same way, one can show the solution (u∗ , v∗ ) of system (6) to depend smoothly on the parameter t0 . To this end we differentiate (7) in t0 : t ∂f ∂v∗ ∂f ∂u∗ ∂u∗ µA(t−t0 ) µA(t−τ ) 2 ∂f1 ∂u∗ 2 ∂f1 ∂v∗ = −e µAu0 + e +µ +µ +µ µ dτ ∂t0 ∂u ∂t0 ∂v ∂t0 ∂u ∂t0 ∂v ∂t0 t0 − eµA(t−t0 ) [µf (u∗ (t0 ), v∗ (t0 )) + µ2 f1 (u∗ (t0 ), v∗ (t0 ), µ, t0 )] +∞ ∂g ∂v∗ ∂g ∂u∗ ∂v∗ µB(t−τ ) 2 ∂g1 ∂u∗ 2 ∂g1 ∂v∗ e +µ +µ +µ µ dτ. ∂t0 = − ∂u ∂t0 ∂v ∂t0 ∂u ∂t0 ∂v ∂t0 t As in the previous case, we shall show that a continuous solution of this system exists. For this, we denote def
∂u∗ (t; u0 , t0 , µ), ∂t0
def
∂v∗ (t; u0 , t0 , µ), ∂t0
U ≡
V ≡
we take: U1 (t) = eµA(t−t0 ) (−µAu0 − µf (u∗ (t0 ), v∗ (t0 )) − µ2 f1 (u∗ (t0 ), v∗ (t0 ), µ, t0 )), V1 (t) ≡ 0.
and will seek for a solution by the successive approximation method. As the first approximation
Carrying out similar calculation one can prove that there exists a continuous limit (U∗ (t), V∗ (t)) for the successive approximations (Un (t), Vn (t))
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as n → +∞. This implies the solution (u∗ , v∗ ) of the system (6) is continuously differentiable in t0 . What is more, the derivatives in t0 for solutions of the full and integrable systems differ by the order µ. Hence local stable manifolds of the periodic solution of integrable and full systems are C 1 -close (close with its derivatives) in the neighborhood of periodic solution. The same holds for local stable manifolds of the fixed point for the Poincar´e map of integrable and full systems. Now, it remains to prove that stable manifolds of the zero solution of integrable and full systems stay C 1 -close up to their intersection with Fix(L1 ) (in fact, one needs to continue them a bit further). To simplify the problem, we introduce in a neighborhood of a segment of the symmetric homoclinic orbit for the integrable system from the local cross-section to the stable manifold till it hits Fix(L1 ), new coordinates in which the orbits of the integrable system are straightened. This can be done, since if one introduces the slow time τ = µt, then the passage time from the local crosssection to the stable manifold in the neighborhood of zero solution till Fix(L1 ) will be finite. Therefore, one can apply the theorem on the straightening trajectories (see, for instance [Arnold, 1992]). We denote these coordinates as (x, y, s, r), where r defines a position of a point on the orbit of the unperturbed integrable system, s enumerates orbits on the unperturbed stable manifold and (x, y) are coordinates in the transverse direction to the stable manifold. Since the full system is nonautonomous, additional t-coordinate appears. Without loss of generality one may regard r = const to be a cross-section for the orbits of the unperturbed system near the symmetric homoclinic orbit. It is more convenient to use the notation def x ≡ (x, y, s). In new coordinates, the perturbed system has the form:
x˙ = µ2 ϕ(x, r, t) (9) r˙ = µ + µ2 ψ(x, r, t), where functions ϕ, ψ are continuously differentiable in x, r and either quasi-periodic or periodic in t (and therefore being bounded). Let us transform the time τ = µt. For this “slow” time the solution reaches Fix(L) for the finite time τ . The system (9) with new time looks as follows:
dx τ dτ = µϕ x, r, µ τ dr = 1 + µψ x, r, . dτ µ
(10)
Observe that in these coordinates, the system for searching for the stable manifold of integrable system casts as follows: dx dτ = 0 dr = 1. dτ int Its solution is xint ∗ (τ ) = x0 , r ∗ (τ ) = τ − τ0 .
Proposition 2. Stable manifold of zero solution for
the full system is C 1 -close up to the terms of order µ to the stable manifold of zero solution of the integrable system till its intersection with Fix(L1 ). Proof. We shall seek solutions of the system (10)
with the initial condition x(τ0 ) = x0 , r(τ0 ) = 0 by the successive approximation method. To this end, we consider the system of integral equation: τ ξ dξ µϕ x(ξ), r(ξ), x(τ ) = x0 + µ τ0 (11) τ ξ dξ, µψ x(ξ), r(ξ), r(τ ) = τ − τ0 + µ τ0 where τ is finite (we take it longer than the exact time of reaching Fix(L)). On the stable manifold of the full system solution of (10) is a solution for (11) and vice versa. As the first approximation, we take r1 (τ ) = τ − τ0 .
x1 (τ ) = x0 ,
To construct the next approximations, we use the formulas: τ ξ dξ µϕ xn (ξ), rn (ξ), xn+1 (τ ) = x0 + µ τ0
rn+1 (τ ) = τ − τ0 +
τ τ0
ξ µψ xn (ξ), rn (ξ), µ
dξ.
It is evident that for finite τ and any n the norm
xn (τ ), rn (τ ) is bounded. Moreover, since ϕ, ψ are
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continuously differentiable then one has: τ ξ ξ µ
xn+1 (τ ) − xn (τ ) ≤ ϕ xn (ξ), rn (ξ), µ − ϕ xn−1 (ξ), rn−1 (ξ), µ dξ τ0 ≤ µ2C4 (τ − τ0 ) max xn (ξ) − xn−1 (ξ), rn (ξ) − rn−1 (ξ) , ξ∈[τ0 ,τ ]
τ ξ ξ dξ µ ψ xn (ξ), rn (ξ), − ψ xn−1 (ξ), rn−1 (ξ),
rn+1 (τ ) − rn (τ ) ≤ µ µ τ0
≤ µ2C4 (τ − τ0 ) max xn (ξ) − xn−1 (ξ), rn (ξ) − rn−1 (ξ) , ξ∈[τ0 ,τ ]
where max ∂(ϕ,ψ) ∂(x,y) ≤ C4 . Hence we have: max xn+1 (ζ) − xn (ζ), rn+1 (ζ) − rn (ζ) ≤
ζ∈[τ0 ,τ ]
1 max xn (ξ) − xn−1 (ξ), rn (ξ) − rn−1 (ξ) 2 ξ∈[τ0 ,τ ]
and the series +∞
(xn+1 (τ ) − xn (τ ), rn+1 (τ ) − rn (τ ))
n=1
converges absolutely and uniformly since the terms of the series +∞
xn+1 (τ ) − xn (τ ), rn+1 (τ ) − rn (τ )
n=1
are bounded by terms of the geometric progression with the ratio 1/2. Thus there is a continuous limit of successive approximations (x∗ (τ ), r∗ (τ )) being a solution of system (11): τ ξ dξ (τ ; τ , x ) = x + µϕ x (ξ; τ , x ), r (ξ; τ , x ), x 0 0 0 ∗ 0 0 ∗ 0 0 ∗ µ τ0 (12) τ ξ dξ. µψ x∗ (ξ; τ0 , x0 ), r∗ (ξ; τ0 , x0 ), r∗ (τ ; τ0 , x0 ) = τ − τ0 + µ τ0 The solution of system (11) is unique. It is worth remarking that on time [τ0 , τ ] solutions of the full and integrable systems differ by the order µ: τ ξ int dξ ≤ µC5 (τ − τ0 ), µ ϕ x (ξ), r (ξ),
x∗ (τ ) − x∗ (τ ) ≤ ∗ ∗ µ τ0 τ ξ int dξ ≤ µC5 (τ − τ0 ), µ ψ x∗ (ξ), r∗ (ξ),
r∗ (τ ) − r ∗ (τ ) ≤ µ τ0 here max (ϕ, ψ) ≤ C5 . Solution (x∗ (τ ; τ0 , x0 ), r∗ (τ ; τ0 , x0 )) depends smoothly on parameters τ0 and x0 . Again, to prove this, we formally differentiate expressions (12) in τ0 : τ ∂ϕ ∂x∗ ∂ϕ ∂r∗ τ0 ∂x∗ = µ + dξ − µϕ x∗ (τ0 ), r∗ (τ0 ), ∂τ0 ∂x ∂τ0 ∂r ∂τ0 µ τ0 (13) τ ∂x ∂r ∂ψ τ ∂ψ ∂r ∗ ∗ ∗ 0 . µ + dξ − µψ x∗ (τ0 ), r∗ (τ0 ), ∂τ0 = −1 + ∂x ∂τ0 ∂r ∂τ0 µ τ0 1440006-15
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As above, we solve this system by the successive approximation method w.r.t. variables ∂x∗ def ∂r∗ , R ≡ . ∂τ0 ∂τ0 Take as the first approximation X1 = 0, R1 = −1. Since τ is finite and µ is small enough, then for any n ˆ Also one has: the norm Xn , Rn is bounded by some constant C. τ ∂ϕ ∂ϕ µ
Xn+1 (τ ) − Xn (τ ) ≤ ∂x (Xn − Xn−1 ) + ∂r (Rn − Rn−1 ) dξ def
X ≡
τ0
≤ µ(τ − τ0 )2C4 max Xn (ξ) − Xn−1 (ξ), Rn (ξ) − Rn−1 (ξ) , ξ∈[τ0 ,τ ]
τ ∂ψ ∂ψ µ
Rn+1 (τ ) − Rn (τ ) ≤ ∂x (Xn − Xn−1 ) + ∂r (Rn − Rn−1 ) dξ τ0 ≤ µ(τ − τ0 )2C4 max Xn (ξ) − Xn−1 (ξ), Rn (ξ) − Rn−1 (ξ) . ξ∈[τ0 ,τ ]
R∗ −
max Xn+1 (ζ) − Xn (ζ), Rn+1 (ζ) − Rn (ζ)
ζ∈[τ0 ,τ ]
≤
1 max Xn (ξ) − Xn−1 (ξ), Rn (ξ) − Rn−1 (ξ) 2 ξ∈[τ0 ,τ ]
and the series +∞
(Xn+1 (τ ) − Xn (τ ), Rn+1 (τ ) − Rn (τ ))
n=1
converges absolutely and uniformly. So, system (13) has a continuous solution (X∗ , R∗ ), this means the solution of system (11) is continuously differentiable in τ0 . Derivative in τ0 for the solution for integrable system is def
X int ∗ ≡ def
Rint ∗ ≡
∂xint ∗ = 0, ∂τ0 ∂r int ∗ = −1. ∂τ0
Respectively, derivatives in τ0 for the full and integrable systems are close up to terms of order µ: ∂ϕ ∂ϕ R∗ dξ ≤ µ X∗ + ∂x ∂r τ0 τ0 + µ ϕ x∗ (τ0 ), r∗ (τ0 ), µ
X∗ − X int ∗
τ
≤ µ(τ − τ0 )2C4 Cˆ + µC5 ≤ µCˆ1 ,
∂ψ ∂ψ X∗ + R∗ ≤ µ dξ ∂x ∂r τ0 τ0 + µ ψ x∗ (τ0 ), r∗ (τ0 ), µ
Hence, the following hold Rint ∗
τ
≤ µ(τ − τ0 )2C4 Cˆ + µC5 ≤ µCˆ1 . In the same way it is shown that the solution (x∗ (τ ; τ0 , x0 ), r∗ (τ ; τ0 , x0 )) depends smoothly on x0 and differs from the related solution of the integrable system by order µ. Thus, we have proved that the stable manifold of the periodic solution for the full system is C 1 close to the stable manifold of the periodic solution of the integrable system up to its intersection with Fix(L). The same is valid for manifolds of the Poincar´e map of the full and integrable systems. The point of intersection of stable manifold of the integrable system with Fix(L) is on a finite distance from the origin and intersects Fix(L) transversely. So, C 1 -close (of order µ) stable manifold of the full system (or for the Poincar´e map of the periodic system) intersects Fix(L1 ) transversely at a close point. This gives a symmetric homoclinic orbit for any such intersection point. But there are four of these points. This completes the proof.
6. Conclusions Thus, we have discovered the following peculiarities of the dynamics at periodic reversible Hamiltonian Hopf bifurcation: • if A > 0 and ε < 0, the autonomous reversible integrable system has at the origin the 1440006-16
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saddle-focus equilibrium O; this equilibrium is symmetric and has homoclinic skirt filled with homoclinic orbits to O. This skirt transversely intersects the set Fix(L) at two different points. Then the full periodic system for any negative ε small enough has the unique symmetric saddlefocus periodic integral curve γε : O × S 1 , its stable and unstable manifolds intersect along at least four symmetric homoclinic curves; • if A < 0 and ε > 0 the reversible integrable system at the level K = k for all sufficiently small k has a saddle symmetric periodic orbit γk , its stable and unstable manifolds coincide forming homoclinic skirt which intersects transversely the set Fix(L) at two different points. Then the full periodic system for any positive ε small enough has a Cantor set of saddle invariant two-dimensional tori with Diophantine rotation numbers whose three-dimensional stable and unstable manifolds intersect each other along at least four symmetric homoclinic orbits of the related torus.
Acknowledgments The authors thank D. Turaev and S. Gonchenko for the useful discussions. We acknowledge a partial support from the Russian Foundation for Basic Research under grants 13-01-00589a (L. Lerman) and 14-01-00344 (A. Markova). We also thank the Russian Ministry of Science and Education for the support (project 1410, June 2014).
References Arnold, V. I. [1992] Ordinary Differential Equations (Springer-Verlag, Berlin-Heidelberg). Arnold, V. I., Kozlov, V. V. & Neishtadt, A. I. [1993] Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems III, Encyclopaedia Math. Sci., Vol. 3 (Springer, Berlin). Bochner, S. [1945] “Compact groups of differentiable transformations,” Ann. Math. 46, 372–381. Bogoliubov, N. N. & Mitropolski, Y. A. [1961] Asymptotic Methods in the Theory of Non-Linear Oscillations (Gordon & Breach, NY). Bridges, T. J., Cushman, R. H. & Mackay, R. S. [1995] “Dynamics near an irrational collision of eigenvalues for symplectic mappings,” Fields Inst. Comm. 4, 61–79. Bryuno, A. D. [1989] “Normalization of a Hamiltonian system near an invariant cycle or torus,” Russ. Math. Surveys 44, 53–89.
Cushman, R. [1978] “Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors,” Trans. Amer. Math. Soc. 238, 45–55. Devaney, R. [1976] “Reversible diffeomorphisms and flows,” Trans. Amer. Math. Soc. 218, 89–113. Fenichel, N. [1974] “Asymptotic stability with rate conditions,” Indiana Univ. Math. J. 23, 1109– 1137. Gaivao, J. P. & Gelfreich, V. [2011] “Splitting of separatrices for the Hamiltonian–Hopf bifurcation with the Swift–Hohenberg equation as an example,” Nonlinearity 24, 677–698. Gantmacher, F. R. [1959] The Theory of Matrices, Vols. 1 and 2 (Chelsea Pub. Co., NY). Glebsky, L. Yu. & Lerman, L. M. [1995] “On small stationary localized solutions for the generalized 1-D Swift–Hohenberg equation,” Chaos 5, 424–431. Glebsky, L. Yu. & Lerman, L. M. [1996] “1:-1 bifurcation for a periodic orbit in a 3D Hamiltonian system,” J. Tech. Phys. 37, 343–346. Haragus, M. & Iooss, G. [2011] Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, UTX Series (Springer, London & EDP Sciences, Les Ulis, France). Heggie, D. C. [1985] “Bifurcation at complex instability,” Celest. Mech. 35, 357–382. Hirsch, M. W., Pugh, C. C. & Shub, M. [1977] Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583 (Springer-Verlag, Berlin-Heidelberg, NY). Howard, J. E., Lichtenberg, A. L., Lieberman, M. A. & Cohen, P. H. [1986] “Four-dimensional mapping model for two-frequency electron cyclotron resonance heating,” Physica D 20, 259–284. Iooss, G. & Perouem´e, M. C. [1993] “Perturbed homoclinic solutions in reversible 1:1 resonance vector fields,” J. Diff. Eqs. 102, 62–88. Ivanov, A. P. & Sokol’skii, A. G. [1980] “On the stability of a nonautonomous Hamiltonian system under second-order resonance,” J. Appl. Math. Mech. 44, 574–581. Jorba, A. & Olle, M. [2004] “Invariant curves near Hamiltonian–Hopf bifurcations of four-dimensional symplectic maps,” Nonlinearity 17, 691–710. Kao, H.-C., Beaume, C. & Knobloch, E. [2014] “Spatial localization in heterogeneous systems,” Phys. Rev. E 89, 012903. Kaper, T. J. & Kovacic, G. [1996] “Multibump orbits homoclinic to resonance bands,” Trans. Amer. Math. Soc. 348, 3835–3887. Koltsova, O. Yu. & Lerman, L. M. [1999] “New criterion of nonintegrability for an N-degrees-of-freedom Hamiltonian system,” in Hamiltonian Systems with Three and More Degrees of Freedom, ed. Sim´ o, C., NATO ASI Series C: Math. and Phys. Sciences, Vol. 533 (Springer, Dordrecht), pp. 458–470.
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Kozlov, V. V. [1983] “Integrability and non-integrability in Hamiltonian mechanics,” Russ. Math. Surv. 38, 1–76. Lamb, J. S. W. & Roberts, J. A. G. [1998] “Time-reversal symmetry in dynamical systems: A survey,” Physica D 112, 1–39. Markeev, A. P. [1978] Libration Points in Celestial Mechanics and Cosmodynamics (Nauka, Moscow) (in Russian). Meyer, K. R. [1981] “Hamiltonian systems with a discrete symmetry,” J. Diff. Eqs. 41, 228–238. Moser, J. K. [1962] “On invariant curves of areapreserving mappings of an annulus,” Nachr. Akad. Wiss. G¨ ottingen, Math. Phys. Kl. IIa, 1–20. Neishtadt, A. I. [1984] “The separation of motions in systems with rapidly rotating phase,” J. Appl. Math. Mech. 48, 133–139. Oll´e, M., Pacha, J. R. & Villanueva, J. [2004] “Motion close to the Hopf bifurcation of the vertical family of periodic orbits of L4 ,” Celest. Mech. Dyn. Astron. 90, 87–107. Olle, M., Pacha, J. R. & Villanueva, J. [2005] “Dynamics close to a non semi-simple 1:-1 resonant periodic orbit,” DCDS-B 5, 799–816. Olle, M., Pacha, J. R. & Villanueva, J. [2008] “Kolmogorov–Arnold–Moser aspects of the periodic Hamiltonian Hopf bifurcation,” Nonlinearity 21, 1759–1812.
Sanders, J. A., Verhulst, F. & Murdock, J. [2007] Averaging Methods in Nonlinear Dynamical Systems, 2nd edition, Applied Math. Sciences, Vol. 59 (Springer, NY, USA). Shilnikov, L. P. [1967] “On a Poincar´e–Birkhoff problem,” Math. USSR Sb. 3, 353–371. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. O. [1998] Methods of Qualitative Theory in Nonlinear Dynamics, Part I, World Scientific Series on Nonlinear Science, Vol. 4 (World Scientific, Singapore). Smale, S. [1965] “Diffeomorphisms with many periodic points,” in Differential and Combinatorial Topology, A Symposium in Honor of Marston Morse, ed. Cairns, S. S., Princeton Math. Series, Vol. 27 (Princeton Univ. Press, Princeton, NJ), pp. 63–80. Sokol’skii, A. G. [1977] “On stability of an autonomous Hamiltonian system with two degrees of freedom under first-order resonance,” J. Appl. Math. Mech. 41, 20–28. Vanderbauwhede, A. & Fiedler, B. [1992] “Homoclinic period blow-up in reversible and conservative systems,” ZAMP 43, 292–318. van der Meer, J. C. [1985] The Hamiltonian Hopf Bifurcation, Lecture Notes in Mathematics, Vol. 1160 (Springer-Verlag, Berlin-Heidelberg-NY-Tokyo).
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International Journal of Bifurcation and Chaos, Vol. 24, No. 8 (2014) 1440006 (18 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414400069
Symmetric Homoclinic Orbits at the Periodic Hamiltonian Hopf Bifurcation Lev Lerman∗ and Anna Markova Department of Differential Equations and Mathematical Analysis, N. I. Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Ave., Nizhny Novgorod 603950, Russia ∗[email protected] Received February 2, 2014 We prove the existence of symmetric homoclinic orbits to a saddle-focus symmetric periodic orbit that appears in a generic family of reversible three degrees of freedom Hamiltonian system due to periodic Hamiltonian Hopf bifurcation, if some coefficient A of the normal form of the fourth order is positive. If this coefficient is negative, then for the opposite side of the bifurcation parameter value, we prove the existence of symmetric homoclinic orbits to saddle invariant 2-tori. Keywords: Periodic Hamiltonian Hopf bifurcation; homoclinic orbit; reversible system; normal form.
1. Introduction and Set-Up Periodic Hamiltonian Hopf Bifurcation (HHB) has received much attention from different angles [Heggie, 1985; Bridges et al., 1995; Glebsky & Lerman, 1996; Jorba & Olle, 2004; Olle et al., 2008]. This bifurcation occurs in many applied systems, see [Heggie, 1985; Howard et al., 1986; Olle et al., 2004; Kao et al., 2014]. The bifurcation was investigated in detail, mainly from the viewpoint of the existence of invariant tori (or invariant curves for the case of a symplectic map). Another restriction (unnecessary in our opinion) in the papers mentioned (except [Glebsky & Lerman, 1996]) was the assumption of an irrational ratio for the frequencies in the critical case. In our study below, we avoid this assumption of irrationality. We think the KAM type results obtained in the papers mentioned should be supplemented with similar results under the assumption of the absence of strong resonances. This would lead to their applicability for an open set of Hamiltonians. Nonetheless, an important aspect of the problem has remained unsolved. This concerns the appearance of homoclinic orbits to a periodic orbit
of the saddle-focus type that arises as a result of the periodic HHB at the positive sign of some characteristic quantity A (see below), as well as homoclinic orbits to periodic orbits and invariant saddle two-dimensional tori which exist in the system for the opposite sign of this quantity A. It is well known that the existence of such orbits in the system is one of the main indicators for the chaotic behavior of the system [Smale, 1965; Shilnikov, 1967] and therefore of its nonintegrability [Cushman, 1978; Kozlov, 1983; Koltsova & Lerman, 1999]. Especially it is clear when we have in addition the transversal intersection of the related stable and unstable manifolds for the periodic orbit [Smale, 1965; Shilnikov, 1967; Koltsova & Lerman, 1999]. The proof of existence of homoclinic orbits at this bifurcation is rather a delicate problem as is seen in [Gaivao & Gelfreich, 2011]. The point is that the splitting of stable and unstable manifolds for the related bifurcation in a two degrees of freedom analytic Hamiltonian system is exponentially small in perturbation. This follows from [Neishtadt, 1984]. It is clear that the problem is not easier for the case of periodic HHB. Nonetheless, sometimes the existence of homoclinic orbits
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can be proved by geometric methods using a feature of the system that appears rather frequently in application problems. This feature is the reversibility of the related Hamiltonian system. Existence of homoclinic orbits in a reversible system (usually not Hamiltonian) at the related reversible Hopf bifurcation was proved in [Iooss & Perouem´e, 1993] using the Lyapunov–Schmidt type method. Here, we extend this result onto the case of three degrees of freedom using the geometric tool. We consider both possible cases of the direct and inverse bifurcations, which corresponds to the case A < 0 and A > 0 below. Let us formulate the problem that we are investigating more precisely. We consider a Hamiltonian system on a smooth1 six-dimensional symplectic manifold M with a closed nondegenerate skew symmetric 2-form Ω with a smooth Hamiltonian H (as an example, one may think of the coordinate linear space R6 with 2-form dx ∧ dy w.r.t. coordinates (x, y) = (x1 , x2 , x3 , y1 , y2 , y3 )). We assume the system to have a periodic orbit γ of period T and the set of multipliers for this orbit consists (except for the common double unit) of two double ones e±iν = ±1 on the unit circle in the complex plane (it is a so-called Krein collision of multipliers [Arnold et al., 1993]). Then this periodic orbit will belong to a smooth two-dimensional symplectic cylinder filled with other closed periodic orbits. This family of periodic orbits is smoothly parameterized by the value h of the Hamiltonian. In a neighborhood of the critical orbit γ in M , one can construct symplectic coordinates in such a way that one variable θ (mod 2π) is angular along the periodic orbit, its conjugated variable I is some smooth function of the Hamiltonian and the remaining four variables go in the transverse directions to the symplectic cylinder [Bryuno, 1989]. Now, if one fixes the level of Hamiltonian H = H(γ), then near γ one can express I as a function of the remaining four variables, θ and h, take angular variable as a new “time” and then one gets a smooth one-parameter family of 2π-periodic nonautonomous Hamiltonian systems with two degrees of freedom (such systems are frequently called systems with two and a half degrees of freedom). Each system of the family obtained has a 2π-periodic solution at each (small enough) value of the parameter ε (this parameter corresponds to the former value H −H(γ)). Without 1
losing in generality, one can regard this periodic solution (x(t, ε), y(t, ε)) as given (x(t, ε), y(t, ε)) ≡ (0, 0) for all small ε. This situation is further studied later on.
2. Normal Form and Strong Resonances Multipliers for the linearized system at zero 2πperiodic solution of 2π-periodic system depend on ε. Now multipliers are four complex numbers and our assumption means that at ε = 0, we have a periodic solution with two double multipliers on the unit circle e±2πiω , 0 < ω < 1, ω = 1/2, distinct from ±1. We assume in addition that at ε = 0 the monodromy matrix of the linearized system has, as its Jordan form, two two-dimensional boxes. By the Lyapunov theorem (see, for instance, [Ivanov & Sokol’skii, 1980; Bryuno, 1989]) this family of linearized systems is reduced to the family of linear systems with real constant coefficients by a linear real 2π-periodic symplectic change of variables with smooth dependence on ε. Our assumption implies that the reduced linear system has at ε = 0 the matrix for which its Jordan normal form consists of two two-dimensional boxes. Then, as the parameter ε varies near zero, the related linear (autonomous) Hamiltonian systems will have as eigenvalues either a quartet of complex numbers for one side of ε = 0 (a saddle-focus equilibrium for the linear system) or two different pairs of pure imaginary numbers for another side from ε = 0 (an elliptic equilibrium) (the dependence of multipliers on the parameter for the related monodromy matrix of linear 2π-periodic system is plotted in Fig. 1). This change of the equilibria types is standard for the autonomous Hamiltonian Hopf bifurcation [van der Meer, 1985]. To study solutions in a neighborhood of the zero periodic solution, we apply the usual normal form method [Ivanov & Sokol’skii, 1980; Bryuno, 1989]. First, we assume the linear part of 2π-periodic system to have been reduced to the linear normal form with constant complex coefficients. After that the normal form transformations with 2π-periodic coefficients are applied in order to eliminate nonresonant terms of the third and fourth orders in the periodic Hamiltonian. In contrast to the autonomous Hamiltonian Hopf bifurcation, resonances can exist here. In order to kill all third order
If another is not specified, by “smooth” we understand C ∞ -smooth. 1440006-2
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Fig. 1.
Multipliers at the periodic HHB.
terms, one needs to require the absence of the resonances ω = 1/3, 2/3, and to normalize the fourth order terms, one needs to demand in addition the absence of resonances ω = 1/4, 3/4. All these resonances are usually said to be strong. In the absence of strong resonances a generic one-parameter family of normalized real Hamiltonians up to the fourth order terms looks like (1) (see [Ivanov & Sokol’skii, 1980] and recent [Olle et al., 2005]): δ ε H4 = (p21 + p22 ) + ω(p1 q2 − p2 q1 ) + (q 21 + q 22 ) 2 2 + (q 21 + q 22 )[A(q 21 + q 22 ) + B(p1 q2 − p2 q1 ) + C(p21 + p22 )],
(1)
where δ, ω, A, B, C depend on ε, δ(0), ω(0) = 0 and we assume a genericity condition A(0) = 0 to hold, ε is the governing parameter. For further use, it is convenient to scale the system in order δ = 1 for every value of ε near the critical ε = 0. More exactly, the scaling allows one to make δ = ±1, in order to change the sign of δ one needs to reverse time (or to pass H → −H). This is assumed to be done later on. Now, as is easily verified, for the vector field XH4 with ε positive, the equilibrium at the origin is elliptic, and it is saddle-focus as ε < 0. In [Ivanov & Sokol’skii, 1980] the normal form was derived using the Deprit–Hori method (see, for instance, [Markeev, 1978]). The normal form can also be constructed by the customary generating function method. Let us remark that the normal form (1) is an autonomous Hamiltonian, though the system is 2π-periodic. Moreover, when the strong resonances are absent, the truncated normal form of the fourth order coincides with the normal form at the standard Hamiltonian Hopf bifurcation. It implies that all dynamical features found in this
integrable two degrees of freedom system [van der Meer, 1985; Arnold et al., 1993; Glebsky & Lerman, 1995] take place for this truncated normal form (for the case of irrational ω/2π these analogies and features are discussed in [Olle et al., 2005]). If we shall try to normalize the Hamiltonian up to terms of higher orders, we need to demand the absence of resonances of higher orders. Then the truncated Hamiltonian will be also autonomous and the normal form coincides with related standard normal form of the Hamiltonian Hopf bifurcation again and the periodic terms will be shifted to the higher terms. For instance, if ω is irrational and Hamiltonian is analytic or C ∞ -smooth, then resonances of any order are absent and the Hamiltonian can be formally transformed to the autonomous normal form being the same as for autonomous Hamiltonian Hopf bifurcation [Sokol’skii, 1977; van der Meer, 1985; Arnold et al., 1993; Glebsky & Lerman, 1995; Olle et al., 2005]. In particular, this autonomous Hamiltonian is integrable and it implies that all homoclinic phenomena that arise for the full analytic Hamiltonian are of an exponentially small order [Neishtadt, 1984; Gaivao & Gelfreich, 2011]. When Hamiltonian is normalized up to the terms of fourth order, the remainder is 2π-periodic and of fifth order in (p, q). The more advanced study of the normal form for the irrational ω, estimates on the remainders and analytic issues related with these transformations, as well as KAM type results for this case, one can find in [Olle et al., 2008], for the numerical simulations related to two models of coupled Hen´ on type symplectic reversible maps, see [Jorba & Olle, 2004]. For 2π-periodic system, it is also natural to consider its Poincar´e map through period 2π. The system with Hamiltonian H4 is integrable and
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autonomous, therefore its Poincar´e map is an integrable symplectic map being 2π-shift along the orbits of the vector field XH4 . As ε < 0, this map has at the origin a fixed point of the saddle-focus type and for ε > 0 this map has an elliptic fixed point. The Poincar´e map for the full system is a perturbation of this integrable map. In order to apply geometric methods for proving the existence of homoclinic orbits, we assume in addition the initial three degree of freedom system to be reversible with respect to involution L and the system has a symmetric critical periodic orbit γ. This assumption makes the related nonautonomous 2π-periodic vector fields to be reversible and related Poincar´e maps be reversible symplectic diffeomorphisms. Recall some proper definitions [Devaney, 1976; Vanderbauwhede & Fiedler, 1992; Lamb & Roberts, 1998]. Definition 2.1. A nonautonomous vector field v(x, t) on a smooth manifold M , x ∈ M , is said to be reversible w.r.t. an action of a smooth involution L : M → M , L2 = idM , if the following identity holds:
DLx (v(x, t)) = −v(L(x), −t). A diffeomorphism f : M → M of the manifold M is said to be reversible w.r.t. an action of a smooth involution L : M → M , L2 = idM , if the following identity holds: L ◦ f ◦ L = f −1 . This definition implies that if x(t) is a solution of a reversible vector field x˙ = v(x, t), then x1 (t) = Lx(−t) is its solution as well. For a reversible diffeomorphism f the related property reads as follows: if {xn } is an orbit: xn+1 = f (xn ), n ∈ Z, then {yn }, yn = Lx−n is an orbit as well. Since the vector field we study is nonautonomous and 2πperiodic, we consider the graphs of solutions (i.e. integral curves) in the extended phase space M ×S 1 , S 1 = [−π, π] mod 2π, and will consider L as acting in this manifold as follows: L1 (x, t) = (L(x), 2π −t). Then in sections t = 0 and t = π = −π(mod 2π) we have fixed points of involution L1 corresponding to Fix(L) ⊂ M . For the case under study, a nonautonomous Hamiltonian vector field is generated by an autonomous Hamiltonian vector field with three degrees of freedom. We assume in addition this
latter vector field to be reversible w.r.t. a smooth anti-symplectic involution L that has a threedimensional smooth submanifold of fixed points Fix(L). Anti-symplectic means the identity L∗ Ω = −Ω holds. Recall that for the case of anti-symplectic involution the set of its fixed points Fix(L) is a Lagrangian submanifold [Meyer, 1981] being threedimensional for our case. We also assume that Hamiltonian H is invariant w.r.t. L : H ◦ L ≡ H and the vector field XH has a symmetric periodic orbit γ of the type described above. Recall an orbit of a reversible (autonomous) vector field to be symmetric, if it is invariant w.r.t. the action of L. There is a characteristic property of a symmetric periodic orbit of a reversible vector field [Devaney, 1976; Vanderbauwhede & Fiedler, 1992]: such an orbit intersects the manifold Fix(L) at exactly two points which are apart from each other through a half period in time. Moreover, if some flow orbit intersects Fix(L) at two points, then this orbit is periodic and its period is twice that of the time distance between these intersection points. When an orbit intersects Fix(L) at a unique point, then this orbit is symmetric nonperiodic (for instance, it can be a symmetric equilibrium, if the orbit coincides with this point, also it can be a symmetric homoclinic orbit to either a symmetric equilibrium or a symmetric periodic orbit, also it can be a symmetric quasi-periodic orbit). Generically, symmetric periodic orbits of a reversible vector field in case, if the dimension of Fix(L) is equal to the half of dim M , form a oneparameter family, even if the system is a nonHamiltonian one. If these conditions are met and the vector field is in addition a Hamiltonian one, then this family of symmetric periodic orbits locally near a basic symmetric periodic orbit γ coincides with that family of periodic orbits for the Hamiltonian vector field which is obtained by continuing in the value of H (if the family is locally unique). To explain the genericity condition mentioned, let us consider a reversible system on a smooth 2n-dimensional manifold and let L be its reversing involution with n-dimensional submanifold of Fix(L). Let γ be a symmetric periodic orbit which intersects Fix(L) at two points m, m1 . It is always possible to choose two cross-sections N, N1 to the flow through the points m, m1 , respectively, such that both cross-sections would contain a related local piece of Fix(L). Then the smooth transition flow map S : N → N1 is defined near point m,
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S(m) = m1 . Denote R, R1 the pieces of Fix(L) in N, N1 , respectively, and consider the image S(R) ⊂ N1 . One has S(R) ∩ R1 = ∅ (it contains m1 ) and assume this intersection be transverse and hence it occurs along some smooth curve σ1 . Every point on σ1 being close enough to m1 has its preimage in R, they make up a curve σ ⊂ R ⊂ Fix(L). All flow orbits through points on σ intersect Fix(L) also at points on σ1 , therefore they intersect Fix(L) twice and all are periodic orbits. Periodic orbits for which this transversality condition is met were called elementary in [Devaney, 1976], we will also use this term. Now let us consider the case, when M is a smooth symplectic 2n-dimensional manifold, H is a smooth Hamiltonian, L is an anti-symplectic involution and XH is reversible w.r.t. L with Fix(L) being a smooth submanifold in M , H ◦ L ≡ H. Let γ be an elementary symmetric periodic orbit for XH and m, m1 be its intersection points with Fix(L). Choose two cross-sections N, N1 at these points which contain pieces R, R1 of Fix(L). Then we have the smooth curves σ ⊂ R, σ1 ⊂ R1 . Suppose, in addition, that the Poincar´e map Nγ → Nγ , Nγ = N ∩ {H = H(γ)}, does not have a unit eigenvalue of its linearization at m. In this case, the cylinder formed by periodic orbits near γ, being the continuation of γ in H, is unique and transverse to the level H = H(m). It implies this cylinder to coincide with the cylinder formed by symmetric periodic orbits through points of σ near γ and hence the tangent vector to σ at the point m (it belongs to Tm R) is not tangent to submanifold H = H(m). Thus submanifold H = H(m) (of dimension 2n − 1) and Fix(L) intersects at m transversely. So, we have proved the following assertion: Lemma 1. Let XH be a smooth reversible Hamiltonian vector field on a smooth symplectic 2n-dimensional manifold, L be a reversible anti-symplectic involution and γ be an elementary symmetric periodic orbit of XH . If γ does not have any unit multipliers except for always existing double unit, then near points {m, m1 } = γ ∩ Fix(L) submanifolds Fix(L) and H = H(γ) are transverse and all periodic orbits of XH being close to γ are symmetric.
In this case, traces of periodic orbits of the family on the submanifold Fix(L) form two smooth
local curves σ, σ1 near two points m, m1 being the traces of γ. We assume γ elementary, then the lemma implies that five-dimensional smooth levels of the Hamiltonian near γ intersect threedimensional disk Fix(L) transversely and thus along two two-dimensional disks. In view of the invariance of the Hamiltonian levels w.r.t. the autonomous Hamiltonian system, one can choose symplectic coordinates near γ in such a way that in these coordinates (x, y, I, θ) periodic solutions of the family are given as x = y = 0, I = I0 , and after transition to the time θ the related 2π-periodic Hamiltonian vector field will be reversible w.r.t. to the involution L1 with its Fix(L1 ) consisting of two two-dimensional disks that cut the periodic integral curve x = y = 0 at two points in sections θ = 0, π (to get this we use results of [Meyer, 1981] and [Bryuno, 1989]). Now let us write down the vector field obtained in a neighborhood of the solution X = (x, y) = 0 (we shall preserve for the time variable here and below its usual notation t). X˙ = I∇H(X, t, ε),
∇H(0, t, ε) ≡ 0.
Since we work in a small neighborhood of the set {0} × S 1 ⊂ R4 × S 1 , one may assume involution L to be linear, due to results [Bochner, 1945; Meyer, 1981]: Vector-function I∇H obeys the identity LI∇H(X, t, ε) = −I∇H(LX, −t, ε), due to the reversibility. Linearized at the zero solution system has the matrix B(t, ε) = ID 2 H(0, t, ε) which in turn satisfies the identity LB(t) = −B(−t)L. Then the normalized fundamental matrix Φ(t) at t = 0 obeys the identity LΦ(t) = Φ(−t)L. It is also a symplectic matrix for any t. By the Lyapunov theorem the linear transformation with the matrix S(t), S(t) = Φ(t)e−Ct , reduces this periodic system to that with constant coefficients. Matrix C is the logarithm of matrix Φ(2π) divided at the period 2π. Since this latter matrix has no negative eigenvalues, matrix Φ(2π) does have a real logarithm [Gantmacher, 1959]. Thus the linearized system by a linear real symplectic 2π-periodic change of variable is reduced to a linear Hamiltonian system with constant matrix C = ID, D = D. Further, similar to [Haragus & Iooss, 2011], we make a linear symplectic change of variables that transforms simultaneously matrix L to the form L = diag (−1, 1, 1, −1), and a quadratic Hamiltonian to
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the form in coordinates (p1 , p2 , q1 , q2 ) 1 H2 = (p21 + p22 ) + ω(p1 q2 − p2 q1 ) 2 ε + (q 21 + q 22 ). 2 After that, following [Bryuno, 1989] we normalize the terms of higher order with the preservation of reversibility w.r.t. to L. Under the assumption of the absence of strong resonances, we come to the normal form (1) up to the fourth order terms.
3. Case A > 0 The main result of this section is the following theorem which we formulate for the initial Hamiltonian system with three degrees of freedom, though the proof will be done for the related one-parameter family of periodic Hamiltonian systems with two degrees of freedom. Theorem 1. Let H be a smooth Hamiltonian on a smooth six-dimensional symplectic manifold (M, Ω) and vector field XH has a periodic orbit γ whose nontrivial multipliers are two double nonsemisimple complex numbers on the unit circle exp[±iν], ν = 0, π (mod 2π). In this case, γ belongs to a symplectic cylinder filled with periodic orbits of XH close to γ and we suppose, to be definite, that for levels H = h < 0 the related periodic orbits on the cylinder are hyperbolic (saddle-focus type) and for levels H = h > 0 the related periodic orbits on the cylinder are elliptic ones. We assume, in addition, the vector field to be reversible w.r.t. smooth antisymplectic involution L having three-dimensional set of fixed points, H be invariant under the action of L and γ be symmetric. Then if some quantity A (see above) is positive and h < 0, then the saddlefocus periodic orbit in the level H = h is symmetric and has at least four symmetric homoclinic orbits.
It is worth remarking that this theorem gives not only the existence of homoclinic orbits themselves, but also localize their intersection points with Fix(L). This section is devoted to the proof of this theorem. Remark. Homoclinic solutions of this theorem are
so-called one-round or one-hump ones, each of them goes around an unperturbed symmetric homoclinic solution only one time. It is well known [Shilnikov,
1967] that if stable and unstable manifolds of the full 2π-periodic system intersect transversely along this persistent homoclinic solution, then there are other multiround homoclinic orbits and many other orbits described by a symbolic system. The system with Hamiltonian H4 is integrable, its additional integral is K = p1 q2 − p2 q1 . Due to this fact, if A > 0 and ε < 0, stable and unstable manifolds of the saddle-focus equilibrium at the origin merge forming a “homoclinic skirt” [van der Meer, 1985; Arnold et al., 1993; Glebsky & Lerman, 1995]. The same is valid for the related Poincar´e map for the Hamiltonian vector field XH4 considered as nonautonomous 2π-periodic. This map is 2π-shift along the orbits of the Hamiltonian system generated by XH4 . The homoclinic skirt is topologically a sphere with two of its points identified, the point of gluing corresponds to the saddle-focus. For the map, this set is filled with discrete homoclinic orbits to the fixed point. For the system with the Hamiltonian H4 considered as 2π-periodic in the space R4 × S 1 this homoclinic skirt corresponds to the set of merged three-dimensional stable and unstable manifolds of zero periodic solution, thus it is topologically a direct product of the homoclinic skirt and S 1 . For the full system with the Hamiltonian H, because of the presence of periodic terms of higher order, stable and unstable manifolds of the periodic solutions are generically split. Our aim is to discover which homoclinic solutions do survive. The idea of the proof is to find an equation for the homoclinic skirt and to show the transversal intersection of the skirt with Fix(L), after that to apply the transversality theorem to the stable manifold of the full system to prove its intersection with Fix(L) to persist. But the case under consideration is more subtle, since the size of the skirt is of the order |ε| and it shrinks to the fixed point as ε → −0. Therefore, to get a regular intersection of homoclinic skirt and Fix(L) one needs to blow up a neighborhood of the fixed point (or the related periodic solution). To do this, we use coordinates depending on ε in such a way that the size of the skirt would become finite. To this end, we make a transformation √ qi → −εQi , pi → −εPi , i = 1, 2 √ −ε. and use a new small parameter µ = After that, the system in coordinates Q, P with respect to 2-form dP ∧ dQ coincides with the
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Hamiltonian H = ω(P1 Q2 − P2 Q1 ) +
µ 2 (P + P 22 ) 2 1
µ − (Q21 + Q22 ) + (Q21 + Q22 )[µA(Q21 + Q22 ) 2 + µ2 B(P1 Q2 − P2 Q1 ) + µ3 C(P 21 + P 22 )] + ···, (2) here dots mean the terms of order five and higher in (P1 , P2 , Q1 , Q2 ) and they are of at least the second order and higher in µ. Now as an integrable approximation, we choose H0 : H0 = ω(P1 Q2 − P2 Q1 ) +
µ 2 (P + P 22 ) 2 1
Fig. 2.
µ − (Q21 + Q22 ) + µA(Q21 + Q22 )2 2
saddle for which (r, P ) obeys the equation:
µ2
and terms of the order and higher we consider as a perturbation. In order to find stable and unstable manifolds of the saddle-focus equilibrium for the vector field XH0 , we use symplectic polar coordinates similar to [Glebsky & Lerman, 1995]: ϕ , r ϕ P2 = P sin ϕ − k cos , r P1 = P cos ϕ + k sin
Q1 = r cos ϕ, Q2 = r sin ϕ,
Phase portrait of the reduced system for K = k = 0.
(3)
dP1 ∧ dQ1 + dP2 ∧ dQ2
P 2 − r 2 + 2Ar 4 = 0. This homoclinic orbit corresponds to the homoclinic skirt Γ of integrable system with two degrees of freedom. In coordinated (P1 , P2 , Q1 , Q2 ) the skirt is written in a parametric form with parameters (P1 , Q2 ): P1 Q2 Γ : P1 , , Q1 (P1 , Q2 ), Q2 , Q1 (P1 , Q2 ) where Q1 (P1 , Q2 ) (1 − 2AQ22 ) + (1 − 2AQ22 )2 − 8AP 21 . =± 4A
= dP ∧ dr + dϕ ∧ dk. In new coordinates H0 casts µ k2 µ 2 ˆ P + 2 − r 2 + µAr 4 . H0 = ωk + 2 r 2
As was remarked, L acts as follows L(P1 , P2 , Q1 , Q2 ) = (−P1 , P2 , Q1 , −Q2 ).
Then we get two differential equations for variables (r, P ) (the reduced system) which do not depend on other variables but depend on k as a parameter. Two more equations are k˙ = 0 and ϕ˙ = −ω −µk/r 2 . Let us set k = 0, keeping in mind that homoclinic orbits to the equilibrium at the origin belong to this level. The reduced system then has a saddle at the origin, it corresponds to the saddle-focus equilibrium of XH0 and to the zero periodic orbit of the related 2π-periodic system. One more equilibrium, center, corresponds to an elliptic periodic solution of XH0 and belongs to the level H0 = 0. The phase portrait of the reduced system is plotted in Fig. 2. The reduced system has a homoclinic orbit to the
Then the set of fixed points for L is given by the equations: Fix(L) = {P1 = 0, Q2 = 0}. Homoclinic skirt intersects the plane of Fix(L) at two points: 1 (4) 0, 0, ± √ , 0 . 2A An explicit calculation shows that the twodimensional tangent plane to the skirt at intersection points (4) is spanned by two vectors (1, 0, 0, 0) and (0, 0, 0, 1). The tangent plane to Fix(L) is
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formed by vectors (0, 1, 0, 0) and (0, 0, 1, 0). Thus the skirt is transverse to Fix(L) at both points. If we consider this integrable system as being 2πperiodic, one need to regard the skirt as threedimensional in coordinates (P1 , P2 , Q1 , Q2 , t). Also for 2π-periodic integrable system, we consider involution L1 instead of L with Fix(L1 ) being two two-dimensional disks at the sections t = 0 and t = ±π. Since for the autonomous integrable system stable manifold of the saddle-focus equilibrium intersects 2-disk Fix(L) √ transversely at two different points (0, 0, ±1/ 2A, 0), then for the same system considered in R4 × S 1 stable manifold of the zero periodic solution intersects Fix(L √1 ) transversely at four √ different points (0, 0, ±1/ 2A, 0, 0) and (0, 0, ±1/ 2A, 0, π). The tangent vectors to the skirt are (1, 0, 0, 0, 1), (0, 0, 0, 1, 1), (0, 0, 0, 0, 1) and to Fix(L1 ) are (0, 1, 0, 0, 0) and (0, 0, 1, 0, 0).
3.1. Existence of homoclinic orbits To prove the theorem we wish to use the averaging method in its classical form [Bogoliubov & Mitropolski, 1961; Sanders et al., 2007]. If the system is in the standard form of the averaging method (we assume the system to be periodic or uniformly almost periodic in t) x˙ = µF (x, t, µ)
coordinate change (P1 , P2 , Q1 , Q2 ) → (ξ1 , ξ2 , η1 , η2 ), dP1 ∧ dQ1 + dP2 ∧ dQ2 = dξ1 ∧ dη1 + dξ2 ∧ dη2 , using a generating function S(P1 , P2 , η1 , η2 , t) = (P1 η1 + P2 η2 ) cos(ωt) − (P1 η2 − P2 η1 ) sin(ωt). In fact, this change of variables is used because all orbits of the linear Hamiltonian system with the Hamiltonian ωK are 2π/ω-periodic (except for the equilibrium). The transformation we make is nothing else but the linear symplectic 2π/ω-periodic transformation with the fundamental matrix of this linear system. This transformation just removes the term ωK in the Hamiltonian. Hamiltonian is transformed in accordance with the formula: H(P1 , P2 , η1 , η2 , t) +
˜ 1 , P2 , η1 , η2 , t). = H(P Then new Hamiltonian casts: ˜ = µ (ξ 21 + ξ 22 ) − µ (η 21 + η 22 ) + µA(η 21 + η 22 )2 H 2 2 ˜ 1 , ξ2 , η1 , η2 , µ, t). + µ2 h(ξ
and its averaged system x˙ = µF0 (x, µ),
∂S (P1 , P2 , η1 , η2 , t) ∂t
F0 (x, µ) = lim
T →∞ 0
T
F (x, t, µ)dt
for a fixed small µ > 0 has a hyperbolic equilibrium xµ , then the full system has in a O(µ2 )neighborhood of x0 a hyperbolic periodic solution (if F is periodic) or an almost periodic solution with the basic of frequencies as for F satisfying the exponential dichotomy of the same type as the equilibrium x0 . These solutions of the full system have their local stable and unstable manifolds which periodically or almost periodically depend on t. But we need to drag these manifolds till the set Fix(L1 ) and to prove their C 1 -proximity to the related manifolds of the averaged system in order to be able to use the transversality theorem. To reduce the problem to the standard form of the averaging method we need first to remove in the Hamiltonian the term ω(P1 Q2 − P2 Q1 ). To this end, we make a symplectic 2πω-periodic in time
(5)
The Hamiltonian contains terms of periods 2π and 2π/ω. This implies that if ω is irrational, the Hamiltonian and its related system will be quasiperiodic in t with two incommensurate frequencies, but if ω is rational, ω = q/p, q, p ∈ N, 0 < q < p and p, q have not common divisors, then the Hamiltonian and the related system will be 2πq-periodic in t. It is worth remarking that in both cases the averaged Hamiltonian up to the terms of the order µ will be the same integrable Hamiltonian H0 without the rotation term ξ1 η2 − ξ2 η1 . After 2π/ω-coordinate change the twodimensional disks which were the set Fix(L1 ) in the extended phase space R4 × S 1 , now become in R4 × R a countable set of 2-disks at the sections t = 2πm/ω and t = π + 2πm/ω. Their coordinate representations become
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and ξ1 cos((2n + 1)πω) − ξ2 sin((2n + 1)πω) = 0, η1 sin((2n + 1)πω) + η2 cos((2n + 1)πω) = 0. For the averaged autonomous system we proved the transversality of stable manifold of zero equilibrium O and Fix(L). In the extended phase space we get the cylindrical surface being the direct product of the stable manifold of O and R, this surface intersects transversally the countable set of 2-disks uniformly. Now, if we prove sufficient uniform C 1 proximity of the stable manifold for quasi-periodic or 2πp-periodic solution of the full system existing near O × R, then we shall get the countable set of intersection points. Returning to the initial system with the Hamiltonian (2), we come to the existence of four homoclinic solutions for the zero periodic solution. Indeed, let us consider for the system after scaling its 2π-Poincar´e map from the section t = 0 to t = 2π. If we consider its approximation up to O(µ2 ) order, this symplectic diffeomorphism is integrable, has a saddle-focus fixed point at the origin and its stable and unstable manifolds coalesce forming the homoclinic skirt. This skirt intersects transversely the set Fix(L1 ) at the section t = 0 at two points. Below, it will be proved that the stable manifold of the saddle-focus fixed point will be sufficiently close in C 1 topology for positive µ small enough to the stable manifold of the integrable approximation up to its intersection with Fix(L1 ) and hence intersects it transversely at a close point giving a symmetric homoclinic orbit to the fixed point. Thus we get two such points. But the same is true for the Poincar´e map constructed from the section t = −π to t = π. Here, we also get two intersection points of the stable manifold and the other piece of Fix(L1 ). Thus, we obtain four symmetric homoclinic orbits to zero periodic solution of 2π-periodic Hamiltonian system for each negative ε small enough in its modulus. Here, we need specially to prove a sufficient C 1 proximity of invariant manifolds till its intersection with the set of disks is obtained, the standard averaging methods do not pay special attention to this question [Bogoliubov & Mitropolski, 1961; Sanders et al., 2007]. It follows only from the standard averaging theory the existence of a periodic solution
being close to the set “{equilibrium}×S 1 ” in the full system and the smoothness of its stable (unstable) manifolds. We shall prove in Sec. 5 C 1 -proximity of these manifolds on the global piece till their crossing with Fix(L1 ) (in fact, on a bit longer piece) in two steps: first, we prove this locally near the periodic solution and after that in times of the order 1/µ in a neighborhood of a symmetric homoclinic orbit of the unperturbed system up to its intersection with Fix(L). Let us remark that for the Poincar´e map defined by 2π-periodic reversible system on the cross-section t = 0, the transversality of stable manifold of the fixed point and Fix(L) does not imply the symmetric homoclinic orbit be transverse, that is stable and unstable manifolds of the fixed point for the Poincar´e map intersect transversely. Here for the case of four-dimensional reversible symplectic diffeomorphism two possible cases can occur: (1) this homoclinic orbit is indeed transverse, and (2) this homoclinic orbit has a first order tangency of these manifolds.
4. Case A < 0 When A is negative, the types of periodic orbits p = q = 0 for 2π-periodic system, as ε varies, are the same as for the case A > 0 (recall that we scale δ = 1). But an essential difference with the case A > 0 is the local bifurcation that occurs here for values ε > 0 [van der Meer, 1985; Arnold et al., 1993; Glebsky & Lerman, 1995]. It is well known that a one-parameter family of periodic orbits branches out from the equilibrium of the integrable √ vector field XH4 in a neighborhood of order ε in this case.2 The collection of these periodic orbits forms a two-dimensional set being as before a topological sphere with two of its points identified. This set is filled with periodic orbits (except for the equilibrium itself), but these orbits are of three types. Namely, locally in a O(ε)-neighborhood of the equilibrium the orbits are elliptic (on the related level of Hamiltonian H4 such an orbit is enclosed by invariant 2-tori), they lie on the related level of the Hamiltonian in pairs, as one from two different local Lyapunov families. When moving along each of these two Lyapunov families further from the equilibrium, one meets a parabolic periodic orbit, two
2
This phenomenon gave one a reason to call this bifurcation as Hamiltonian Hopf bifurcation [van der Meer, 1985] in analogy with the usual Andronov–Hopf bifurcation [Shilnikov et al., 1998]. Following the historical sequence it would be better to name this bifurcation as Hamiltonian Andronov–Hopf bifurcation. 1440006-9
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parabolic orbits marked cut off from the singular sphere an annulus filled with hyperbolic (saddle) periodic orbits. Stable and unstable manifolds of each such orbit belong to the same level K = k as the periodic orbit itself, and two sheets of stable and unstable manifolds merge forming a homoclinic skirt, two√other sheets leave the neighborhood. As |k| → 1/6 −A, the hyperbolic periodic orbit at the level K = k with its skirt and the elliptic periodic orbit lying inside the skirt approach and coalesce into the related parabolic periodic orbit and disappear for greater |k|. At the level K = 0, there is a unique hyperbolic orbit, two sheets of stable and unstable manifolds coalesce forming a homoclinic skirt for this periodic orbit. The period of this orbit is equal to 2π/ω, so, if ω is rational, we get here a resonance with the period of the system. In this case, for the full system in the related 2πPoincar´e map instead of a closed invariant curve we have on the cylinder a resonance band with periodic points of elliptic and hyperbolic types forming a heteroclinic chain [Kaper & Kovacic, 1996]. When the system under study is reversible, then periodic orbits of the integrable system are symmetric and their skirts intersect transversely the set Fix(L1 ) at two points giving two symmetric homoclinic orbits for each corresponding periodic orbit. All this will be shown below for the reader’s convenience. For the periodic Hamiltonian Hopf bifurcation in the absence of strong resonances, the vector field XH0 describes only the behavior of the averaged system (in the scaled variables). This integrable system has the whole family of saddle periodic orbits for all values k close to zero of integral K. It means that we cannot apply directly the conclusions of the averaging theory, instead we have to use results of the theory of normally hyperbolic invariant sets [Fenichel, 1974; Hirsch et al., 1977]. Our main result in this section is the following theorem which we again formulate for the initial autonomous Hamiltonian system with three degrees of freedom. Theorem 2. Let H be a smooth Hamiltonian on a smooth six-dimensional symplectic manifold (M, Ω) and vector field XH has a periodic orbit γ whose nontrivial multipliers are two double nonsemisimple complex numbers on the unit circle exp[±iν], ν = 0, π (mod 2π). In this case, γ belongs to a symplectic cylinder filled with periodic orbits of XH close to γ and we suppose, to be definite, that for levels H = h < 0 the related periodic orbits on the
cylinder are hyperbolic (saddle-focus type) and for levels H = h > 0 the related periodic orbits on the cylinder are elliptic ones. We assume, in addition, the vector field to be reversible w.r.t. smooth antisymplectic involution L having a three-dimensional set of fixed points, H be invariant under the action of L and γ be symmetric. Then if some quantity A (see above) is negative and h > 0, then near the elliptic periodic orbit at the level H = h, there exists a Cantor set of saddle symmetric invariant twodimensional tori whose stable and unstable manifolds intersect each other along at least four symmetric homoclinic solutions. We outline only the main points of the proof, since it follows along the same lines as above for the case A > 0. After the same √ transformations as before, only with ε > 0, µ = ε, the Hamiltonian casts: µ H = ω(P1 Q2 − P2 Q1 ) + (P 21 + P 22 ) 2 µ + (Q21 + Q22 ) + µA(Q21 + Q22 )2 + O(µ2 ) 2 and we consider the terms of order µ2 and higher as a perturbation. In symplectic polar coordinates (3) we get a Hamiltonian up to O(µ2 ) terms µ k2 2 2 4 ˆ H0 = ωk + P + 2 + r + 2Ar , 2 r which generates a reduced system with one degree of freedom, its phase portrait is as in Fig. 3. This system has a saddle equilibrium at the point P = 0, r = −1/4A whose two separatrices coalesce forming the homoclinic loop. This homoclinic
Fig. 3. Phase portrait of the reduced system for K = k = 0, A < 0.
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loop belongs to the level of the reduced Hamiltonian ˆ 0 = µ [P 2 + r 2 + 2Ar 4 ] = − µ . H 2 16A Using this relation, the equality k = 0, we express P via r and after integration of the equation r˙ = µP we derive expressions for the solutions on the homoclinic skirt to the symmetric periodic orbit of the system with the truncated Hamiltonian of order µ in coordinates (P1 , P2 , Q1 , Q2 ): P1 =
cos(ωt − ϕ0 ) , √ √ 2µt −8A cosh2 2
sin(ωt − ϕ0 ) , √ √ 2µt −8A cosh2 2 √ 2µt 1 cos(ωt − ϕ0 ), Q1 = √ tanh 2 −4A √ 1 2µt tanh sin(ωt − ϕ0 ), Q2 = − √ 2 −4A
Fig. 4. Phase portrait of the reduced system for K = k = 0, A < 0.
P2 = −
from where we deduce that for the integrable system, the homoclinic skirt of this symmetric saddle periodic orbit intersects Fix(L) transversely giving homoclinic orbits with initial points at t = 0 through the points with ϕ0 = π/2, 3π/2. The periodic orbit has the period 2πω, when k = 0, since 2 for the integrable system, √ one has 2ϕ˙ = −ω 4− µk/r with √ r = r(k) = 1/ −4A + αk + O(k ), α = 2A −4A < 0. Here P = 0, r = r(k) are coordinates of the saddle equilibria of the reduced system for a fixed k. Symmetric saddle periodic orbits of the integrable system for k = 0 but small also have homoclinic skirts, since the related saddle equilibria P = 0, r = r(k) of the reduced system have homoclinic orbits (see Fig. 4). Now we consider the 2π-Poincar´e map of the full system being a perturbation of order µ2 for the Poincar´e map of the integrable approximation. Since the integrable system has the normally hyperbolic cylinder filled with periodic orbits, then the Poincar´e map of the full system also has the normally hyperbolic cylinder for k sufficiently close to zero (the smallness of k depends on ω and µ) [Fenichel, 1974]. The cylinder of the unperturbed Poincar´e map is foliated with closed symmetric
invariant curves, each of them intersects Fix(L) at rotation numbers are 2π(ω + √ two points, their −4Aµk + O(k3 )), so vary strictly monotonically in k (k plays the role of the action variable on the cylinder). Due to the Moser theorem [Moser, 1962], the majority of invariant curves on the cylinder, whose rotation numbers are Diophantine, persist. Every such persisting invariant curve is symmetric and possesses smooth two-dimensional stable and unstable manifolds [Fenichel, 1974] which can be drawn till their intersection with Fix(L). Then we get for the Poincar´e map of the full system the existence of symmetric homoclinic orbits to every such persistent invariant curve.
5. C 1 -Proximity of Stable Manifolds of Integrable and Full Systems In this section, for the case A > 0, we prove C 1 proximity for stable manifolds of integrable and full systems till their intersection with the set of fixed points of involution. This will be proved in two steps: first, we prove this locally near the periodic solution and after that in times of order 1/µ in a neighborhood of a symmetric homoclinic orbit of the unperturbed system up to its intersection with Fix(L). The related system for Hamiltonian (5) system after some additional transformations can be written in the following form:
u˙ = µAu + µf (u, v) + µ2 f1 (u, v, µ, t) (6) v˙ = µBv + µg(u, v) + µ2 g1 (u, v, µ, t),
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where u = (u1 , u2 ), v = (v1 , v2 ), A, B are 2 × 2 real matrices such that eigenvalues of A have negative real parts, and those of B have positive real parts, f , g, f1 , g1 are smooth vector-functions of the third order in (u, v). Integrable system corresponds to (6) with omitting terms f1 , g1 . The proof of existence and smoothness of the stable manifold for a periodic solution for system (6) and its C 1 -proximity to the stable manifold of the integrable system follow the lines in [Shilnikov et al., 1998]. In fact, the local part was proved in [Bogoliubov & Mitropolski, 1961] but there the manifolds were proved to be only Lipschitz continuous. Proposition 1. Zero periodic solution of system (6) has local smooth stable manifold which is C 1 -close to the stable manifold of zero periodic solution of the integrable system up to terms of order µ. Proof. Let us consider the following system of
integral equations with parameters t0 , u0 , µ: t µA(t−t ) 0 u0 + eµA(t−τ ) [µf (u(τ ), v(τ )) u(t) = e t 0 2 + µ f1 (u(τ ), v(τ ), µ, τ )]dτ +∞ v(t) = − eµB(t−τ ) [µg(u(τ ), v(τ )) t + µ2 g1 (u(τ ), v(τ ), µ, τ )]dτ. (7) On the stable manifold every solution of the system (6) is also a solution of the system (7) and vice versa. To prove the existence of solutions for this system, one uses the Banach contraction map principle considering the r.h.s. of (7) as an operator acting in the space of bounded continuous vector-functions (u(t), v(t)) defined on the semi-line t ≥ t0 with sufficiently small norm max u , v . This operator depends continuously on the parameter u0 , so its fixed point, if exists, also depends continuously on u0 .
At the first step, it is proved that this operator transforms the space into itself and is contracting. Therefore, there is a unique solution for u0 small enough. Let us notice that the solution obtained (u(t, t0 , u0 ), v(t, t0 , u0 )) at the section t = t0 gives the function v0 = v(t0 , t0 , u0 ). In R × R4 , its graph in the tube around solution (u, v) = (0, 0) is the stable manifold of this solution. One needs to prove its smoothness and uniform C 1 -proximity to the stable manifold of the integrable system considered as nonautonomous. At the next step, we prove that the stable manifold of zero solution for the full system deviates from that for the integrable system by the quantity of order µ locally. Indeed, solutions for the full sysint tem satisfy (7) and a solution (uint ∗ (t), v ∗ (t)) for integrable system with the same initial condition satisfies the same system with omitting the O(µ2 ) terms. Subtracting them on both sides and making similar estimates as before we get ˜ 2) (1 − Cρ
max
t∈[t0 ,+∞)
int
u∗ (t) − uint ∗ (t), v∗ (t) − v ∗ (t)
≤ µC˜1 . Since ρ is small, the solutions of the integrable and full systems are distinct by order µ, so the stable manifold of periodic solution for the full system deviates from that for the integrable system by order µ. Further, we show the derivatives for the stable manifolds for the integrable and full systems are also close in a neighborhood of zero solution. The solution of system (6) depends on three parameters: (u∗ (t; u0 , t0 , µ), v∗ (t; u0 , t0 , µ)). First we shall show that there is a continuous derivative (u∗ , v∗ ) in u0 . To this end, we formally differentiate (7) in u0 and derive for derivatives def
∂u∗ (t; u0 , t0 , µ), ∂u0
def
∂v∗ (t; u0 , t0 , µ). ∂u0
U ≡
V ≡
The approximation scheme is then as follows t ∂f ∂f µA(t−τ ) 2 ∂f1 2 ∂f1 µA(t−t ) 0 Un (τ ) + µ Vn (τ ) dτ (t) = e + e µ Un (τ ) + µ Vn (τ ) + µ U n+1 ∂u ∂v ∂u ∂v t0 +∞ ∂g ∂g µB(t−τ ) 2 ∂g1 2 ∂g1 Un (τ ) + µ Vn (τ ) dτ. e µ Un (τ ) + µ Vn (τ ) + µ Vn+1 (t) = − ∂u ∂v ∂u ∂v t 1440006-12
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We seek for solutions of the system (8) using the method of successive approximations as well. If we show the existence of a continuous solution for this system w.r.t. variables (U, V ), this will imply the solution for system (6) depends smoothly on u0 . As a first approximate solution, we take U1 (t) = eµA(t−t0 ) ,
V1 (t) ≡ 0.
The estimates on the r.h.s. of (8) allow one to assert that the sequence (Un (t), Vn (t))
converges and a continuous limit exists (U∗ (t), V∗ (t)) for the approximations (Un (t), Vn (t)) as n → +∞. It means that solution (u∗ , v∗ ) for the system (6) depends smoothly on u0 . The same is valid for the integrable system. Next, we show the derivatives in u0 for solutions of the full and integrable systems differ by order µ. Indeed, if we omit perturbation terms f1 , g1 , it is seen that the derivative in u0 of the solution for the integrable system has to satisfy the following system:
t ∂f int ∂f int µA(t−τ ) int µA(t−t ) 0 + e µ U ∗ (τ ) + µ V ∗ (τ ) dτ U ∗ (t) = e ∂u ∂v t0 +∞ ∂g int ∂g int µB(t−τ ) int e µ U ∗ (τ ) + µ V ∗ (τ ) dτ, V ∗ (t) = − t ∂u ∂v where def
U int ∗ ≡
∂uint ∗ (t; u0 , t0 , µ), ∂u0
def
V int ∗ ≡
∂v int ∗ (t; u0 , t0 , µ), ∂u0
∂f ∂g ∂g int int int int related Jacobi matrices ∂f ∂u , ∂v , ∂u , ∂v are taken at (u∗ , v ∗ ) and U ∗ , V ∗ ≤ 2C0 . Again we prove the differences int max max U∗ (t) − U int ∗ (t) , max V∗ (t) − V ∗ (t) t
t
are of the order µ. So, derivatives in u0 for solutions of integrable and full systems are close up to the terms of order µ. In the same way, one can show the solution (u∗ , v∗ ) of system (6) to depend smoothly on the parameter t0 . To this end we differentiate (7) in t0 : t ∂f ∂v∗ ∂f ∂u∗ ∂u∗ µA(t−t0 ) µA(t−τ ) 2 ∂f1 ∂u∗ 2 ∂f1 ∂v∗ = −e µAu0 + e +µ +µ +µ µ dτ ∂t0 ∂u ∂t0 ∂v ∂t0 ∂u ∂t0 ∂v ∂t0 t0 − eµA(t−t0 ) [µf (u∗ (t0 ), v∗ (t0 )) + µ2 f1 (u∗ (t0 ), v∗ (t0 ), µ, t0 )] +∞ ∂g ∂v∗ ∂g ∂u∗ ∂v∗ µB(t−τ ) 2 ∂g1 ∂u∗ 2 ∂g1 ∂v∗ e +µ +µ +µ µ dτ. ∂t0 = − ∂u ∂t0 ∂v ∂t0 ∂u ∂t0 ∂v ∂t0 t As in the previous case, we shall show that a continuous solution of this system exists. For this, we denote def
∂u∗ (t; u0 , t0 , µ), ∂t0
def
∂v∗ (t; u0 , t0 , µ), ∂t0
U ≡
V ≡
we take: U1 (t) = eµA(t−t0 ) (−µAu0 − µf (u∗ (t0 ), v∗ (t0 )) − µ2 f1 (u∗ (t0 ), v∗ (t0 ), µ, t0 )), V1 (t) ≡ 0.
and will seek for a solution by the successive approximation method. As the first approximation
Carrying out similar calculation one can prove that there exists a continuous limit (U∗ (t), V∗ (t)) for the successive approximations (Un (t), Vn (t))
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as n → +∞. This implies the solution (u∗ , v∗ ) of the system (6) is continuously differentiable in t0 . What is more, the derivatives in t0 for solutions of the full and integrable systems differ by the order µ. Hence local stable manifolds of the periodic solution of integrable and full systems are C 1 -close (close with its derivatives) in the neighborhood of periodic solution. The same holds for local stable manifolds of the fixed point for the Poincar´e map of integrable and full systems. Now, it remains to prove that stable manifolds of the zero solution of integrable and full systems stay C 1 -close up to their intersection with Fix(L1 ) (in fact, one needs to continue them a bit further). To simplify the problem, we introduce in a neighborhood of a segment of the symmetric homoclinic orbit for the integrable system from the local cross-section to the stable manifold till it hits Fix(L1 ), new coordinates in which the orbits of the integrable system are straightened. This can be done, since if one introduces the slow time τ = µt, then the passage time from the local crosssection to the stable manifold in the neighborhood of zero solution till Fix(L1 ) will be finite. Therefore, one can apply the theorem on the straightening trajectories (see, for instance [Arnold, 1992]). We denote these coordinates as (x, y, s, r), where r defines a position of a point on the orbit of the unperturbed integrable system, s enumerates orbits on the unperturbed stable manifold and (x, y) are coordinates in the transverse direction to the stable manifold. Since the full system is nonautonomous, additional t-coordinate appears. Without loss of generality one may regard r = const to be a cross-section for the orbits of the unperturbed system near the symmetric homoclinic orbit. It is more convenient to use the notation def x ≡ (x, y, s). In new coordinates, the perturbed system has the form:
x˙ = µ2 ϕ(x, r, t) (9) r˙ = µ + µ2 ψ(x, r, t), where functions ϕ, ψ are continuously differentiable in x, r and either quasi-periodic or periodic in t (and therefore being bounded). Let us transform the time τ = µt. For this “slow” time the solution reaches Fix(L) for the finite time τ . The system (9) with new time looks as follows:
dx τ dτ = µϕ x, r, µ τ dr = 1 + µψ x, r, . dτ µ
(10)
Observe that in these coordinates, the system for searching for the stable manifold of integrable system casts as follows: dx dτ = 0 dr = 1. dτ int Its solution is xint ∗ (τ ) = x0 , r ∗ (τ ) = τ − τ0 .
Proposition 2. Stable manifold of zero solution for
the full system is C 1 -close up to the terms of order µ to the stable manifold of zero solution of the integrable system till its intersection with Fix(L1 ). Proof. We shall seek solutions of the system (10)
with the initial condition x(τ0 ) = x0 , r(τ0 ) = 0 by the successive approximation method. To this end, we consider the system of integral equation: τ ξ dξ µϕ x(ξ), r(ξ), x(τ ) = x0 + µ τ0 (11) τ ξ dξ, µψ x(ξ), r(ξ), r(τ ) = τ − τ0 + µ τ0 where τ is finite (we take it longer than the exact time of reaching Fix(L)). On the stable manifold of the full system solution of (10) is a solution for (11) and vice versa. As the first approximation, we take r1 (τ ) = τ − τ0 .
x1 (τ ) = x0 ,
To construct the next approximations, we use the formulas: τ ξ dξ µϕ xn (ξ), rn (ξ), xn+1 (τ ) = x0 + µ τ0
rn+1 (τ ) = τ − τ0 +
τ τ0
ξ µψ xn (ξ), rn (ξ), µ
dξ.
It is evident that for finite τ and any n the norm
xn (τ ), rn (τ ) is bounded. Moreover, since ϕ, ψ are
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continuously differentiable then one has: τ ξ ξ µ
xn+1 (τ ) − xn (τ ) ≤ ϕ xn (ξ), rn (ξ), µ − ϕ xn−1 (ξ), rn−1 (ξ), µ dξ τ0 ≤ µ2C4 (τ − τ0 ) max xn (ξ) − xn−1 (ξ), rn (ξ) − rn−1 (ξ) , ξ∈[τ0 ,τ ]
τ ξ ξ dξ µ ψ xn (ξ), rn (ξ), − ψ xn−1 (ξ), rn−1 (ξ),
rn+1 (τ ) − rn (τ ) ≤ µ µ τ0
≤ µ2C4 (τ − τ0 ) max xn (ξ) − xn−1 (ξ), rn (ξ) − rn−1 (ξ) , ξ∈[τ0 ,τ ]
where max ∂(ϕ,ψ) ∂(x,y) ≤ C4 . Hence we have: max xn+1 (ζ) − xn (ζ), rn+1 (ζ) − rn (ζ) ≤
ζ∈[τ0 ,τ ]
1 max xn (ξ) − xn−1 (ξ), rn (ξ) − rn−1 (ξ) 2 ξ∈[τ0 ,τ ]
and the series +∞
(xn+1 (τ ) − xn (τ ), rn+1 (τ ) − rn (τ ))
n=1
converges absolutely and uniformly since the terms of the series +∞
xn+1 (τ ) − xn (τ ), rn+1 (τ ) − rn (τ )
n=1
are bounded by terms of the geometric progression with the ratio 1/2. Thus there is a continuous limit of successive approximations (x∗ (τ ), r∗ (τ )) being a solution of system (11): τ ξ dξ (τ ; τ , x ) = x + µϕ x (ξ; τ , x ), r (ξ; τ , x ), x 0 0 0 ∗ 0 0 ∗ 0 0 ∗ µ τ0 (12) τ ξ dξ. µψ x∗ (ξ; τ0 , x0 ), r∗ (ξ; τ0 , x0 ), r∗ (τ ; τ0 , x0 ) = τ − τ0 + µ τ0 The solution of system (11) is unique. It is worth remarking that on time [τ0 , τ ] solutions of the full and integrable systems differ by the order µ: τ ξ int dξ ≤ µC5 (τ − τ0 ), µ ϕ x (ξ), r (ξ),
x∗ (τ ) − x∗ (τ ) ≤ ∗ ∗ µ τ0 τ ξ int dξ ≤ µC5 (τ − τ0 ), µ ψ x∗ (ξ), r∗ (ξ),
r∗ (τ ) − r ∗ (τ ) ≤ µ τ0 here max (ϕ, ψ) ≤ C5 . Solution (x∗ (τ ; τ0 , x0 ), r∗ (τ ; τ0 , x0 )) depends smoothly on parameters τ0 and x0 . Again, to prove this, we formally differentiate expressions (12) in τ0 : τ ∂ϕ ∂x∗ ∂ϕ ∂r∗ τ0 ∂x∗ = µ + dξ − µϕ x∗ (τ0 ), r∗ (τ0 ), ∂τ0 ∂x ∂τ0 ∂r ∂τ0 µ τ0 (13) τ ∂x ∂r ∂ψ τ ∂ψ ∂r ∗ ∗ ∗ 0 . µ + dξ − µψ x∗ (τ0 ), r∗ (τ0 ), ∂τ0 = −1 + ∂x ∂τ0 ∂r ∂τ0 µ τ0 1440006-15
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As above, we solve this system by the successive approximation method w.r.t. variables ∂x∗ def ∂r∗ , R ≡ . ∂τ0 ∂τ0 Take as the first approximation X1 = 0, R1 = −1. Since τ is finite and µ is small enough, then for any n ˆ Also one has: the norm Xn , Rn is bounded by some constant C. τ ∂ϕ ∂ϕ µ
Xn+1 (τ ) − Xn (τ ) ≤ ∂x (Xn − Xn−1 ) + ∂r (Rn − Rn−1 ) dξ def
X ≡
τ0
≤ µ(τ − τ0 )2C4 max Xn (ξ) − Xn−1 (ξ), Rn (ξ) − Rn−1 (ξ) , ξ∈[τ0 ,τ ]
τ ∂ψ ∂ψ µ
Rn+1 (τ ) − Rn (τ ) ≤ ∂x (Xn − Xn−1 ) + ∂r (Rn − Rn−1 ) dξ τ0 ≤ µ(τ − τ0 )2C4 max Xn (ξ) − Xn−1 (ξ), Rn (ξ) − Rn−1 (ξ) . ξ∈[τ0 ,τ ]
R∗ −
max Xn+1 (ζ) − Xn (ζ), Rn+1 (ζ) − Rn (ζ)
ζ∈[τ0 ,τ ]
≤
1 max Xn (ξ) − Xn−1 (ξ), Rn (ξ) − Rn−1 (ξ) 2 ξ∈[τ0 ,τ ]
and the series +∞
(Xn+1 (τ ) − Xn (τ ), Rn+1 (τ ) − Rn (τ ))
n=1
converges absolutely and uniformly. So, system (13) has a continuous solution (X∗ , R∗ ), this means the solution of system (11) is continuously differentiable in τ0 . Derivative in τ0 for the solution for integrable system is def
X int ∗ ≡ def
Rint ∗ ≡
∂xint ∗ = 0, ∂τ0 ∂r int ∗ = −1. ∂τ0
Respectively, derivatives in τ0 for the full and integrable systems are close up to terms of order µ: ∂ϕ ∂ϕ R∗ dξ ≤ µ X∗ + ∂x ∂r τ0 τ0 + µ ϕ x∗ (τ0 ), r∗ (τ0 ), µ
X∗ − X int ∗
τ
≤ µ(τ − τ0 )2C4 Cˆ + µC5 ≤ µCˆ1 ,
∂ψ ∂ψ X∗ + R∗ ≤ µ dξ ∂x ∂r τ0 τ0 + µ ψ x∗ (τ0 ), r∗ (τ0 ), µ
Hence, the following hold Rint ∗
τ
≤ µ(τ − τ0 )2C4 Cˆ + µC5 ≤ µCˆ1 . In the same way it is shown that the solution (x∗ (τ ; τ0 , x0 ), r∗ (τ ; τ0 , x0 )) depends smoothly on x0 and differs from the related solution of the integrable system by order µ. Thus, we have proved that the stable manifold of the periodic solution for the full system is C 1 close to the stable manifold of the periodic solution of the integrable system up to its intersection with Fix(L). The same is valid for manifolds of the Poincar´e map of the full and integrable systems. The point of intersection of stable manifold of the integrable system with Fix(L) is on a finite distance from the origin and intersects Fix(L) transversely. So, C 1 -close (of order µ) stable manifold of the full system (or for the Poincar´e map of the periodic system) intersects Fix(L1 ) transversely at a close point. This gives a symmetric homoclinic orbit for any such intersection point. But there are four of these points. This completes the proof.
6. Conclusions Thus, we have discovered the following peculiarities of the dynamics at periodic reversible Hamiltonian Hopf bifurcation: • if A > 0 and ε < 0, the autonomous reversible integrable system has at the origin the 1440006-16
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saddle-focus equilibrium O; this equilibrium is symmetric and has homoclinic skirt filled with homoclinic orbits to O. This skirt transversely intersects the set Fix(L) at two different points. Then the full periodic system for any negative ε small enough has the unique symmetric saddlefocus periodic integral curve γε : O × S 1 , its stable and unstable manifolds intersect along at least four symmetric homoclinic curves; • if A < 0 and ε > 0 the reversible integrable system at the level K = k for all sufficiently small k has a saddle symmetric periodic orbit γk , its stable and unstable manifolds coincide forming homoclinic skirt which intersects transversely the set Fix(L) at two different points. Then the full periodic system for any positive ε small enough has a Cantor set of saddle invariant two-dimensional tori with Diophantine rotation numbers whose three-dimensional stable and unstable manifolds intersect each other along at least four symmetric homoclinic orbits of the related torus.
Acknowledgments The authors thank D. Turaev and S. Gonchenko for the useful discussions. We acknowledge a partial support from the Russian Foundation for Basic Research under grants 13-01-00589a (L. Lerman) and 14-01-00344 (A. Markova). We also thank the Russian Ministry of Science and Education for the support (project 1410, June 2014).
References Arnold, V. I. [1992] Ordinary Differential Equations (Springer-Verlag, Berlin-Heidelberg). Arnold, V. I., Kozlov, V. V. & Neishtadt, A. I. [1993] Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems III, Encyclopaedia Math. Sci., Vol. 3 (Springer, Berlin). Bochner, S. [1945] “Compact groups of differentiable transformations,” Ann. Math. 46, 372–381. Bogoliubov, N. N. & Mitropolski, Y. A. [1961] Asymptotic Methods in the Theory of Non-Linear Oscillations (Gordon & Breach, NY). Bridges, T. J., Cushman, R. H. & Mackay, R. S. [1995] “Dynamics near an irrational collision of eigenvalues for symplectic mappings,” Fields Inst. Comm. 4, 61–79. Bryuno, A. D. [1989] “Normalization of a Hamiltonian system near an invariant cycle or torus,” Russ. Math. Surveys 44, 53–89.
Cushman, R. [1978] “Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors,” Trans. Amer. Math. Soc. 238, 45–55. Devaney, R. [1976] “Reversible diffeomorphisms and flows,” Trans. Amer. Math. Soc. 218, 89–113. Fenichel, N. [1974] “Asymptotic stability with rate conditions,” Indiana Univ. Math. J. 23, 1109– 1137. Gaivao, J. P. & Gelfreich, V. [2011] “Splitting of separatrices for the Hamiltonian–Hopf bifurcation with the Swift–Hohenberg equation as an example,” Nonlinearity 24, 677–698. Gantmacher, F. R. [1959] The Theory of Matrices, Vols. 1 and 2 (Chelsea Pub. Co., NY). Glebsky, L. Yu. & Lerman, L. M. [1995] “On small stationary localized solutions for the generalized 1-D Swift–Hohenberg equation,” Chaos 5, 424–431. Glebsky, L. Yu. & Lerman, L. M. [1996] “1:-1 bifurcation for a periodic orbit in a 3D Hamiltonian system,” J. Tech. Phys. 37, 343–346. Haragus, M. & Iooss, G. [2011] Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, UTX Series (Springer, London & EDP Sciences, Les Ulis, France). Heggie, D. C. [1985] “Bifurcation at complex instability,” Celest. Mech. 35, 357–382. Hirsch, M. W., Pugh, C. C. & Shub, M. [1977] Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583 (Springer-Verlag, Berlin-Heidelberg, NY). Howard, J. E., Lichtenberg, A. L., Lieberman, M. A. & Cohen, P. H. [1986] “Four-dimensional mapping model for two-frequency electron cyclotron resonance heating,” Physica D 20, 259–284. Iooss, G. & Perouem´e, M. C. [1993] “Perturbed homoclinic solutions in reversible 1:1 resonance vector fields,” J. Diff. Eqs. 102, 62–88. Ivanov, A. P. & Sokol’skii, A. G. [1980] “On the stability of a nonautonomous Hamiltonian system under second-order resonance,” J. Appl. Math. Mech. 44, 574–581. Jorba, A. & Olle, M. [2004] “Invariant curves near Hamiltonian–Hopf bifurcations of four-dimensional symplectic maps,” Nonlinearity 17, 691–710. Kao, H.-C., Beaume, C. & Knobloch, E. [2014] “Spatial localization in heterogeneous systems,” Phys. Rev. E 89, 012903. Kaper, T. J. & Kovacic, G. [1996] “Multibump orbits homoclinic to resonance bands,” Trans. Amer. Math. Soc. 348, 3835–3887. Koltsova, O. Yu. & Lerman, L. M. [1999] “New criterion of nonintegrability for an N-degrees-of-freedom Hamiltonian system,” in Hamiltonian Systems with Three and More Degrees of Freedom, ed. Sim´ o, C., NATO ASI Series C: Math. and Phys. Sciences, Vol. 533 (Springer, Dordrecht), pp. 458–470.
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10:32
WSPC/S0218-1274
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Kozlov, V. V. [1983] “Integrability and non-integrability in Hamiltonian mechanics,” Russ. Math. Surv. 38, 1–76. Lamb, J. S. W. & Roberts, J. A. G. [1998] “Time-reversal symmetry in dynamical systems: A survey,” Physica D 112, 1–39. Markeev, A. P. [1978] Libration Points in Celestial Mechanics and Cosmodynamics (Nauka, Moscow) (in Russian). Meyer, K. R. [1981] “Hamiltonian systems with a discrete symmetry,” J. Diff. Eqs. 41, 228–238. Moser, J. K. [1962] “On invariant curves of areapreserving mappings of an annulus,” Nachr. Akad. Wiss. G¨ ottingen, Math. Phys. Kl. IIa, 1–20. Neishtadt, A. I. [1984] “The separation of motions in systems with rapidly rotating phase,” J. Appl. Math. Mech. 48, 133–139. Oll´e, M., Pacha, J. R. & Villanueva, J. [2004] “Motion close to the Hopf bifurcation of the vertical family of periodic orbits of L4 ,” Celest. Mech. Dyn. Astron. 90, 87–107. Olle, M., Pacha, J. R. & Villanueva, J. [2005] “Dynamics close to a non semi-simple 1:-1 resonant periodic orbit,” DCDS-B 5, 799–816. Olle, M., Pacha, J. R. & Villanueva, J. [2008] “Kolmogorov–Arnold–Moser aspects of the periodic Hamiltonian Hopf bifurcation,” Nonlinearity 21, 1759–1812.
Sanders, J. A., Verhulst, F. & Murdock, J. [2007] Averaging Methods in Nonlinear Dynamical Systems, 2nd edition, Applied Math. Sciences, Vol. 59 (Springer, NY, USA). Shilnikov, L. P. [1967] “On a Poincar´e–Birkhoff problem,” Math. USSR Sb. 3, 353–371. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. O. [1998] Methods of Qualitative Theory in Nonlinear Dynamics, Part I, World Scientific Series on Nonlinear Science, Vol. 4 (World Scientific, Singapore). Smale, S. [1965] “Diffeomorphisms with many periodic points,” in Differential and Combinatorial Topology, A Symposium in Honor of Marston Morse, ed. Cairns, S. S., Princeton Math. Series, Vol. 27 (Princeton Univ. Press, Princeton, NJ), pp. 63–80. Sokol’skii, A. G. [1977] “On stability of an autonomous Hamiltonian system with two degrees of freedom under first-order resonance,” J. Appl. Math. Mech. 41, 20–28. Vanderbauwhede, A. & Fiedler, B. [1992] “Homoclinic period blow-up in reversible and conservative systems,” ZAMP 43, 292–318. van der Meer, J. C. [1985] The Hamiltonian Hopf Bifurcation, Lecture Notes in Mathematics, Vol. 1160 (Springer-Verlag, Berlin-Heidelberg-NY-Tokyo).
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