bifurcations of generic heteroclinic loop ... - Semantic Scholar
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International Journal of Bifurcation and Chaos, Vol. 18, No. 4 (2008) 1069–1083 c World Scientific Publishing Company
BIFURCATIONS OF GENERIC HETEROCLINIC LOOP ACCOMPANIED BY TRANSCRITICAL BIFURCATION* FENGJIE GENG† School of Information Engineering, China University of Geosciences (Beijing), Beijing 100083, P. R. China gengfengjie−[email protected] DAN LIU and DEMING ZHU Department of Mathematics, East China Normal University, Shanghai 200062, P. R. China Received January 22, 2007; Revised August 28, 2007 The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1 and one hyperbolic saddle p2 are investigated, where p1 is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1 (resp. p2 ) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1 splits into two hyperbolic saddles p01 and p11 , a heteroclinic loop connecting p11 and p2 , homoclinic loop with p11 (resp. p2 ) and heteroclinic orbit joining p01 and p11 (resp. p11 and p2 ; p2 and p01 ) are found. The results achieved here can be extended to higher dimensional systems. Keywords: Local coordinates; Poincar´e map; transcritical bifurcation; homoclinic loop; periodic orbit.
1. Introduction Homoclinic and heteroclinic orbits have tremendous potential for applications in many important areas. Therefore, bifurcations of homoclinic and heteroclinic orbits have been studied extensively in the literature, see [Homburg & Krauskopf, 2000; Jin & Zhu, 2000, 2003; Morales & Pacifico, 2001; Wiggins, 1990; Zhang & Zhu, 2004; Zhu, 1998; Zhu & Xia, 1998]. However, most of the papers considered bifurcation problems of orbits connecting hyperbolic equilibria, and the corresponding problems with
nonhyperbolic equilibria are rarely investigated. It is well known that nonhyperbolic equilibrium is unstable and always undergoes saddle-node (transcritical or pitchfork) bifurcation. Obviously, the bifurcation problems of orbits joining nonhyperbolic equilibria are much more difficult and challenging. To the best of our knowledge, the research on homoclinic and heteroclinic bifurcations with nonhyperbolic equilibria are relatively less. Chow and Lin [1990] investigated bifurcations of homoclinic orbit with a saddle-node equilibrium. Deng [1990]
∗
Supported by NNSF of China ( 10671069), Shanghai Leading Academic Discipline Project (B 407). Author for correspondence
†
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studied bifurcations of codimensional-1 homoclinic orbit with a nonhyperbolic equilibrium. Liu and Zhu [2004] discussed bifurcations of generic homoclinic orbit for high dimensional system with nonhyperbolic equilibrium, they obtained the persistence of homoclinic orbit and the existence of periodic orbit bifurcated from the homoclinic orbit accompanied by transcritical bifurcation. Klaus and Knobloch [2003] considered bifurcations of homoclinic orbits to saddle-enter, they obtained the existence of orbit homoclinic to equilibrium and orbit homoclinic to center manifold. Sun and Luo [1994] investigated the local and global bifurcations for a generic (d + 1)-parameter family of threedimensional system with a heteroclinic cycle connecting a hyperbolic saddle and a nonhyperbolic equilibrium. Zhu [1994] achieved the persistence of generic heteroclinic orbits joining a saddle-node and a saddle. For other research on the bifurcations of orbits with nonhyperbolic equilibria, the readers are referred to [Liu & Zhu, 2004a, 2004b; Rademacher, 2005; Shilnikov et al., 2001; Zhu, 1994a, 1994b]. In view of less work on heteroclinic loop with nonhyperbolic equilibria, we investigate bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1 and one hyperbolic saddle p2 for the four-dimensional system. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1 (resp. p2 ) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1 splits into two hyperbolic saddles p01 and p11 , a heteroclinic loop connecting p11 and p2 , homoclinic loop with p11 (resp. p2 ) and heteroclinic orbit joining p01 and p11 (resp. p11 and p2 ; p2 and p01 ) are found. It is worthy of note that homoclinic loop connecting p01 and heteroclinic loop joining p01 and p2 cannot be bifurcated from the original heteroclinic loop, which is exactly determined by generic condition (H1 ). The results achieved here can be extended to higher dimensional systems. We introduce the method originally established in [Zhu, 1998; Zhu & Xia, 1998] and then improved in [Jin & Zhu, 2000, 2003; Zhang & Zhu, 2004], that is, choosing fundamental solutions of variational equations as a new local active system, taking a coordinate change and then constructing a Poincar´e map to induce bifurcation equations, we
attain the results by means of solutions for bifurcation equations. The method is more applicable and bifurcation equations obtained in this paper are easy to compute. Consider the following C r (r ≥ 5) system z˙ = g(z, λ, µ),
(1)
and its unperturbed system z˙ = f (z),
(2)
R4 ,
the vector field g depends on paramwhere z ∈ eters (λ, µ), λ ∈ R, µ ∈ Rl , l ≥ 2, 0 ≤ λ, |µ| 1, g(z, 0, 0) = f (z), g(p1 , 0, µ) = 0, g(p2 , λ, µ) = 0. Furthermore, suppose the parameters are generic in the sense that λ governs bifurcations of equilibria while µ controls bifurcations of orbits. Assume system (2) has a heteroclinic loop Γ connecting its two critical points p1 , p2 , where Γ = Γ1 ∪ Γ2 , Γi = {z = ri (t) : t ∈ R}, ri (+∞) = ri+1 (−∞) = pi+1 , i = 1, 2, r3 (t) = r1 (t), p3 = p1 . Moreover, the linearization Df (p1 ) has real eigenvalues 0, λ11 , −ρ11 and −ρ21 satisfying −ρ21 < −ρ11 < 0 < λ11 , Df (p2 ) has simple real eigenvalues λ12 , λ22 , −ρ12 and −ρ22 fulfilling −ρ22 < −ρ12 < 0 < λ12 < λ22 . It is easy to see dim(W 1u ) = 1, dim(W 1s ) = 2, dim(W 1c ) = 1 and dim(W 2u ) = dim(W 2s ) = 2, where W iu and W is denote unstable manifold and stable manifold of pi , respectively, W 1c is the center manifold of p1 . In addition, the following conditions hold: H1 : generic condition dim(Tr1 (t) W 1c ∩ Tr1 (t) W 2s ) = dim(Tr1 (t) W 1cu ∩ Tr1 (t) W 2s ) = 1, where W 1cu is the center unstable manifold of p1 . H2 : generic condition lim (Tr1 (t) W 1cu + Tr1 (t) W 2s ) = Tp2 W 2uu + Tp2 W 2s ,
t→+∞
lim (Tr2 (t) W 2u + Tr2 (t) W 1s ) = Tp1 W 1u + Tp1 W 1s ,
t→+∞
lim (Tr1 (t) W 1cu + Tr1 (t) W 2s ) = Tp1 W 1cu + Tp1 W 1ss ,
t→−∞
lim (Tr2 (t) W 2u + Tr2 (t) W 1s ) = Tp2 W 2u + Tp2 W 2ss ,
t→−∞
where W iuu and W iss are strong unstable manifold and strong stable manifold of pi , i = 1, 2, respectively. See Fig. 1 (we draw the manifolds W 1cu and W 2u only). H2 is called the strong inclination property. It is worthy of note that, for any integer m ≥ 1, n ≥ 1, if we assume dim(W 1u ) = dim(W 2uu ) = m, dim(W 1ss ) = dim(W 2ss ) = n, then all the results achieved in this paper are still valid.
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Bifurcations of Generic Heteroclinic Loop Accompanied by Transcritical Bifurcation
W pc 1 W pu2 W ps2 W puu 2 W pss2
= {(x, y, u, v)∗ = {(x, y, u, v)∗ = {(x, y, u, v)∗ = {(x, y, u, v)∗ = {(x, y, u, v)∗
1071
∈ U1 |y = u = v = 0}, ∈ U2 |y = v = 0}, ∈ U2 |x = u = 0}, ∈ U2 |x = y = v = 0}, ∈ U2 |x = y = u = 0}.
Based on the invariance of these manifolds, intro−1 (x , 0, 0)x, duce a scale transformation x → θxx p1 −1 λ → −θxλ (xp1 , 0, 0)λ, system (1) has the following expression in U1 :
Fig. 1.
Let λ ∈ R be a parameter to control the transcritical bifurcation of system (1), and the x-axis be the tangent space of the center manifold at p1 , θ(x, λ, µ) be the vector field defined on the center manifold, then by [Wiggins, 1990], we may assume H3 : θ(xp1 , λ, µ) = 0, (∂θ/∂x)(xp1 , 0, 0) = 0, (∂ 2 θ/∂x2 )(xp1 , 0, 0) > 0, (∂ 2 θ/∂x∂λ)(xp1 , 0, 0) < 0, (∂ 2 θ/∂x∂µ)(xp1 , 0, µ) = 0, where xp1 is the x component of p1 . If H3 is true, then system (1) exhibits the transcritical bifurcation, i.e. when λ > 0 (or λ < 0, in this paper, we only consider the case λ > 0, for the case λ < 0, one may discussed similarly), there are two hyperbolic saddles p01 , p11 bifurcated from p1 . Denote by p01 = p1 = (0, 0, 0, 0)∗ , p11 = p1 + (λp , 0, 0, 0)∗ , where λp = θ0 λ + O(λ2 ) + O(λµ), θ0 = −(∂ 2 θ/∂x∂λ)(xp1 , 0, 0)/(∂ 2 θ/∂x2 )(xp1 , 0, 0). Moreover, dim(W ps0 ) = 3, dim(W pu0 ) = 1 and 1 1 dim(W pu1 ) = dim(W ps1 ) = 2. 1
1
Remark. Conditions H1 –H3 imply that local bifurcation is codimensional 1 and global bifurcation is codimensional 2.
2. Local Coordinates and Successor Functions Suppose the neighborhood Ui of pi is small enough, there exist successively two C r , C r−1 transformations such that W 1c , W iu , W is , W 2uu , W iss , i = 1, 2, are straightened in the neighborhood Ui . The local invariant manifolds take the forms as W pu1 W ps1 W pcu1 W pss1
∗
= {(x, y, u, v) = {(x, y, u, v)∗ = {(x, y, u, v)∗ = {(x, y, u, v)∗
∈ U1 |x = y = v = 0}, ∈ U1 |x = u = 0}, ∈ U1 |y = v = 0}, ∈ U1 |x = y = u = 0},
x˙ = −λp x + x2 + O(u)[O(y) + O(v)] + O(x)[O(y) + O(u) + O(v)] + O(x)O(x2 ), y˙ = [−ρ11 (α) + · · ·]y + O(v)[O(x) + O(u)], u˙ = [λ11 (α) + · · ·]u + O(x)[O(y) + O(v)], v˙ = [−ρ21 (α) + · · ·]v + O(y)[O(x) + O(y) + O(u)], (3) and in U2 it takes the form: x˙ = [λ12 (α) + · · ·]x + O(u)[O(y) + O(v)], y˙ = [−ρ12 (α) + · · ·]y + O(v)[O(x) + O(u)], u˙ = [λ22 (α) + · · ·]u + O(x)[O(x) + O(y) + O(v)], v˙ = [−ρ22 (α) + · · ·]v + O(y)[O(x) + O(y) + O(u)], (4) where α = (λ, µ), λp = λ + O(λ2 ) + O(λµ), λ11 (0) = λ11 , ρji (0) = ρji , j = 1, 2, i = 1, 2, λj2 (0) = λj2 , j = 1, 2. The resulting systems (3) and (4) are C r−2 . Due to normal form (3), (4) and H1 , one may choose −Ti and Ti such that r1 (−T1 ) = (δ, 0, 0, 0)∗ , r1 (−T2 ) = (δ, 0, 0, δ2u , 0)∗ , ri (Ti ) = (0, δ, 0, δiv )∗ , i = 1, 2, where δ > 0 is small enough such that {(x, y, u, v) : |x|, |y|, |u|, |v| < 2δ} ⊂ Ui . Obviously, |δ2u | = o(δ), |δiv | = o(δ), i = 1, 2. Take into account the linear variational system z˙ = Df (ri (t))z and its adjoint system φ˙ = −(Df (ri (t)))∗ φ,
(5)i (6)i
i = 1, 2, where (Df (ri (t)))∗ is the transpose of Df (ri (t)). Let Zi (t) = (zi1 (t), zi2 (t), zi3 (t), zi4 (t)) be a fundamental solution matrix of (5)i , then similarly to [Jin & Zhu, 2000], we achieve the following lemma. Lemma 2.1.
If conditions H1–H3 are satisfied,
then (1) there exists a fundamental solution matrix of (5)1 satisfying z11 (t) ∈ (Tr1 (t) W 1cu )c ∩ (Tr1 (t) W 2s )c ,
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z12 (t) =
r˙1 (t) ∈ Tr1 (t) W 1c ∩ Tr1 (t) W 2s , |r˙1 (−T1 )|
z13 (t) ∈ Tr1 (t) W 1cu ∩ (Tr1 (t) W 2s )c , z14 (t) ∈ (Tr1 (t) W 1cu )c ∩ Tr1 (t) W 2s , such that
11 w1 12 w1 Z1 (−T1 ) = 13 w1 0
1
0 Z1 (T1 ) = 0 w114
w141
1
0
0
0
0
1
w142 , w143
0
0
w144
0
w131
w122
w132
0
w133
0 0 . 0
w124
w134
1
(2) (5)2 has a fundamental solution matrix fulfilling z21 (t) ∈ (Tr2 (t) W 2u )c ∩ (Tr2 (t) W 1s )c , z22 (t) =
r˙2 (t) ∈ Tr2 (t) W 2u ∩ Tr2 (t) W 1s , |r˙2 (−T2 )|
z23 (t) ∈ Tr2 (t) W 2u ∩ (Tr2 (t) W 1s )c , z24 (t) ∈ (Tr2 (t) W 2u )c ∩ Tr2 (t) W 1s , such that
w211
0
0
0
w223
1
w242 , w243
0
0
0
w244
1
0
w231
w222
w232
0
w233
0 , 0
w224
w234
1
12 w2 Z2 (−T2 ) = 13 w2
w241
1
0 Z2 (T2 ) = 0 w214
0
where wi12 = 0, wi22 < 0, wijj = 0, j = 3, 4, i = 1, 2. Now, let (zi1 (t), zi2 (t), zi3 (t), zi4 (t)) be a new local active coordinate system along Γi . Set Φi (t) = (φ1i (t), φ2i (t), φ3i (t), φ4i (t)) = (Zi−1 (t))∗ , then Φi (t) is the fundamental solution matrix of (6)i , i = 1, 2. Take a coordinate change z = ri (t) + Zi (t)Ni (t) hi (t), where Ni (t) = (n1i , 0, n3i , n4i )∗ , i = 1, 2. Define S 0i = {z = hi (−Ti ) : |x|, |y|, |u|, |v| < 2δ}, S 1i = {z = hi (Ti ) : |x|, |y|, |u|, |v| < 2δ}
Fig. 2.
as the cross-sections of Γi at t = −Ti and t = Ti , respectively, i = 1, 2. (See Fig. 2.) Notice that if q 0i ∈ S 0i , q 1i ∈ S 1i , then q 0i = (x0i , yi0 , u0i , vi0 )∗ = ri (−Ti ) + Z1 (−Ti )Ni (−Ti ),
0,3 0,4 ∗ Ni (−Ti ) = (n0,1 i , 0, ni , ni ) ,
q 1i = (x1i , yi1 , u1i , vi1 )∗ = ri (Ti ) + Zi (Ti )Ni (Ti ),
1,3 1,4 ∗ Ni (Ti ) = (n1,1 i , 0, ni , ni ) .
Based on the expressions of Zi (−Ti ) and Zi (Ti ), one derives the relationship between the old coordinates of q 0i (x0i , yi0 , u0i , vi0 )∗ , q 1i (x1i , yi1 , u1i , vi1 )∗ 0,3 0,4 ∗ and their new coordinates q,i0 (n0,1 i , 0, ni , ni ) , 1,3 1,4 ∗ q 1i (n1,1 i , 0, ni , ni ) , that is 12 −1 0 42 44 −1 0 n0,1 1 = (w1 ) [y1 − w1 (w1 ) v1 ], n0,3 = u01 − w113 (w112 )−1 y10 1 + [w113 w142 (w112 )−1 − w143 ](w144 )−1 v10 , (7)1 44 −1 0 n0,4 1 = (w1 ) v1 , x0 = δ + w11 n0,1 + w41 n0,4 ≈ δ, 1 1 1 1 1 1,1 n1 = x11 − w131 (w133 )−1 u11 , 1,3 33 −1 1 n1 = (w1 ) u1 , 1 v 14 1 n1,4 1 = v1 − δ1 − w1 x1 + (w114 w131 − w134 )(w133 )−1 u11 , 1 y1 = δ + w132 n1,3 1 ≈ δ.
0,1 12 −1 0 42 44 −1 0 n2 = (w2 ) [y2 − w2 (w2 ) v2 ], n0,3 = u02 − δ2u − w213 (w212 )−1 y20 2 + [w213 w242 (w212 )−1 − w243 ](w244 )−1 v20 , 44 −1 0 n0,4 2 = (w2 ) v2 , x0 = δ + w11 n0,1 + w41 n0,4 ≈ δ, 2 2 2 2 2
(8)1
(7)2
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Bifurcations of Generic Heteroclinic Loop Accompanied by Transcritical Bifurcation
1 31 33 −1 1 n1,1 2 = x0 − w2 (w2 ) u0 , n1,3 = (w233 )−1 u10 , 2 1 v 14 1 n1,4 2 = v0 − δ2 − w2 x0 + (w214 w231 − w234 )(w233 )−1 u10 , y 1 = δ + w32 n1,3 ≈ δ. 0
2
(8)2
2
Next, we shall take three steps to establish the Poincar´e map in new coordinate system. Step 1. Consider the map F i1 : S 0i → S 1i . Put z = hi (t) into (1), we have r˙i (t) + Z˙ i (t)Ni (t) + Zi (t)N˙ i (t)
+ gλ (ri (t), 0, 0)λ + gµ (ri (t), 0, 0)µ + h.o.t. = f (ri (t)) + Df (ri (t))Zi (t)Ni (t) + gλ (ri (t), 0, 0)λ + gµ (ri (t), 0, 0)µ + h.o.t. By way of the fact r˙i (t) = f (ri (t)) and Z˙ i (t) = Df (ri (t))Zi (t), it then follows that N˙ i (t) = Zi−1 (t)[gλ (ri (t), 0, 0)λ + gµ (ri (t), 0, 0)µ] + h.o.t. Integrating the above equation from −Ti to Ti , we arrive at Ti Zi−1 (t)gλ (ri (t), 0, 0)λdt Ni (Ti ) = Ni (−Ti ) + −Ti
Ti
+ −Ti
x10 ≈ h(s1 )x01 , u10
λ1 1 (α) ρ1 (α) 1
≈ s1
u01 ,
y10 ≈ s1 y01 , v10
u11 ≈ s2
u02 ,
ρ1 2 (α) λ1 2 (α)
y20 ≈ s2
v20 ≈ s2
s1 = 0,
M jiµ
Ti
−Ti Ti
Zi−1 (t)gµ (ri (t), 0, 0)µdt + h.o.t.
j = 1, 3, 4,
= −Ti
φj∗ i gλ (ri (t), 0, 0)dt, (10)i φj∗ i gµ (ri (t), 0, 0)dt,
j = 1, 3, 4.
Together with (7)i , (8)i and (10)i , (9)i define 0,3 0,4 the map F i1 : S 0i → S 1i , (n0,1 i , 0, ni , ni ) → 1,3 1,4 (n1,1 i , 0, ni , ni ).
v11 ,
if x10 ∈ [−β, λp ).
where =
(12)
Since p1 undergoes transcritical bifurcation based on the structure of orbits in U1 , one may see that the equation x10 ≈ h(s1 )x01 holds only when x10 ≥ λp . While for x10 ∈ [−β, λp )(0 < β 1), the map F 10 is well defined only if s1 = 0 (see Fig. 3). Therefore, the extension for the domain of F 10 is defined as
(9)i
y11 ,
ρ2 2 (α) λ1 (α) 2
Applying Φ∗i (t) = Z −1 i (t), it then produces
M jiλ
≈ s1
(11) v01
where the higher order terms are neglected, 1 ), i = 1, 2, are called Shilnikov coordi(si , u0i , vi−1 nates, and λp ρ1 (α) −1 0 0 1 ] , λp = 0, (13) h(s) = λp [x1 − (x1 − λp )s −1 1 −1 0 λp = 0. [1 − (ρ1 (α)) x1 ln s] ,
x01 = δ,
0,j j j n1,j i = ni + M iλ λ + M iµ µ + h.o.t,
ρ2 1 (α) ρ1 (α) 1
and F 02 : S 11 → S 12 :
λ2 2 (α) λ1 (α) 2
= g(ri (t), 0, 0) + gz (ri (t), 0, 0)Zi (t)Ni (t)
Step 2. To construct the map F i0 : S 1i−1 → S 0i (where S 10 = S 12 ). Let τi , i = 1, 2 be 1 , u1 , v 1 )∗ to the flying time from q 1i−1 (x1i−1 , yi−1 i−1 i−1 1 1 q 0i (x0i , yi0 , u0i , vi0 )∗ , set s1 = e−ρ1 (α)τ1 , s2 = e−λ2 (α)τ2 . Utilizing approximate solution of (3) and (4), we can easily obtain the expression of F 10 : S 10 → S 01 :
x11 ≈ s2 x02 ,
= g(ri (t) + Zi (t)Ni (t), λ, µ)
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Fig. 3.
(14)
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Step 3. Composing the maps F i0 and F 1i , then F1 = F 11 ◦ F 10 : S 10 → S 11 can be expressed as ρ2 1 (α) ρ1 1 (α)
12 −1 12 −1 42 44 −1 n1,1 1 = (w1 ) δs1 − (w1 ) w1 (w1 ) s1
+ M 11λ λ
+
M 11µ µ
44 −1 n1,4 1 = (w1 ) s1
12 −1 n1,1 2 = (w2 ) δs2
G42 = (w244 )−1 s2
v01 + M 41λ λ + M 41µ µ + h.o.t. (15)
ρ2 2 (α) λ1 (α) 2
− (w212 )−1 w242 (w244 )−1 s2
v11
G11 = (w112 )−1 δs1 − δs2 + M 11λ λ + M 11µ µ + h.o.t., λ2 2 (α) λ1 (α) 2
=
u02
=
w113 (w112 )−1 δs1 + − M 31µ µ + h.o.t.,
λ2 2 1
λ (w133 )−1 δs2 2
− M 31λ λ λ1 1
ρ1 + w213 (w112 )−1 δsβ2 2 + (w233 )−1 s1 1 − M 32λ λ − M 32µ µ + h.o.t.,
δ2u
u01
ρ2 1 ρ1 1
v11 = δ1v + (w144 )−1 s1 v01 + w114 δs2 + M 41λ λ + M 41µ µ + h.o.t., ρ2 2 λ1 2
v01 = δ2v + (w244 )−1 s2 v11 + w214 δh(s1 ) λ1 1 ρ1 1
+ (w214 w231 − w234 )(w233 )−1 s1 u01 + M 42λ λ + M 42µ µ + h.o.t. Putting the above solution into (G11 , G12 ) = 0, then we obtain the following bifurcation equations
G31 = u01 − w113 (w112 )−1 δs1 − (w133 )−1 δs2 + M 31λ λ + M 31µ µ + h.o.t.,
(w112 )−1 δs1 − δs2 + M 11λ λ + M 11µ µ + h.o.t. = 0, (w212 )−1 δsβ2 2 − x10 + M 12λ λ + M 12µ µ
v01 − v11 + δ1v + w114 δs2
λ1 1 +1 ρ1 1
+ M 41λ λ + M 41µ µ + h.o.t., λ1 1 (α) ρ1 (α) 1
− x10 + w231 (w233 )−1 s1
+ M 12λ λ + M 12µ µ + h.o.t.,
For convenience, denote β2 = ρ12 /λ12 . By implicit function theorem, we know that the equation (G3i , G4i ) = 0 has a unique solution
ρ2 2 (α) 1
Let Gi = Fi (q 1i−1 ) − q 1i , i = 1, 2. Owing to (8)i , (11), (12), (15), (16), we have the successor functions Gji as follows:
G12 = (w212 )−1 δs2
(17)
with s1 ≥ 0, s2 ≥ 0.
u01
v11 + M 42λ λ + M 42µ µ + h.o.t. (16)
ρ1 2 (α) λ1 (α) 2
(G11 , G31 , G41 , G12 , G32 , G42 ) = 0
3. Bifurcations Analysis
λ (α) + [w213 w242 (w212 )−1 − w243 ](w244 )−1 s2 2 v11 + M 32λ λ + M 32µ µ + h.o.t.,
G41 = (w144 )−1 s1
+ M 42λ λ
Clearly, to study the bifurcations near Γ, we should consider the solutions of
0 u 13 12 −1 n1,3 2 = u2 − δ2 − w2 (w2 ) δs2
ρ2 1 (α) ρ1 1 (α)
v11 − v01 + δ2v + w214 x10 λ1 1 (α) 1
ρ1 2 (α) λ1 (α) 2
44 −1 n1,4 2 = (w2 ) s2
u01 + M 32λ λ + M 32µ µ + h.o.t.,
ρ (α) + (w234 − w214 w231 )(w233 )−1 s1 1 u01 + M 42µ µ + h.o.t.
v01
+ M 12λ λ + M 12µ µ + h.o.t.,
ρ2 2 (α) λ1 (α) 2
−
ρ1 2 (α) 1
λ (α) w213 (w112 )−1 δs2 2
ρ2 2 (α) λ1 (α) 2
and F2 = F 21 ◦ F 20 : S 11 → S 12 (= S 10 ) as ρ1 2 (α) λ1 (α) 2
−
δ2u
− (w233 )−1 s1
+ M 31λ λ + M 31µ µ + h.o.t., ρ2 1 (α) ρ1 (α) 1
=
v01
0 13 12 −1 13 42 12 −1 n1,3 1 = u1 − w1 (w1 ) δs1 + [w1 w1 (w1 )
− w143 ](w144 )−1 s1
u02
λ1 1 (α) ρ1 (α) 1
+ h.o.t.,
ρ2 1 (α) ρ1 (α) 1
G32
+ δw113 w231 (w112 w233 )−1 s1 u01
λ2 2 1
λ × [(w133 )−1 δs2 2
λ1 1 ρ1 1
+ w231 (w233 )−1 s1
−M 31λ λ − M 31µ µ] + h.o.t. = 0. (18)
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Firstly, we assume the transcritical bifurcation does not happen, namely, λ = 0. Then based on (11) and (13), (18) becomes
Proof.
If s1 = s2 = 0 in (20), then δ−1 M 11µ µ + h.o.t. = 0, δ−1 M 12µ µ + h.o.t. = 0,
(w112 )−1 δs1 − δs2 + M 11µ µ + h.o.t. = 0, (w212 )−1 δsβ2 2 −
δ + M 12µ µ 1 − (ρ11 )−1 δ ln s1 λ1 1 ρ1 1
+ w231 (w233 )−1 s1 λ2 2 1
λ + (w133 )−1 δs2 2
−
(19)
s2 = δ−1 M 11µ µ + h.o.t., (w212 )−1 sβ2 2 + δ−1 M 12µ µ + h.o.t. = 0.
h.o.t. = 0. λ1 1 ρ1 1
Since λ11 /ρ11 > 0, we have lims1 →0 s1 (1 − (ρ11 )−1 δ ln s1 ) = 0, it then follows that (w112 )−1 s1 − s2 + δ−1 M 11µ µ + h.o.t. = 0, (w212 )−1 sβ2 2 −
1− + h.o.t. = 0.
which immediately produces result (1). If we assume s1 = 0, s2 > 0 in (20), then
[w113 (w112 )−1 δs1 M 31µ µ] +
1 (ρ11 )−1 δ ln s1
+ δ−1 M 12µ µ
(20)
Theorem 3.1. Let conditions H1 –H3 be true and M 1iµ = 0, i = 1, 2. Then for λ = 0 and 0 < |µ| 1, we have the following propositions.
(1) If rank(M 11µ , M 12µ ) = 2, then there exists a codimension 2 surface L12 = {µ : M 11µ µ + h.o.t. = M 12µ µ + h.o.t. = 0} such that the heteroclinic loop Γ persists if and only if µ ∈ L12 , where the surface L12 has a normal plane span{M 11µ , M 12µ } at µ = 0. (2) There exists an (l − 1)-dimensional surface L21 = {µ : δ−1 w212 M 12µ µ + (δ−1 M 11µ µ)β2 + h.o.t. = 0, M 11µ µ > 0}
1075
(21)
such that system (1) has a unique homoclinic loop Γ21 connecting p1 near Γ if and only if µ ∈ L21 , where L21 has a normal vector M 12µ (resp. M 11µ or M 11µ + w212 M 12µ ) as β2 > 1 (resp. β2 < 1 or β2 = 1) at µ = 0. (3) There exists an (l − 1)-dimensional surface δ 1 L2 = µ : 1 −1 1 − (ρ1 ) δ ln(−δ−1 w112 M 11µ µ)
− M 12µ µ + h.o.t. = 0, w112 M 11µ µ < 0 (22) such that system (1) has a unique homoclinic loop Γ12 joining p2 in the neighborhood of Γ if and only if µ ∈ L12 .
It follows that there exists an (l − 1)dimensional surface L21 given by (21) such that (20) has a unique solution s1 = 0, s2 = s2 (µ) > 0 as µ ∈ L21 and 0 < |µ| 1. This implies system (1) has a homoclinic loop Γ21 connecting p1 . There is no difficulty to see that L21 has a normal vector M 12µ at µ = 0 as β2 > 1, while for β2 < 1 (resp. β2 = 1) it has a normal vector M 11µ (resp. M 11µ + w212 M 12µ ) at µ = 0. The existence of L12 can be obtained similarly. This completes the proof. Remark 3.1. Set θ = −δ −1 w112 M 11µ µ, then for µ ∈
L12 , we have θ > 0 and M 12µ µ =
ρ11
ρ1 δρ11 + h.o.t. = − 1 + h.o.t., ln θ − δ ln θ
that is, M 12µ µ ln θ = −ρ11 + o(1), which means |M 11µ µ| = o([M 12µ µ]α ) for α > 1 as µ ∈ L12 . Theorem 3.2. Assume hypotheses H1 –H3 hold and M 1iµ = 0, i = 1, 2. Then for λ = 0, µ ∈ L21 and 0 < |µ| 1, system (1) has no periodic orbits except homoclinic loop Γ21 .
Due to Theorem 3.1, µ ∈ L21 and 0 < |µ| 1 mean that system (1) has a homoclinic loop Γ21 . Now restricting s1 ≥ 0, s2 = (w112 )−1 s1 + δ−1 M 11µ µ + h.o.t. > 0 and µ ∈ L21 , then (20) is reduced to Proof.
V1 (s1 ) [(w112 )−1 s1 + δ−1 M 11µ µ]β2 + δ−1 w212 M 12µ µ + h.o.t. =
w212 1 − (ρ11 )−1 δ ln s1
N1 (s1 ).
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Thus V1 (0) = N1 (0) and V 1 (s1 ) = β2 (w112 )−1 [(w112 )−1 s1 + δ−1 M 11µ µ]β2 −1 , N1 (s1 ) =
w212 (ρ11 )−1 δ . (1 − (ρ11 )−1 δ ln s1 )2 s1
If w112 w212 < 0, then V 1 (s1 )N1 (s1 ) < 0, it is clear that V1 (s1 ) = N1 (s1 ) has no sufficiently small positive solutions. If β2 > 1, then |V 1 (s1 )| 1 and |N1 (s1 )| 1 hold for 0 < s1 1, which shows that V1 (s1 ) = N1 (s1 ) has no sufficiently small positive solutions. Next, we only consider the case β2 < 1 and w112 w212 > 0. There are two cases to study, namely, (i) β2 < 1, w112 > 0, w212 > 0, (ii) β2 < 1, w112 < 0, w212 < 0. Case (i). In this case, it is easy to see that V 1 (s1 ) = β2 (w112 )−1 [(w112 )−1 s1 + δ−1 M 11µ µ]β2 −1 ≤ β2 (w112 )−β2 s1β2 −1 .
(23)
Fig. 4.
β2 < 1, w112 < 0, w212 < 0.
for µ ∈ L21 . Combining with the fact V1 (0) = N1 (0) and V 1 (0) N1 (0), we see that there does not exist a small positive solution for V1 (s1 ) = N1 (s1 ). (See Fig. 4.) The proof is then completed.
Due to lim sβ2 −1 s1 →0 1
= ∞,
lim N1 (s1 ) = ∞
s1 →0
and
(1 − (ρ11 )−1 δ ln s1 )2 sβ1 2 s1β2 −1 = 0, = s1 →0 N1 (s1 ) w212 (ρ11 )−1 δ
w212 (ρ11 )−1 δ (1 − (ρ11 )−1 δ ln s1 )3 s21
(1) If w212 > 0 and β2 > 1 (resp. β2 = 1; β2 < 1), then system (1) has a unique periodic orbit near Γ as µ is situated in the neighborhood of L21 and confined to the side pointed by w212 M 12µ (resp. w212 M 12µ + M 11µ ; M 11µ ); system (1) has no periodic orbits near Γ as µ is situated in the neighborhood of L21 and confined to the side pointed by −w212 M 12µ (resp. −[w212 M 12µ +M 11µ ]; −M 11µ ). (2) If w212 < 0 and β2 > 1 (resp. β2 = 1; β2 < 1), then system (1) has exactly one periodic orbit as µ situated in the neighborhood of L21 and confined to the side pointed by −w212 M 12µ (resp. −[w212 M 12µ + M 11µ ]; −M 11µ ); system (1) has no periodic orbits as µ is situated in the neighborhood of L21 and confined to the side pointed by w212 M 12µ (resp. w212 M 12µ + M 11µ ; M 11µ ).
× [2(ρ11 )−1 δ − (1 − (ρ11 )−1 δ ln s1 )] > 0,
Proof.
lim
it then follows from (23) that V 1 (s1 ) < N1 (s1 ) for 0 < s1 1, this means V1 (s1 ) = N1 (s1 ) has no sufficiently small positive solutions. Obviously, the conclusion holds for β2 = 1. Case (ii). Notice that for w112 < 0, w212 < 0, s1 should subject to 0 < s1 < δ−1 |w112 |M 11µ µ, then one achieves V 1 (s1 ) < 0, N1 (s1 ) < 0, V 1 (s1 ) = β2 (β2 − 1)(w112 )−2 × [(w112 )−1 s1 + δ−1 M 11µ µ]β2 −2 < 0, N1 (s1 ) =
Corollary 3.1. Suppose the conditions of Theorem 3.2 are fulfilled, then the following propositions are true.
and V1 (δ−1 |w112 |M 11µ µ) = δ−1 w212 M 12µ µ + h.o.t. = −[δ−1 M 11µ µ]β2
1−
w212 (ρ11 )−1 δ ln(δ−1 |w112 |M 11µ µ)
= N1 (δ
−1
|w112 |M 11µ µ)
Let
W (µ) = V1 (0) − N1 (0) = δ−1 w212 M 12µ µ + (δ−1 M 11µ µ)β2 + h.o.t. Then
∂W (µ) = δ−1 w212 M 12µ ∂µ L2 1
+ β2 (δ−1 M 11µ µ)β2 −1 δ−1 M 11µ + h.o.t.,
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(a) w112 > 0, w212 < 0
(b) w112 < 0, w212 > 0
(c) w112 > 0, w212 > 0
(d) w112 < 0, w212 < 0 Fig. 5.
β2 > 1.
clearly, the gradient direction (∂W (µ)/∂µ)|L21 at µ = 0 is determined by w212 M 12µ as β2 > 1, by w212 M 12µ + M 11µ as β2 = 1 and by M 11µ as β2 < 1. Combining the proof of Theorem 3.2 with relative positions of the curves V1 (s1 ) and N1 (s1 ) for 0 < s1 1 and w212 > 0 (resp. w212 < 0), Proposition 1 (resp. (2)) follows immediately. For relative positions of the curves V1 (s1 ) and N1 (s1 ), one may see Fig. 5. Theorem 3.3. Assume that H1–H3 hold and M 1iµ =
0, i = 1, 2. Let λ = 0, µ ∈ L12 and 0 < |µ| 1, then in addition to the homoclinic loop Γ12 , system (1)
(1) has no periodic orbits near Γ as β2 ≥ 1 or w112 w212 < 0; (2) has at least one periodic orbit near Γ as β2 < 1, w112 > 0 and w212 > 0; (3) has a unique periodic orbit near Γ as β2 < 1, w112 < 0 and w212 < 0. Proof. Based on Theorem 3.1, µ ∈ L12 and 0
0. Notice that V2 (0) = N2 (0) as µ ∈ L12 . Moreover, V 2 (s2 ) = β2 s2β2 −1 , N2 (s2 ) =
w212 (ρ11 )−1 δ ·
[1 −
1
s2 − δ−1 M 11µ µ . (ρ11 )−1 δ ln(w112 (s2 − δ−1 M 11µ µ))]2
With similar arguments to proof of Theorem 3.2, one knows that V2 (s2 ) = N2 (s2 ) has no sufficiently small positive solutions as β2 ≥ 1 or w112 < 0. Next, we investigate the case β2 < 1 and w112 w212 > 0. 1 For β2 < 1, define s2 = e−ρ2 τ2 , then functions V2 (s2 ) and N2 (s2 ) are changed to the following forms:
N2 (s2 ) =
w212
.
1 β
1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ)) 1 β
=
1077
w212 (ρ11 )−1 δs2 2 1 β
−1 1 β
β2 [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))]2 (s2 2 − δ−1 M 11µ µ)
.
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There are still two cases to consider: (i) β2 < 1, w112 > 0, w212 > 0, (ii) β2 < 1, w112 < 0, w212 < 0.
Case (i). V 2 (0) = 1 > 0 = N2 (0) implies that there exists an 0 < s˜2 1 such that V2 (s2 ) > N2 (s2 ) for 0 < s2 < s˜2 . Choosing s2 = δ−1 w212 M 12µ µ > 0, then V2 (s2 ) = 2s2 + h.o.t., N2 (s2 ) =
w212
1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))
V2 ((δ−1 M 11µ µ)β2 ) = (δ−1 M 11µ µ)β2 + δ−1 w212 M 12µ µ
In view of w212 1
1
β − (ρ11 )−1 δ ln(w112 (s2 2 w212
−
1
β (ρ11 )−1 δ ln(w112 s2 2 )
1− which shows that V2 (s2 ) < N2 (s2 ).
=
+ h.o.t. < 0 = N2 ((δ−1 M 11µ µ)β2 )
δ−1 M 11µ µ))
>
N2 (s2 )
Case (ii). Under these circumstances, s2 must fulfil 0 < s2 < (δ−1 M 11µ µ)β2 , it then follows that N2 (s2 ) > 0. Owing to V 2 (0) = 1 > 0 = N2 (0), there exists an 0 < s˜2 1 such that V2 (s2 ) > N2 (s2 ) as 0 < s2 < s˜2 < (δ−1 M 11µ µ)β2 . On the other hand, by Remark 3.1, we have
.
1 β
As a result, N2 (s2 ) = V2 (s2 ) has at least one solution sˆ2 satisfying 0 < s˜2 < sˆ2 < s2 1.
for µ ∈ L12 and 0 < |µ| 1. Based on the continuity of the functions, there exists an 0 < s∗2 1 such that V2 (s∗2 ) = N2 (s∗2 ) for 0 < s˜2 < s∗2 < (δ−1 M 11µ µ)β2 1. Next, we shall show s∗2 is unique. Note that
2s2 ,
2 β
2w212 (ρ11 )−2 δ2 1 β
s2 2
1 β
−2
β22 [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))]3 (s2 2 − δ−1 M 11µ µ)2 1 2 −2 β1 −2 1 β2 −1 M 1 µ) − 1 s β2 2 − 1 s (s − δ 1µ 2 2 2 12 1 −1 w2 (ρ1 ) δ β2 β2 + 1 1 β β β2 [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))]2 (s2 2 − δ−1 M 11µ µ)2 1 β
=
w212 (ρ11 )−1 δs2 2
−2
1 β
1 β
1 β
β22 [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))]3 (s2 2 − δ−1 M 11µ µ)2 1 β
{2(ρ11 )−1 δs2 2
1 β
+ [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))](−β2 s2 2 − (1 − β2 )δ−1 M 11µ µ)}. Obviously, N2 (s2 ) > 0 for w212 < 0, 0 < s2 < (δ−1 M 11µ µ)β2 and µ ∈ L12 . Combining with the fact N2 (s2 ) > 0 and V 2 (s2 ) = 1, one immediately knows that s∗2 is unique. This completes the proof. Corollary 3.2. Let all conditions of Theorem 3.3 be satisfied, then the following results are true.
(1) For w112 < 0, w212 < 0 and β2 < 1, if µ is situated in the neighborhood of L12 and confined to the side pointed by −w112 w212 M 11µ , then system (1) has two periodic orbits near Γ; if µ is situated in the neighborhood of L12 and confined to the side pointed by w112 w212 M 11µ , then system (1) has one periodic orbit near Γ.
(2) For w112 > 0, w212 > 0 and β2 < 1, if µ is situated in the neighborhood of L12 and confined to the side pointed by −w112 w212 M 11µ , then system (1) has at least two periodic orbits near Γ; if µ is situated in the neighborhood of L12 and confined to the side pointed by w112 w212 M 11µ , then system (1) has at least one periodic orbit near Γ. (3) For w112 w212 > 0 and β2 ≥ 1, if µ is situated in the neighborhood of L12 and confined to the side pointed by −w112 w212 M 11µ , then system (1) has no periodic orbits near Γ; if µ is situated in the neighborhood of L12 and confined to the side pointed by w112 w212 M 11µ , then system (1) has one periodic orbit near Γ.
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1079
(4) For w112 w212 < 0, if µ is situated in the neighborhood of L12 and confined to the side pointed by −w112 w212 M 11µ , then system (1) has one periodic orbit near Γ; if µ is situated in the neighborhood of L12 and confined to the side pointed by w112 w212 M 11µ , then system (1) has no periodic orbits near Γ. Proof. Set
W (µ) = N2 (0) − V2 (0) =
w212 − δ−1 w212 M 12µ µ + h.o.t. 1 − (ρ11 )1 δ ln(−δ−1 w112 M 11µ µ)
If µ ∈ L12 , we have [1 − (ρ11 )1 δ ln(−δ−1 w112 M 11µ µ)]−1 = (M 12µ µ/δ) + h.o.t., which means −δw212 w112 M 11µ ∂W (µ) = 1 − δ−1 w212 M 12µ + h.o.t.|L12 ∂µ L1 ρ1 [1 − (ρ11 )1 δ ln(−δ−1 w112 M 11µ µ)]2 (−w112 M 11µ µ) 2
=
−w212 w112 M 11µ (M 12µ µ)2 δρ11 (−w112 M 11µ µ)
− δ−1 w212 M 12µ + h.o.t.|L12 .
By Remark 3.1, −w112 w212 M 11µ is approximately the gradient direction of W (µ). Combining the relative positions of the curves N2 (s2 ) and V2 (s2 ) in the interval s2 ∈ [0, sˆ2 ) for some 0 < sˆ2 1, we claim that the non-negative solution s2 (µ) of V2 (s2 ) = N2 (s2 ) satisfying s2 (µ) = 0 as µ ∈ L12 , and for w112 < 0, w212 < 0, β2 < 1, if µ moves along the direction −w112 w212 M 11µ from L12 , then s2 (µ) = 0 will become s2 (µ) > 0, if µ moves along the direction w112 w212 M 11µ from L12 , then s2 (µ) will disappear. On the other hand, the curves N2 (s2 ) and V2 (s2 ) are intersected transversally at the unique positive solution s∗2 (µ), then it will survive under any small perturbation of µ. The proof of (1) is then completed. The proof (2)–(4) can be completed with similar arguments. For the relative positions of curves N2 (s2 ) and V2 (s2 ) as w112 w212 > 0, β2 < 1, one may see Fig. 6.
Now, we turn to discussing the bifurcations of the heteroclinic loop when p1 undergoes transcritical bifurcation, namely, λ > 0. One knows that when λ > 0, after the creation of p01 and p11 , there always exists a straight segment orbit heteroclinic to p11 and p01 , its length is λp , we denote this heteroclinic orbit by Γ∗ . Moreover, based on Fig. 3, we may see that x10 = λp is a critical position. First, we investigate the case x10 ≥ λP . For this case, (18) becomes (w112 )−1 δs1 − δs2 + M 11λ λ + M 11µ µ + h.o.t. = 0, (w212 )−1 δsβ2 2
λp ρ1 1
− δλp [δ − (δ − λp )s1 ]−1 + M 12λ λ
+ M 12µ µ
+
(24) λ2 2 λ1 2
× [w113 (w112 )−1 δs1 + (w133 )−1 δs2 − M 31λ λ − M 31µ µ] + h.o.t. = 0.
(a) w112 > 0, w212 > 0
(b) w112 < 0, w212 < 0 Fig. 6.
λ1 1 1
ρ w231 (w233 )−1 s1 1
β2 < 1.
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1080
0 < |µ| 1, where L21λ has a normal vector M 12µ (resp. M 11µ or (w212 )−1 M 11µ + M 12µ ) at µ = 0 as β2 > 1 (resp. β2 < 1 or β2 = 1); (2) there exists an (l − 1)-dimensional surface
λp ρ1 1
Let r = s1 , we have δλp λp ρ1 1
δ − (δ − λp )s1
=
δλp δ − (δ − λp )r
L12λ = {µ(λ) : W 21 (λ, µ) = δλp + λp (δ − λp )
λp (δ − λp ) r + h.o.t. = λp + δ
λp 1
× [−δ−1 w112 (M 11λ λ + M 11µ µ)] ρ1
− δM 12λ λ − δM 12µ µ + h.o.t. = 0,
It then follows from (24) that ρ1 1
(w112 )−1 δr λp
− δs2 + M 11λ λ + M 11µ µ + h.o.t. = 0,
λp (δ − λp ) r + M 12λ λ δ + M 12µ µ + h.o.t. = 0.
(w212 )−1 δsβ2 2 − λp −
(25)
Theorem 3.4. Suppose the conditions H1 –H3 hold,
rank(M 11µ , M 12µ ) = 2 and 0 < λ 1, then there exists an (l − 2)-dimensional surface Lλ12 = {µ(λ) : M 11µ µ+M 11λ λ+h.o.t. = M 12µ µ+M 12λ λ−λ+h.o.t. = 0} such that system (1) has a unique heteroclinic loop near Γ if and only if µ ∈ Lλ12 and 0 < |µ| 1, where Lλ12 has a normal plane span{M 11µ , M 12µ } at µ = 0. Proof.
Assume r = s2 = 0 in (25), then M 11µ µ + M 11λ λ + h.o.t. = 0, M 12µ µ + M 12λ λ − λp + h.o.t. = 0.
If rank(M 11µ , M 12µ ) = 2, the above equation determines an (l − 2)-dimensional surface Lλ12 = {µ(λ) : M 11µ µ + M 11λ λ + h.o.t. = M 12µ µ + M 12λ λ − λp + h.o.t. = 0} such that (25) has a solution r = s2 = 0 as µ ∈ Lλ12 and 0 < |µ| 1, i.e. system (1) has a unique heteroclinic loop near Γ. Clearly, Lλ12 has a normal plane spanned by M 11µ and M 12µ at µ = 0. This completes the proof. Theorem 3.5. Let H1 –H3 be true and 0 < λ 1, M 1iµ = 0, i = 1, 2, then
(1) there exists an (l − 1)-dimensional surface L21λ = {µ(λ) : W 12 (λ, µ) = (w212 )−1 [δ−1 (M 11µ µ + M 11λ λ)]β2 + δ−1 (M 12µ µ + M 12λ λ) − δ−1 λp + h.o.t. = 0, M 11µ µ + M 11λ λ > 0} such that system (1) has one homoclinic loop connecting p11 if and only if µ ∈ L21λ and
w112 (M 11λ λ + M 11µ µ) < 0} such that system (1) has a homoclinic loop joining p2 if and only if µ ∈ L12λ and 0 < |µ| 1, where L12λ has a normal vector M 11µ at µ = 0. Theorem 3.6. Suppose hypotheses H1 –H3 hold, M 1iµ = 0, i = 1, 2, 0 < λ, |µ| 1 and w112 w212 < 0. Then except the homoclinic loop connecting p11 , system (1) has no periodic orbits as µ ∈ L21λ . Corollary 3.3. Let all conditions in Theorem 3.6 be
satisfied, for w112 w212 < 0 and β2 > 1 (resp. β2 < 1 or β2 = 1), if µ is situated in the neighborhood of L21λ and confined to the side pointed by M 12µ (resp. M 11µ or (w212 )−1 M 11µ + M 12µ ), then system (1) has a unique periodic orbit near Γ; if µ is situated in the neighborhood of L21λ and confined to the side pointed by −M 12µ (resp. −M 11µ or −[(w212 )−1 M 11µ + M 12µ ]), then system (1) has no periodic orbits near Γ. Theorem 3.7.
Assume conditions H1–H3 are
= 0, i = 1, 2, 0 < λ, |µ| 1 and satisfied, w112 w212 < 0. Then in addition to the homoclinic loop joining p2 , system has no periodic orbits as µ ∈ L12λ . M 1iµ
Corollary 3.4. Let all hypotheses of Theorem 3.7 be
fulfilled, then the following propositions are true. (1) For w212 > 0 and w112 < 0, if µ is situated in the neighborhood of L12λ and confined to the side pointed by M 11µ , then system (1) has a unique periodic orbit near Γ; if µ is situated in the neighborhood of L12λ and confined to the side pointed by −M 11µ , then system (1) has no periodic orbits near Γ. (2) For w212 < 0 and w112 > 0, if µ is situated in the neighborhood of L12λ and confined to the side pointed by M 11µ , then system (1) has no periodic orbits near Γ; if µ is situated in the neighborhood of L12λ and confined to the side pointed by −M 11µ , then system (1) has one periodic orbit near Γ.
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(a) λ = 0, µ ∈ L12
(b) λ = 0, µ ∈ L21
(c) λ = 0, µ ∈ L12
(d) λ > 0, µ ∈ Lλ 12
(e) λ > 0, µ ∈ L21λ
(f) λ > 0, µ ∈ L12λ
1081
Fig. 7.
To illustrate Theorems 3.1–3.7, we draw some of the figures, see Fig. 7. Remark 3.2. By virtue of Fig. 7, one may see that homoclinic loop connecting p01 and heteroclinic loop joining p01 , p2 cannot be bifurcated from Γ, which is exactly determined by the generic condition H1 .
In the following, −β ≤ x10 < λp (0 < β 1) will be considered. Due to Fig. 3 and (14), it follows from (18) that s2 = δ−1 (M 11λ λ + M 11µ µ) + h.o.t., x10 = (w212 )−1 δsβ2 2 + M 12λ λ + M 12µ µ + h.o.t.
(26)
Theorem 3.8. Assume the conditions H1–H3 are true, rank(M 11λ , M 11µ ) > 0 and rank(M 12λ , M 12µ ) > 0. Then
(1) there exists a surface Σ1 (µ, λ) = {µ(λ) : M 11λ λ + M 11µ µ + h.o.t. = 0, − β ≤ M 12λ λ + M 12µ µ + h.o.t. < λp , 0 < |µ|, λ 1}, (27) such that system (1) has two orbits heteroclinic to p11 , p2 , p01 as µ ∈ Σ1 (µ, λ), see Figs. 8(a)–8(c);
(2) there exists a region in the (λ, µ) space ∆ = {(λ, µ) : −β ≤ (w212 )−1 δ1−β2 (M 11λ λ + M 11µ µ)β2 + M 12λ λ + M 12µ µ + h.o.t. < λp , M 11λ λ + M 11µ µ > 0, 0 < |µ|, λ 1},
(28)
such that system (1) has a heteroclinic orbit connecting p11 and p01 for (λ, µ) ∈ ∆, see Figs. 8(d )–8(f ). Proof.
If s2 = 0 in (26), then 0 = δ−1 (M 11λ λ + M 11µ µ) + h.o.t., x10 = M 12λ λ + M 12µ µ + h.o.t.
(29)
(29) shows there exists a surface Σ1 (µ, λ) given by (27) such that (26) has a solution s2 = 0 and −β ≤ x10 < λp for µ ∈ Σ1 (µ, λ), which means system (1) has two heteroclinic orbits Γ1 and Γ2 , where Γ1 is heteroclinic to p11 and p2 , Γ2 is heteroclinic to p2 and p01 .
(2) If s2 > 0 in (26), one attains M 11λ λ + M 11µ µ > 0. Eliminating s2 in (26), we achieve x10 = (w212 )−1 δ1−β2 (M 11λ λ + M 11µ µ)β2 + M 12λ λ + M 12µ µ + h.o.t.,
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(a) µ ∈ Σ11 (µ, λ)
(b) µ ∈ Σ21 (µ, λ)
(c) µ ∈ Σ31 (µ, λ)
(d) (λ, µ) ∈ ∆1
(e) (λ, µ) ∈ ∆2
(f) (λ, µ) ∈ ∆3
Fig. 8.
which shows that there exists a region ∆ given by (28) such that when (λ, µ) ∈ ∆, system (1) has one heteroclinic orbit Γ3 , where Γ3 is heteroclinic to p11 and p01 .
∆1 = {(λ, µ) : 0 < (w212 )−1 δ1−β2 × (M 11λ λ + M 11µ µ)β2 + M 12λ λ
Remark. The heteroclinic orbit Γ2 and Γ3 will go into p01 in different ways according to different fields of x10 , see Fig. 8, where we denote
∆2 = {(λ, µ) : (w212 )−1 δ1−β2 (M 11λ λ + M 11µ µ)β2
Σ11 (µ, λ) = {µ(λ) : M 11λ λ + M 11µ µ + h.o.t. = 0, 0 < M 12λ λ + M 12µ µ + h.o.t. < λp , 0 < |µ|, λ 1}, 2 Σ1 (µ, λ) = {µ(λ) : M 11λ λ + M 11µ µ + h.o.t. = 0, M 12λ λ + M 12µ µ + h.o.t. = 0, 0 < |µ|, λ 1}, 3 Σ1 (µ, λ) = {µ(λ) : M 11λ λ + M 11µ µ + h.o.t. = 0, −β ≤ M 12λ λ + M 12µ µ + h.o.t. < 0, 0 < |µ|, λ 1}.
+ M 12µ µ + h.o.t. < λp , M 11λ λ + M 11µ µ > 0, 0 < |µ|, λ 1}, + M 12λ λ + M 12µ µ + h.o.t. = 0, M 11λ λ + M 11µ µ > 0, 0 < |µ|, λ 1}, ∆3 = {(λ, µ) : −β ≤ (w212 )−1 δ1−β2 × (M 11λ λ + M 11µ µ)β2 + M 12λ λ + M 12µ µ + h.o.t. < 0, M 11λ λ + M 11µ µ > 0, 0 < |µ|, λ 1}.
References Chow, S. N. & Lin, X. B. [1990] “Bifurcation of a homoclinic orbit with a saddle-node equilibrium,” Diff. Integ. Eqs. 3, 435–466.
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Deng, B. [1990] “Homoclinic bifurcations with nonhyperbolic equilibria,” SIAM J. Math. Anal. 21, 693–720. Homburg, A. J. & Krauskopf, B. [2000] “Resonant homoclinic flip bifurcations,” J. Dyn. Diff. Eq. 12, 807– 850. Jin, Y. L. & Zhu, D. M. [2000] “Degenerated homoclinic bifurcations with higher dimensions,” Chin. Ann. Math. B 21, 201–210. Jin, Y. L. & Zhu, D. M. [2003] “Bifurcations of rough heteroclinic loop with two saddle points,” Sci. China, Series A 46, 459–468. Klaus, J. & Knobloch, J. [2003] “Bifurcation of homoclinic orbits to a saddle-center in reversible systems,” Int. J. Bifurcation and Chaos 9, 2603–2622. Liu, X. B. & Zhu, D. M. [2004a] “Bifurcation of generic homoclinic orbit accompanied with transcritical bifurcation,” Acta Math. Sin. (in Chinese) 49, 957–964. Liu, X. B. & Zhu, D. M. [2004b] “Bifurcation of nongeneric homoclinic orbit accompanied with transcritical bifurcation,” Appl. Math. J. Chinese Univ. Ser. A (in Chinese) 19, 401–408. Morales, C. A. & Pacifico, M. J. [2001] “Inclination-flip homoclinic orbits arising from orbit-flip,” Nonlinearity 14, 379–393. Rademacher, J. D. M. [2005] “Homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit,” J. Diff. Eqs. 218, 390–443.
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Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V. & Chua, L. O. [2001] Methods of Qualitative Theory in Nonlinear Dynamics (World Scientific, Singapore). Sun, J. H. & Luo, D. J. [1994] “Local and global bifurcations with nonhyperbolic equilibria,” Sci. China Series A 37, 523–534. Wiggins, S. [1990] Introduction to Applied Nonlinear Dynamical System and Chaos (Spinger-Verlag, NY). Zhang, T. S. & Zhu, D. M. [2004] “Codimension 3 homoclinic bifurcation of orbit flip with resonant eigenvalues corresponding to the tangent directions,” Int. J. Bifurcation and Chaos 14, 4161–4175. Zhu, D. M. [1994a] “The existence of nongeneric heteroclinic orbits accompanied with transcritical bifurcation,” Sci. China, Series A 24, 912–916. Zhu, D. M. [1994b] “The persistence of generic heteroclinic orbits connecting non-hyperbolic equilibrium points,” Chinese Sci. Bull. (in Chinese) 39, 1731– 1733. Zhu, D. M. [1998] “Problems in homoclinic bifurcation with higher dimensions,” Acta Math. Sinica, English Series 14, 341–352. Zhu, D. M. & Xia, Z. H. [1998] “Bifurcation of heteroclinic loops,” Sci. China, Series A 41, 837–848.
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International Journal of Bifurcation and Chaos, Vol. 18, No. 4 (2008) 1069–1083 c World Scientific Publishing Company
BIFURCATIONS OF GENERIC HETEROCLINIC LOOP ACCOMPANIED BY TRANSCRITICAL BIFURCATION* FENGJIE GENG† School of Information Engineering, China University of Geosciences (Beijing), Beijing 100083, P. R. China gengfengjie−[email protected] DAN LIU and DEMING ZHU Department of Mathematics, East China Normal University, Shanghai 200062, P. R. China Received January 22, 2007; Revised August 28, 2007 The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1 and one hyperbolic saddle p2 are investigated, where p1 is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1 (resp. p2 ) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1 splits into two hyperbolic saddles p01 and p11 , a heteroclinic loop connecting p11 and p2 , homoclinic loop with p11 (resp. p2 ) and heteroclinic orbit joining p01 and p11 (resp. p11 and p2 ; p2 and p01 ) are found. The results achieved here can be extended to higher dimensional systems. Keywords: Local coordinates; Poincar´e map; transcritical bifurcation; homoclinic loop; periodic orbit.
1. Introduction Homoclinic and heteroclinic orbits have tremendous potential for applications in many important areas. Therefore, bifurcations of homoclinic and heteroclinic orbits have been studied extensively in the literature, see [Homburg & Krauskopf, 2000; Jin & Zhu, 2000, 2003; Morales & Pacifico, 2001; Wiggins, 1990; Zhang & Zhu, 2004; Zhu, 1998; Zhu & Xia, 1998]. However, most of the papers considered bifurcation problems of orbits connecting hyperbolic equilibria, and the corresponding problems with
nonhyperbolic equilibria are rarely investigated. It is well known that nonhyperbolic equilibrium is unstable and always undergoes saddle-node (transcritical or pitchfork) bifurcation. Obviously, the bifurcation problems of orbits joining nonhyperbolic equilibria are much more difficult and challenging. To the best of our knowledge, the research on homoclinic and heteroclinic bifurcations with nonhyperbolic equilibria are relatively less. Chow and Lin [1990] investigated bifurcations of homoclinic orbit with a saddle-node equilibrium. Deng [1990]
∗
Supported by NNSF of China ( 10671069), Shanghai Leading Academic Discipline Project (B 407). Author for correspondence
†
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studied bifurcations of codimensional-1 homoclinic orbit with a nonhyperbolic equilibrium. Liu and Zhu [2004] discussed bifurcations of generic homoclinic orbit for high dimensional system with nonhyperbolic equilibrium, they obtained the persistence of homoclinic orbit and the existence of periodic orbit bifurcated from the homoclinic orbit accompanied by transcritical bifurcation. Klaus and Knobloch [2003] considered bifurcations of homoclinic orbits to saddle-enter, they obtained the existence of orbit homoclinic to equilibrium and orbit homoclinic to center manifold. Sun and Luo [1994] investigated the local and global bifurcations for a generic (d + 1)-parameter family of threedimensional system with a heteroclinic cycle connecting a hyperbolic saddle and a nonhyperbolic equilibrium. Zhu [1994] achieved the persistence of generic heteroclinic orbits joining a saddle-node and a saddle. For other research on the bifurcations of orbits with nonhyperbolic equilibria, the readers are referred to [Liu & Zhu, 2004a, 2004b; Rademacher, 2005; Shilnikov et al., 2001; Zhu, 1994a, 1994b]. In view of less work on heteroclinic loop with nonhyperbolic equilibria, we investigate bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1 and one hyperbolic saddle p2 for the four-dimensional system. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1 (resp. p2 ) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1 splits into two hyperbolic saddles p01 and p11 , a heteroclinic loop connecting p11 and p2 , homoclinic loop with p11 (resp. p2 ) and heteroclinic orbit joining p01 and p11 (resp. p11 and p2 ; p2 and p01 ) are found. It is worthy of note that homoclinic loop connecting p01 and heteroclinic loop joining p01 and p2 cannot be bifurcated from the original heteroclinic loop, which is exactly determined by generic condition (H1 ). The results achieved here can be extended to higher dimensional systems. We introduce the method originally established in [Zhu, 1998; Zhu & Xia, 1998] and then improved in [Jin & Zhu, 2000, 2003; Zhang & Zhu, 2004], that is, choosing fundamental solutions of variational equations as a new local active system, taking a coordinate change and then constructing a Poincar´e map to induce bifurcation equations, we
attain the results by means of solutions for bifurcation equations. The method is more applicable and bifurcation equations obtained in this paper are easy to compute. Consider the following C r (r ≥ 5) system z˙ = g(z, λ, µ),
(1)
and its unperturbed system z˙ = f (z),
(2)
R4 ,
the vector field g depends on paramwhere z ∈ eters (λ, µ), λ ∈ R, µ ∈ Rl , l ≥ 2, 0 ≤ λ, |µ| 1, g(z, 0, 0) = f (z), g(p1 , 0, µ) = 0, g(p2 , λ, µ) = 0. Furthermore, suppose the parameters are generic in the sense that λ governs bifurcations of equilibria while µ controls bifurcations of orbits. Assume system (2) has a heteroclinic loop Γ connecting its two critical points p1 , p2 , where Γ = Γ1 ∪ Γ2 , Γi = {z = ri (t) : t ∈ R}, ri (+∞) = ri+1 (−∞) = pi+1 , i = 1, 2, r3 (t) = r1 (t), p3 = p1 . Moreover, the linearization Df (p1 ) has real eigenvalues 0, λ11 , −ρ11 and −ρ21 satisfying −ρ21 < −ρ11 < 0 < λ11 , Df (p2 ) has simple real eigenvalues λ12 , λ22 , −ρ12 and −ρ22 fulfilling −ρ22 < −ρ12 < 0 < λ12 < λ22 . It is easy to see dim(W 1u ) = 1, dim(W 1s ) = 2, dim(W 1c ) = 1 and dim(W 2u ) = dim(W 2s ) = 2, where W iu and W is denote unstable manifold and stable manifold of pi , respectively, W 1c is the center manifold of p1 . In addition, the following conditions hold: H1 : generic condition dim(Tr1 (t) W 1c ∩ Tr1 (t) W 2s ) = dim(Tr1 (t) W 1cu ∩ Tr1 (t) W 2s ) = 1, where W 1cu is the center unstable manifold of p1 . H2 : generic condition lim (Tr1 (t) W 1cu + Tr1 (t) W 2s ) = Tp2 W 2uu + Tp2 W 2s ,
t→+∞
lim (Tr2 (t) W 2u + Tr2 (t) W 1s ) = Tp1 W 1u + Tp1 W 1s ,
t→+∞
lim (Tr1 (t) W 1cu + Tr1 (t) W 2s ) = Tp1 W 1cu + Tp1 W 1ss ,
t→−∞
lim (Tr2 (t) W 2u + Tr2 (t) W 1s ) = Tp2 W 2u + Tp2 W 2ss ,
t→−∞
where W iuu and W iss are strong unstable manifold and strong stable manifold of pi , i = 1, 2, respectively. See Fig. 1 (we draw the manifolds W 1cu and W 2u only). H2 is called the strong inclination property. It is worthy of note that, for any integer m ≥ 1, n ≥ 1, if we assume dim(W 1u ) = dim(W 2uu ) = m, dim(W 1ss ) = dim(W 2ss ) = n, then all the results achieved in this paper are still valid.
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W pc 1 W pu2 W ps2 W puu 2 W pss2
= {(x, y, u, v)∗ = {(x, y, u, v)∗ = {(x, y, u, v)∗ = {(x, y, u, v)∗ = {(x, y, u, v)∗
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∈ U1 |y = u = v = 0}, ∈ U2 |y = v = 0}, ∈ U2 |x = u = 0}, ∈ U2 |x = y = v = 0}, ∈ U2 |x = y = u = 0}.
Based on the invariance of these manifolds, intro−1 (x , 0, 0)x, duce a scale transformation x → θxx p1 −1 λ → −θxλ (xp1 , 0, 0)λ, system (1) has the following expression in U1 :
Fig. 1.
Let λ ∈ R be a parameter to control the transcritical bifurcation of system (1), and the x-axis be the tangent space of the center manifold at p1 , θ(x, λ, µ) be the vector field defined on the center manifold, then by [Wiggins, 1990], we may assume H3 : θ(xp1 , λ, µ) = 0, (∂θ/∂x)(xp1 , 0, 0) = 0, (∂ 2 θ/∂x2 )(xp1 , 0, 0) > 0, (∂ 2 θ/∂x∂λ)(xp1 , 0, 0) < 0, (∂ 2 θ/∂x∂µ)(xp1 , 0, µ) = 0, where xp1 is the x component of p1 . If H3 is true, then system (1) exhibits the transcritical bifurcation, i.e. when λ > 0 (or λ < 0, in this paper, we only consider the case λ > 0, for the case λ < 0, one may discussed similarly), there are two hyperbolic saddles p01 , p11 bifurcated from p1 . Denote by p01 = p1 = (0, 0, 0, 0)∗ , p11 = p1 + (λp , 0, 0, 0)∗ , where λp = θ0 λ + O(λ2 ) + O(λµ), θ0 = −(∂ 2 θ/∂x∂λ)(xp1 , 0, 0)/(∂ 2 θ/∂x2 )(xp1 , 0, 0). Moreover, dim(W ps0 ) = 3, dim(W pu0 ) = 1 and 1 1 dim(W pu1 ) = dim(W ps1 ) = 2. 1
1
Remark. Conditions H1 –H3 imply that local bifurcation is codimensional 1 and global bifurcation is codimensional 2.
2. Local Coordinates and Successor Functions Suppose the neighborhood Ui of pi is small enough, there exist successively two C r , C r−1 transformations such that W 1c , W iu , W is , W 2uu , W iss , i = 1, 2, are straightened in the neighborhood Ui . The local invariant manifolds take the forms as W pu1 W ps1 W pcu1 W pss1
∗
= {(x, y, u, v) = {(x, y, u, v)∗ = {(x, y, u, v)∗ = {(x, y, u, v)∗
∈ U1 |x = y = v = 0}, ∈ U1 |x = u = 0}, ∈ U1 |y = v = 0}, ∈ U1 |x = y = u = 0},
x˙ = −λp x + x2 + O(u)[O(y) + O(v)] + O(x)[O(y) + O(u) + O(v)] + O(x)O(x2 ), y˙ = [−ρ11 (α) + · · ·]y + O(v)[O(x) + O(u)], u˙ = [λ11 (α) + · · ·]u + O(x)[O(y) + O(v)], v˙ = [−ρ21 (α) + · · ·]v + O(y)[O(x) + O(y) + O(u)], (3) and in U2 it takes the form: x˙ = [λ12 (α) + · · ·]x + O(u)[O(y) + O(v)], y˙ = [−ρ12 (α) + · · ·]y + O(v)[O(x) + O(u)], u˙ = [λ22 (α) + · · ·]u + O(x)[O(x) + O(y) + O(v)], v˙ = [−ρ22 (α) + · · ·]v + O(y)[O(x) + O(y) + O(u)], (4) where α = (λ, µ), λp = λ + O(λ2 ) + O(λµ), λ11 (0) = λ11 , ρji (0) = ρji , j = 1, 2, i = 1, 2, λj2 (0) = λj2 , j = 1, 2. The resulting systems (3) and (4) are C r−2 . Due to normal form (3), (4) and H1 , one may choose −Ti and Ti such that r1 (−T1 ) = (δ, 0, 0, 0)∗ , r1 (−T2 ) = (δ, 0, 0, δ2u , 0)∗ , ri (Ti ) = (0, δ, 0, δiv )∗ , i = 1, 2, where δ > 0 is small enough such that {(x, y, u, v) : |x|, |y|, |u|, |v| < 2δ} ⊂ Ui . Obviously, |δ2u | = o(δ), |δiv | = o(δ), i = 1, 2. Take into account the linear variational system z˙ = Df (ri (t))z and its adjoint system φ˙ = −(Df (ri (t)))∗ φ,
(5)i (6)i
i = 1, 2, where (Df (ri (t)))∗ is the transpose of Df (ri (t)). Let Zi (t) = (zi1 (t), zi2 (t), zi3 (t), zi4 (t)) be a fundamental solution matrix of (5)i , then similarly to [Jin & Zhu, 2000], we achieve the following lemma. Lemma 2.1.
If conditions H1–H3 are satisfied,
then (1) there exists a fundamental solution matrix of (5)1 satisfying z11 (t) ∈ (Tr1 (t) W 1cu )c ∩ (Tr1 (t) W 2s )c ,
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z12 (t) =
r˙1 (t) ∈ Tr1 (t) W 1c ∩ Tr1 (t) W 2s , |r˙1 (−T1 )|
z13 (t) ∈ Tr1 (t) W 1cu ∩ (Tr1 (t) W 2s )c , z14 (t) ∈ (Tr1 (t) W 1cu )c ∩ Tr1 (t) W 2s , such that
11 w1 12 w1 Z1 (−T1 ) = 13 w1 0
1
0 Z1 (T1 ) = 0 w114
w141
1
0
0
0
0
1
w142 , w143
0
0
w144
0
w131
w122
w132
0
w133
0 0 . 0
w124
w134
1
(2) (5)2 has a fundamental solution matrix fulfilling z21 (t) ∈ (Tr2 (t) W 2u )c ∩ (Tr2 (t) W 1s )c , z22 (t) =
r˙2 (t) ∈ Tr2 (t) W 2u ∩ Tr2 (t) W 1s , |r˙2 (−T2 )|
z23 (t) ∈ Tr2 (t) W 2u ∩ (Tr2 (t) W 1s )c , z24 (t) ∈ (Tr2 (t) W 2u )c ∩ Tr2 (t) W 1s , such that
w211
0
0
0
w223
1
w242 , w243
0
0
0
w244
1
0
w231
w222
w232
0
w233
0 , 0
w224
w234
1
12 w2 Z2 (−T2 ) = 13 w2
w241
1
0 Z2 (T2 ) = 0 w214
0
where wi12 = 0, wi22 < 0, wijj = 0, j = 3, 4, i = 1, 2. Now, let (zi1 (t), zi2 (t), zi3 (t), zi4 (t)) be a new local active coordinate system along Γi . Set Φi (t) = (φ1i (t), φ2i (t), φ3i (t), φ4i (t)) = (Zi−1 (t))∗ , then Φi (t) is the fundamental solution matrix of (6)i , i = 1, 2. Take a coordinate change z = ri (t) + Zi (t)Ni (t) hi (t), where Ni (t) = (n1i , 0, n3i , n4i )∗ , i = 1, 2. Define S 0i = {z = hi (−Ti ) : |x|, |y|, |u|, |v| < 2δ}, S 1i = {z = hi (Ti ) : |x|, |y|, |u|, |v| < 2δ}
Fig. 2.
as the cross-sections of Γi at t = −Ti and t = Ti , respectively, i = 1, 2. (See Fig. 2.) Notice that if q 0i ∈ S 0i , q 1i ∈ S 1i , then q 0i = (x0i , yi0 , u0i , vi0 )∗ = ri (−Ti ) + Z1 (−Ti )Ni (−Ti ),
0,3 0,4 ∗ Ni (−Ti ) = (n0,1 i , 0, ni , ni ) ,
q 1i = (x1i , yi1 , u1i , vi1 )∗ = ri (Ti ) + Zi (Ti )Ni (Ti ),
1,3 1,4 ∗ Ni (Ti ) = (n1,1 i , 0, ni , ni ) .
Based on the expressions of Zi (−Ti ) and Zi (Ti ), one derives the relationship between the old coordinates of q 0i (x0i , yi0 , u0i , vi0 )∗ , q 1i (x1i , yi1 , u1i , vi1 )∗ 0,3 0,4 ∗ and their new coordinates q,i0 (n0,1 i , 0, ni , ni ) , 1,3 1,4 ∗ q 1i (n1,1 i , 0, ni , ni ) , that is 12 −1 0 42 44 −1 0 n0,1 1 = (w1 ) [y1 − w1 (w1 ) v1 ], n0,3 = u01 − w113 (w112 )−1 y10 1 + [w113 w142 (w112 )−1 − w143 ](w144 )−1 v10 , (7)1 44 −1 0 n0,4 1 = (w1 ) v1 , x0 = δ + w11 n0,1 + w41 n0,4 ≈ δ, 1 1 1 1 1 1,1 n1 = x11 − w131 (w133 )−1 u11 , 1,3 33 −1 1 n1 = (w1 ) u1 , 1 v 14 1 n1,4 1 = v1 − δ1 − w1 x1 + (w114 w131 − w134 )(w133 )−1 u11 , 1 y1 = δ + w132 n1,3 1 ≈ δ.
0,1 12 −1 0 42 44 −1 0 n2 = (w2 ) [y2 − w2 (w2 ) v2 ], n0,3 = u02 − δ2u − w213 (w212 )−1 y20 2 + [w213 w242 (w212 )−1 − w243 ](w244 )−1 v20 , 44 −1 0 n0,4 2 = (w2 ) v2 , x0 = δ + w11 n0,1 + w41 n0,4 ≈ δ, 2 2 2 2 2
(8)1
(7)2
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1 31 33 −1 1 n1,1 2 = x0 − w2 (w2 ) u0 , n1,3 = (w233 )−1 u10 , 2 1 v 14 1 n1,4 2 = v0 − δ2 − w2 x0 + (w214 w231 − w234 )(w233 )−1 u10 , y 1 = δ + w32 n1,3 ≈ δ. 0
2
(8)2
2
Next, we shall take three steps to establish the Poincar´e map in new coordinate system. Step 1. Consider the map F i1 : S 0i → S 1i . Put z = hi (t) into (1), we have r˙i (t) + Z˙ i (t)Ni (t) + Zi (t)N˙ i (t)
+ gλ (ri (t), 0, 0)λ + gµ (ri (t), 0, 0)µ + h.o.t. = f (ri (t)) + Df (ri (t))Zi (t)Ni (t) + gλ (ri (t), 0, 0)λ + gµ (ri (t), 0, 0)µ + h.o.t. By way of the fact r˙i (t) = f (ri (t)) and Z˙ i (t) = Df (ri (t))Zi (t), it then follows that N˙ i (t) = Zi−1 (t)[gλ (ri (t), 0, 0)λ + gµ (ri (t), 0, 0)µ] + h.o.t. Integrating the above equation from −Ti to Ti , we arrive at Ti Zi−1 (t)gλ (ri (t), 0, 0)λdt Ni (Ti ) = Ni (−Ti ) + −Ti
Ti
+ −Ti
x10 ≈ h(s1 )x01 , u10
λ1 1 (α) ρ1 (α) 1
≈ s1
u01 ,
y10 ≈ s1 y01 , v10
u11 ≈ s2
u02 ,
ρ1 2 (α) λ1 2 (α)
y20 ≈ s2
v20 ≈ s2
s1 = 0,
M jiµ
Ti
−Ti Ti
Zi−1 (t)gµ (ri (t), 0, 0)µdt + h.o.t.
j = 1, 3, 4,
= −Ti
φj∗ i gλ (ri (t), 0, 0)dt, (10)i φj∗ i gµ (ri (t), 0, 0)dt,
j = 1, 3, 4.
Together with (7)i , (8)i and (10)i , (9)i define 0,3 0,4 the map F i1 : S 0i → S 1i , (n0,1 i , 0, ni , ni ) → 1,3 1,4 (n1,1 i , 0, ni , ni ).
v11 ,
if x10 ∈ [−β, λp ).
where =
(12)
Since p1 undergoes transcritical bifurcation based on the structure of orbits in U1 , one may see that the equation x10 ≈ h(s1 )x01 holds only when x10 ≥ λp . While for x10 ∈ [−β, λp )(0 < β 1), the map F 10 is well defined only if s1 = 0 (see Fig. 3). Therefore, the extension for the domain of F 10 is defined as
(9)i
y11 ,
ρ2 2 (α) λ1 (α) 2
Applying Φ∗i (t) = Z −1 i (t), it then produces
M jiλ
≈ s1
(11) v01
where the higher order terms are neglected, 1 ), i = 1, 2, are called Shilnikov coordi(si , u0i , vi−1 nates, and λp ρ1 (α) −1 0 0 1 ] , λp = 0, (13) h(s) = λp [x1 − (x1 − λp )s −1 1 −1 0 λp = 0. [1 − (ρ1 (α)) x1 ln s] ,
x01 = δ,
0,j j j n1,j i = ni + M iλ λ + M iµ µ + h.o.t,
ρ2 1 (α) ρ1 (α) 1
and F 02 : S 11 → S 12 :
λ2 2 (α) λ1 (α) 2
= g(ri (t), 0, 0) + gz (ri (t), 0, 0)Zi (t)Ni (t)
Step 2. To construct the map F i0 : S 1i−1 → S 0i (where S 10 = S 12 ). Let τi , i = 1, 2 be 1 , u1 , v 1 )∗ to the flying time from q 1i−1 (x1i−1 , yi−1 i−1 i−1 1 1 q 0i (x0i , yi0 , u0i , vi0 )∗ , set s1 = e−ρ1 (α)τ1 , s2 = e−λ2 (α)τ2 . Utilizing approximate solution of (3) and (4), we can easily obtain the expression of F 10 : S 10 → S 01 :
x11 ≈ s2 x02 ,
= g(ri (t) + Zi (t)Ni (t), λ, µ)
1073
Fig. 3.
(14)
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Step 3. Composing the maps F i0 and F 1i , then F1 = F 11 ◦ F 10 : S 10 → S 11 can be expressed as ρ2 1 (α) ρ1 1 (α)
12 −1 12 −1 42 44 −1 n1,1 1 = (w1 ) δs1 − (w1 ) w1 (w1 ) s1
+ M 11λ λ
+
M 11µ µ
44 −1 n1,4 1 = (w1 ) s1
12 −1 n1,1 2 = (w2 ) δs2
G42 = (w244 )−1 s2
v01 + M 41λ λ + M 41µ µ + h.o.t. (15)
ρ2 2 (α) λ1 (α) 2
− (w212 )−1 w242 (w244 )−1 s2
v11
G11 = (w112 )−1 δs1 − δs2 + M 11λ λ + M 11µ µ + h.o.t., λ2 2 (α) λ1 (α) 2
=
u02
=
w113 (w112 )−1 δs1 + − M 31µ µ + h.o.t.,
λ2 2 1
λ (w133 )−1 δs2 2
− M 31λ λ λ1 1
ρ1 + w213 (w112 )−1 δsβ2 2 + (w233 )−1 s1 1 − M 32λ λ − M 32µ µ + h.o.t.,
δ2u
u01
ρ2 1 ρ1 1
v11 = δ1v + (w144 )−1 s1 v01 + w114 δs2 + M 41λ λ + M 41µ µ + h.o.t., ρ2 2 λ1 2
v01 = δ2v + (w244 )−1 s2 v11 + w214 δh(s1 ) λ1 1 ρ1 1
+ (w214 w231 − w234 )(w233 )−1 s1 u01 + M 42λ λ + M 42µ µ + h.o.t. Putting the above solution into (G11 , G12 ) = 0, then we obtain the following bifurcation equations
G31 = u01 − w113 (w112 )−1 δs1 − (w133 )−1 δs2 + M 31λ λ + M 31µ µ + h.o.t.,
(w112 )−1 δs1 − δs2 + M 11λ λ + M 11µ µ + h.o.t. = 0, (w212 )−1 δsβ2 2 − x10 + M 12λ λ + M 12µ µ
v01 − v11 + δ1v + w114 δs2
λ1 1 +1 ρ1 1
+ M 41λ λ + M 41µ µ + h.o.t., λ1 1 (α) ρ1 (α) 1
− x10 + w231 (w233 )−1 s1
+ M 12λ λ + M 12µ µ + h.o.t.,
For convenience, denote β2 = ρ12 /λ12 . By implicit function theorem, we know that the equation (G3i , G4i ) = 0 has a unique solution
ρ2 2 (α) 1
Let Gi = Fi (q 1i−1 ) − q 1i , i = 1, 2. Owing to (8)i , (11), (12), (15), (16), we have the successor functions Gji as follows:
G12 = (w212 )−1 δs2
(17)
with s1 ≥ 0, s2 ≥ 0.
u01
v11 + M 42λ λ + M 42µ µ + h.o.t. (16)
ρ1 2 (α) λ1 (α) 2
(G11 , G31 , G41 , G12 , G32 , G42 ) = 0
3. Bifurcations Analysis
λ (α) + [w213 w242 (w212 )−1 − w243 ](w244 )−1 s2 2 v11 + M 32λ λ + M 32µ µ + h.o.t.,
G41 = (w144 )−1 s1
+ M 42λ λ
Clearly, to study the bifurcations near Γ, we should consider the solutions of
0 u 13 12 −1 n1,3 2 = u2 − δ2 − w2 (w2 ) δs2
ρ2 1 (α) ρ1 1 (α)
v11 − v01 + δ2v + w214 x10 λ1 1 (α) 1
ρ1 2 (α) λ1 (α) 2
44 −1 n1,4 2 = (w2 ) s2
u01 + M 32λ λ + M 32µ µ + h.o.t.,
ρ (α) + (w234 − w214 w231 )(w233 )−1 s1 1 u01 + M 42µ µ + h.o.t.
v01
+ M 12λ λ + M 12µ µ + h.o.t.,
ρ2 2 (α) λ1 (α) 2
−
ρ1 2 (α) 1
λ (α) w213 (w112 )−1 δs2 2
ρ2 2 (α) λ1 (α) 2
and F2 = F 21 ◦ F 20 : S 11 → S 12 (= S 10 ) as ρ1 2 (α) λ1 (α) 2
−
δ2u
− (w233 )−1 s1
+ M 31λ λ + M 31µ µ + h.o.t., ρ2 1 (α) ρ1 (α) 1
=
v01
0 13 12 −1 13 42 12 −1 n1,3 1 = u1 − w1 (w1 ) δs1 + [w1 w1 (w1 )
− w143 ](w144 )−1 s1
u02
λ1 1 (α) ρ1 (α) 1
+ h.o.t.,
ρ2 1 (α) ρ1 (α) 1
G32
+ δw113 w231 (w112 w233 )−1 s1 u01
λ2 2 1
λ × [(w133 )−1 δs2 2
λ1 1 ρ1 1
+ w231 (w233 )−1 s1
−M 31λ λ − M 31µ µ] + h.o.t. = 0. (18)
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Firstly, we assume the transcritical bifurcation does not happen, namely, λ = 0. Then based on (11) and (13), (18) becomes
Proof.
If s1 = s2 = 0 in (20), then δ−1 M 11µ µ + h.o.t. = 0, δ−1 M 12µ µ + h.o.t. = 0,
(w112 )−1 δs1 − δs2 + M 11µ µ + h.o.t. = 0, (w212 )−1 δsβ2 2 −
δ + M 12µ µ 1 − (ρ11 )−1 δ ln s1 λ1 1 ρ1 1
+ w231 (w233 )−1 s1 λ2 2 1
λ + (w133 )−1 δs2 2
−
(19)
s2 = δ−1 M 11µ µ + h.o.t., (w212 )−1 sβ2 2 + δ−1 M 12µ µ + h.o.t. = 0.
h.o.t. = 0. λ1 1 ρ1 1
Since λ11 /ρ11 > 0, we have lims1 →0 s1 (1 − (ρ11 )−1 δ ln s1 ) = 0, it then follows that (w112 )−1 s1 − s2 + δ−1 M 11µ µ + h.o.t. = 0, (w212 )−1 sβ2 2 −
1− + h.o.t. = 0.
which immediately produces result (1). If we assume s1 = 0, s2 > 0 in (20), then
[w113 (w112 )−1 δs1 M 31µ µ] +
1 (ρ11 )−1 δ ln s1
+ δ−1 M 12µ µ
(20)
Theorem 3.1. Let conditions H1 –H3 be true and M 1iµ = 0, i = 1, 2. Then for λ = 0 and 0 < |µ| 1, we have the following propositions.
(1) If rank(M 11µ , M 12µ ) = 2, then there exists a codimension 2 surface L12 = {µ : M 11µ µ + h.o.t. = M 12µ µ + h.o.t. = 0} such that the heteroclinic loop Γ persists if and only if µ ∈ L12 , where the surface L12 has a normal plane span{M 11µ , M 12µ } at µ = 0. (2) There exists an (l − 1)-dimensional surface L21 = {µ : δ−1 w212 M 12µ µ + (δ−1 M 11µ µ)β2 + h.o.t. = 0, M 11µ µ > 0}
1075
(21)
such that system (1) has a unique homoclinic loop Γ21 connecting p1 near Γ if and only if µ ∈ L21 , where L21 has a normal vector M 12µ (resp. M 11µ or M 11µ + w212 M 12µ ) as β2 > 1 (resp. β2 < 1 or β2 = 1) at µ = 0. (3) There exists an (l − 1)-dimensional surface δ 1 L2 = µ : 1 −1 1 − (ρ1 ) δ ln(−δ−1 w112 M 11µ µ)
− M 12µ µ + h.o.t. = 0, w112 M 11µ µ < 0 (22) such that system (1) has a unique homoclinic loop Γ12 joining p2 in the neighborhood of Γ if and only if µ ∈ L12 .
It follows that there exists an (l − 1)dimensional surface L21 given by (21) such that (20) has a unique solution s1 = 0, s2 = s2 (µ) > 0 as µ ∈ L21 and 0 < |µ| 1. This implies system (1) has a homoclinic loop Γ21 connecting p1 . There is no difficulty to see that L21 has a normal vector M 12µ at µ = 0 as β2 > 1, while for β2 < 1 (resp. β2 = 1) it has a normal vector M 11µ (resp. M 11µ + w212 M 12µ ) at µ = 0. The existence of L12 can be obtained similarly. This completes the proof. Remark 3.1. Set θ = −δ −1 w112 M 11µ µ, then for µ ∈
L12 , we have θ > 0 and M 12µ µ =
ρ11
ρ1 δρ11 + h.o.t. = − 1 + h.o.t., ln θ − δ ln θ
that is, M 12µ µ ln θ = −ρ11 + o(1), which means |M 11µ µ| = o([M 12µ µ]α ) for α > 1 as µ ∈ L12 . Theorem 3.2. Assume hypotheses H1 –H3 hold and M 1iµ = 0, i = 1, 2. Then for λ = 0, µ ∈ L21 and 0 < |µ| 1, system (1) has no periodic orbits except homoclinic loop Γ21 .
Due to Theorem 3.1, µ ∈ L21 and 0 < |µ| 1 mean that system (1) has a homoclinic loop Γ21 . Now restricting s1 ≥ 0, s2 = (w112 )−1 s1 + δ−1 M 11µ µ + h.o.t. > 0 and µ ∈ L21 , then (20) is reduced to Proof.
V1 (s1 ) [(w112 )−1 s1 + δ−1 M 11µ µ]β2 + δ−1 w212 M 12µ µ + h.o.t. =
w212 1 − (ρ11 )−1 δ ln s1
N1 (s1 ).
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Thus V1 (0) = N1 (0) and V 1 (s1 ) = β2 (w112 )−1 [(w112 )−1 s1 + δ−1 M 11µ µ]β2 −1 , N1 (s1 ) =
w212 (ρ11 )−1 δ . (1 − (ρ11 )−1 δ ln s1 )2 s1
If w112 w212 < 0, then V 1 (s1 )N1 (s1 ) < 0, it is clear that V1 (s1 ) = N1 (s1 ) has no sufficiently small positive solutions. If β2 > 1, then |V 1 (s1 )| 1 and |N1 (s1 )| 1 hold for 0 < s1 1, which shows that V1 (s1 ) = N1 (s1 ) has no sufficiently small positive solutions. Next, we only consider the case β2 < 1 and w112 w212 > 0. There are two cases to study, namely, (i) β2 < 1, w112 > 0, w212 > 0, (ii) β2 < 1, w112 < 0, w212 < 0. Case (i). In this case, it is easy to see that V 1 (s1 ) = β2 (w112 )−1 [(w112 )−1 s1 + δ−1 M 11µ µ]β2 −1 ≤ β2 (w112 )−β2 s1β2 −1 .
(23)
Fig. 4.
β2 < 1, w112 < 0, w212 < 0.
for µ ∈ L21 . Combining with the fact V1 (0) = N1 (0) and V 1 (0) N1 (0), we see that there does not exist a small positive solution for V1 (s1 ) = N1 (s1 ). (See Fig. 4.) The proof is then completed.
Due to lim sβ2 −1 s1 →0 1
= ∞,
lim N1 (s1 ) = ∞
s1 →0
and
(1 − (ρ11 )−1 δ ln s1 )2 sβ1 2 s1β2 −1 = 0, = s1 →0 N1 (s1 ) w212 (ρ11 )−1 δ
w212 (ρ11 )−1 δ (1 − (ρ11 )−1 δ ln s1 )3 s21
(1) If w212 > 0 and β2 > 1 (resp. β2 = 1; β2 < 1), then system (1) has a unique periodic orbit near Γ as µ is situated in the neighborhood of L21 and confined to the side pointed by w212 M 12µ (resp. w212 M 12µ + M 11µ ; M 11µ ); system (1) has no periodic orbits near Γ as µ is situated in the neighborhood of L21 and confined to the side pointed by −w212 M 12µ (resp. −[w212 M 12µ +M 11µ ]; −M 11µ ). (2) If w212 < 0 and β2 > 1 (resp. β2 = 1; β2 < 1), then system (1) has exactly one periodic orbit as µ situated in the neighborhood of L21 and confined to the side pointed by −w212 M 12µ (resp. −[w212 M 12µ + M 11µ ]; −M 11µ ); system (1) has no periodic orbits as µ is situated in the neighborhood of L21 and confined to the side pointed by w212 M 12µ (resp. w212 M 12µ + M 11µ ; M 11µ ).
× [2(ρ11 )−1 δ − (1 − (ρ11 )−1 δ ln s1 )] > 0,
Proof.
lim
it then follows from (23) that V 1 (s1 ) < N1 (s1 ) for 0 < s1 1, this means V1 (s1 ) = N1 (s1 ) has no sufficiently small positive solutions. Obviously, the conclusion holds for β2 = 1. Case (ii). Notice that for w112 < 0, w212 < 0, s1 should subject to 0 < s1 < δ−1 |w112 |M 11µ µ, then one achieves V 1 (s1 ) < 0, N1 (s1 ) < 0, V 1 (s1 ) = β2 (β2 − 1)(w112 )−2 × [(w112 )−1 s1 + δ−1 M 11µ µ]β2 −2 < 0, N1 (s1 ) =
Corollary 3.1. Suppose the conditions of Theorem 3.2 are fulfilled, then the following propositions are true.
and V1 (δ−1 |w112 |M 11µ µ) = δ−1 w212 M 12µ µ + h.o.t. = −[δ−1 M 11µ µ]β2
1−
w212 (ρ11 )−1 δ ln(δ−1 |w112 |M 11µ µ)
= N1 (δ
−1
|w112 |M 11µ µ)
Let
W (µ) = V1 (0) − N1 (0) = δ−1 w212 M 12µ µ + (δ−1 M 11µ µ)β2 + h.o.t. Then
∂W (µ) = δ−1 w212 M 12µ ∂µ L2 1
+ β2 (δ−1 M 11µ µ)β2 −1 δ−1 M 11µ + h.o.t.,
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(a) w112 > 0, w212 < 0
(b) w112 < 0, w212 > 0
(c) w112 > 0, w212 > 0
(d) w112 < 0, w212 < 0 Fig. 5.
β2 > 1.
clearly, the gradient direction (∂W (µ)/∂µ)|L21 at µ = 0 is determined by w212 M 12µ as β2 > 1, by w212 M 12µ + M 11µ as β2 = 1 and by M 11µ as β2 < 1. Combining the proof of Theorem 3.2 with relative positions of the curves V1 (s1 ) and N1 (s1 ) for 0 < s1 1 and w212 > 0 (resp. w212 < 0), Proposition 1 (resp. (2)) follows immediately. For relative positions of the curves V1 (s1 ) and N1 (s1 ), one may see Fig. 5. Theorem 3.3. Assume that H1–H3 hold and M 1iµ =
0, i = 1, 2. Let λ = 0, µ ∈ L12 and 0 < |µ| 1, then in addition to the homoclinic loop Γ12 , system (1)
(1) has no periodic orbits near Γ as β2 ≥ 1 or w112 w212 < 0; (2) has at least one periodic orbit near Γ as β2 < 1, w112 > 0 and w212 > 0; (3) has a unique periodic orbit near Γ as β2 < 1, w112 < 0 and w212 < 0. Proof. Based on Theorem 3.1, µ ∈ L12 and 0
0. Notice that V2 (0) = N2 (0) as µ ∈ L12 . Moreover, V 2 (s2 ) = β2 s2β2 −1 , N2 (s2 ) =
w212 (ρ11 )−1 δ ·
[1 −
1
s2 − δ−1 M 11µ µ . (ρ11 )−1 δ ln(w112 (s2 − δ−1 M 11µ µ))]2
With similar arguments to proof of Theorem 3.2, one knows that V2 (s2 ) = N2 (s2 ) has no sufficiently small positive solutions as β2 ≥ 1 or w112 < 0. Next, we investigate the case β2 < 1 and w112 w212 > 0. 1 For β2 < 1, define s2 = e−ρ2 τ2 , then functions V2 (s2 ) and N2 (s2 ) are changed to the following forms:
N2 (s2 ) =
w212
.
1 β
1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ)) 1 β
=
1077
w212 (ρ11 )−1 δs2 2 1 β
−1 1 β
β2 [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))]2 (s2 2 − δ−1 M 11µ µ)
.
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There are still two cases to consider: (i) β2 < 1, w112 > 0, w212 > 0, (ii) β2 < 1, w112 < 0, w212 < 0.
Case (i). V 2 (0) = 1 > 0 = N2 (0) implies that there exists an 0 < s˜2 1 such that V2 (s2 ) > N2 (s2 ) for 0 < s2 < s˜2 . Choosing s2 = δ−1 w212 M 12µ µ > 0, then V2 (s2 ) = 2s2 + h.o.t., N2 (s2 ) =
w212
1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))
V2 ((δ−1 M 11µ µ)β2 ) = (δ−1 M 11µ µ)β2 + δ−1 w212 M 12µ µ
In view of w212 1
1
β − (ρ11 )−1 δ ln(w112 (s2 2 w212
−
1
β (ρ11 )−1 δ ln(w112 s2 2 )
1− which shows that V2 (s2 ) < N2 (s2 ).
=
+ h.o.t. < 0 = N2 ((δ−1 M 11µ µ)β2 )
δ−1 M 11µ µ))
>
N2 (s2 )
Case (ii). Under these circumstances, s2 must fulfil 0 < s2 < (δ−1 M 11µ µ)β2 , it then follows that N2 (s2 ) > 0. Owing to V 2 (0) = 1 > 0 = N2 (0), there exists an 0 < s˜2 1 such that V2 (s2 ) > N2 (s2 ) as 0 < s2 < s˜2 < (δ−1 M 11µ µ)β2 . On the other hand, by Remark 3.1, we have
.
1 β
As a result, N2 (s2 ) = V2 (s2 ) has at least one solution sˆ2 satisfying 0 < s˜2 < sˆ2 < s2 1.
for µ ∈ L12 and 0 < |µ| 1. Based on the continuity of the functions, there exists an 0 < s∗2 1 such that V2 (s∗2 ) = N2 (s∗2 ) for 0 < s˜2 < s∗2 < (δ−1 M 11µ µ)β2 1. Next, we shall show s∗2 is unique. Note that
2s2 ,
2 β
2w212 (ρ11 )−2 δ2 1 β
s2 2
1 β
−2
β22 [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))]3 (s2 2 − δ−1 M 11µ µ)2 1 2 −2 β1 −2 1 β2 −1 M 1 µ) − 1 s β2 2 − 1 s (s − δ 1µ 2 2 2 12 1 −1 w2 (ρ1 ) δ β2 β2 + 1 1 β β β2 [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))]2 (s2 2 − δ−1 M 11µ µ)2 1 β
=
w212 (ρ11 )−1 δs2 2
−2
1 β
1 β
1 β
β22 [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))]3 (s2 2 − δ−1 M 11µ µ)2 1 β
{2(ρ11 )−1 δs2 2
1 β
+ [1 − (ρ11 )−1 δ ln(w112 (s2 2 − δ−1 M 11µ µ))](−β2 s2 2 − (1 − β2 )δ−1 M 11µ µ)}. Obviously, N2 (s2 ) > 0 for w212 < 0, 0 < s2 < (δ−1 M 11µ µ)β2 and µ ∈ L12 . Combining with the fact N2 (s2 ) > 0 and V 2 (s2 ) = 1, one immediately knows that s∗2 is unique. This completes the proof. Corollary 3.2. Let all conditions of Theorem 3.3 be satisfied, then the following results are true.
(1) For w112 < 0, w212 < 0 and β2 < 1, if µ is situated in the neighborhood of L12 and confined to the side pointed by −w112 w212 M 11µ , then system (1) has two periodic orbits near Γ; if µ is situated in the neighborhood of L12 and confined to the side pointed by w112 w212 M 11µ , then system (1) has one periodic orbit near Γ.
(2) For w112 > 0, w212 > 0 and β2 < 1, if µ is situated in the neighborhood of L12 and confined to the side pointed by −w112 w212 M 11µ , then system (1) has at least two periodic orbits near Γ; if µ is situated in the neighborhood of L12 and confined to the side pointed by w112 w212 M 11µ , then system (1) has at least one periodic orbit near Γ. (3) For w112 w212 > 0 and β2 ≥ 1, if µ is situated in the neighborhood of L12 and confined to the side pointed by −w112 w212 M 11µ , then system (1) has no periodic orbits near Γ; if µ is situated in the neighborhood of L12 and confined to the side pointed by w112 w212 M 11µ , then system (1) has one periodic orbit near Γ.
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1079
(4) For w112 w212 < 0, if µ is situated in the neighborhood of L12 and confined to the side pointed by −w112 w212 M 11µ , then system (1) has one periodic orbit near Γ; if µ is situated in the neighborhood of L12 and confined to the side pointed by w112 w212 M 11µ , then system (1) has no periodic orbits near Γ. Proof. Set
W (µ) = N2 (0) − V2 (0) =
w212 − δ−1 w212 M 12µ µ + h.o.t. 1 − (ρ11 )1 δ ln(−δ−1 w112 M 11µ µ)
If µ ∈ L12 , we have [1 − (ρ11 )1 δ ln(−δ−1 w112 M 11µ µ)]−1 = (M 12µ µ/δ) + h.o.t., which means −δw212 w112 M 11µ ∂W (µ) = 1 − δ−1 w212 M 12µ + h.o.t.|L12 ∂µ L1 ρ1 [1 − (ρ11 )1 δ ln(−δ−1 w112 M 11µ µ)]2 (−w112 M 11µ µ) 2
=
−w212 w112 M 11µ (M 12µ µ)2 δρ11 (−w112 M 11µ µ)
− δ−1 w212 M 12µ + h.o.t.|L12 .
By Remark 3.1, −w112 w212 M 11µ is approximately the gradient direction of W (µ). Combining the relative positions of the curves N2 (s2 ) and V2 (s2 ) in the interval s2 ∈ [0, sˆ2 ) for some 0 < sˆ2 1, we claim that the non-negative solution s2 (µ) of V2 (s2 ) = N2 (s2 ) satisfying s2 (µ) = 0 as µ ∈ L12 , and for w112 < 0, w212 < 0, β2 < 1, if µ moves along the direction −w112 w212 M 11µ from L12 , then s2 (µ) = 0 will become s2 (µ) > 0, if µ moves along the direction w112 w212 M 11µ from L12 , then s2 (µ) will disappear. On the other hand, the curves N2 (s2 ) and V2 (s2 ) are intersected transversally at the unique positive solution s∗2 (µ), then it will survive under any small perturbation of µ. The proof of (1) is then completed. The proof (2)–(4) can be completed with similar arguments. For the relative positions of curves N2 (s2 ) and V2 (s2 ) as w112 w212 > 0, β2 < 1, one may see Fig. 6.
Now, we turn to discussing the bifurcations of the heteroclinic loop when p1 undergoes transcritical bifurcation, namely, λ > 0. One knows that when λ > 0, after the creation of p01 and p11 , there always exists a straight segment orbit heteroclinic to p11 and p01 , its length is λp , we denote this heteroclinic orbit by Γ∗ . Moreover, based on Fig. 3, we may see that x10 = λp is a critical position. First, we investigate the case x10 ≥ λP . For this case, (18) becomes (w112 )−1 δs1 − δs2 + M 11λ λ + M 11µ µ + h.o.t. = 0, (w212 )−1 δsβ2 2
λp ρ1 1
− δλp [δ − (δ − λp )s1 ]−1 + M 12λ λ
+ M 12µ µ
+
(24) λ2 2 λ1 2
× [w113 (w112 )−1 δs1 + (w133 )−1 δs2 − M 31λ λ − M 31µ µ] + h.o.t. = 0.
(a) w112 > 0, w212 > 0
(b) w112 < 0, w212 < 0 Fig. 6.
λ1 1 1
ρ w231 (w233 )−1 s1 1
β2 < 1.
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1080
0 < |µ| 1, where L21λ has a normal vector M 12µ (resp. M 11µ or (w212 )−1 M 11µ + M 12µ ) at µ = 0 as β2 > 1 (resp. β2 < 1 or β2 = 1); (2) there exists an (l − 1)-dimensional surface
λp ρ1 1
Let r = s1 , we have δλp λp ρ1 1
δ − (δ − λp )s1
=
δλp δ − (δ − λp )r
L12λ = {µ(λ) : W 21 (λ, µ) = δλp + λp (δ − λp )
λp (δ − λp ) r + h.o.t. = λp + δ
λp 1
× [−δ−1 w112 (M 11λ λ + M 11µ µ)] ρ1
− δM 12λ λ − δM 12µ µ + h.o.t. = 0,
It then follows from (24) that ρ1 1
(w112 )−1 δr λp
− δs2 + M 11λ λ + M 11µ µ + h.o.t. = 0,
λp (δ − λp ) r + M 12λ λ δ + M 12µ µ + h.o.t. = 0.
(w212 )−1 δsβ2 2 − λp −
(25)
Theorem 3.4. Suppose the conditions H1 –H3 hold,
rank(M 11µ , M 12µ ) = 2 and 0 < λ 1, then there exists an (l − 2)-dimensional surface Lλ12 = {µ(λ) : M 11µ µ+M 11λ λ+h.o.t. = M 12µ µ+M 12λ λ−λ+h.o.t. = 0} such that system (1) has a unique heteroclinic loop near Γ if and only if µ ∈ Lλ12 and 0 < |µ| 1, where Lλ12 has a normal plane span{M 11µ , M 12µ } at µ = 0. Proof.
Assume r = s2 = 0 in (25), then M 11µ µ + M 11λ λ + h.o.t. = 0, M 12µ µ + M 12λ λ − λp + h.o.t. = 0.
If rank(M 11µ , M 12µ ) = 2, the above equation determines an (l − 2)-dimensional surface Lλ12 = {µ(λ) : M 11µ µ + M 11λ λ + h.o.t. = M 12µ µ + M 12λ λ − λp + h.o.t. = 0} such that (25) has a solution r = s2 = 0 as µ ∈ Lλ12 and 0 < |µ| 1, i.e. system (1) has a unique heteroclinic loop near Γ. Clearly, Lλ12 has a normal plane spanned by M 11µ and M 12µ at µ = 0. This completes the proof. Theorem 3.5. Let H1 –H3 be true and 0 < λ 1, M 1iµ = 0, i = 1, 2, then
(1) there exists an (l − 1)-dimensional surface L21λ = {µ(λ) : W 12 (λ, µ) = (w212 )−1 [δ−1 (M 11µ µ + M 11λ λ)]β2 + δ−1 (M 12µ µ + M 12λ λ) − δ−1 λp + h.o.t. = 0, M 11µ µ + M 11λ λ > 0} such that system (1) has one homoclinic loop connecting p11 if and only if µ ∈ L21λ and
w112 (M 11λ λ + M 11µ µ) < 0} such that system (1) has a homoclinic loop joining p2 if and only if µ ∈ L12λ and 0 < |µ| 1, where L12λ has a normal vector M 11µ at µ = 0. Theorem 3.6. Suppose hypotheses H1 –H3 hold, M 1iµ = 0, i = 1, 2, 0 < λ, |µ| 1 and w112 w212 < 0. Then except the homoclinic loop connecting p11 , system (1) has no periodic orbits as µ ∈ L21λ . Corollary 3.3. Let all conditions in Theorem 3.6 be
satisfied, for w112 w212 < 0 and β2 > 1 (resp. β2 < 1 or β2 = 1), if µ is situated in the neighborhood of L21λ and confined to the side pointed by M 12µ (resp. M 11µ or (w212 )−1 M 11µ + M 12µ ), then system (1) has a unique periodic orbit near Γ; if µ is situated in the neighborhood of L21λ and confined to the side pointed by −M 12µ (resp. −M 11µ or −[(w212 )−1 M 11µ + M 12µ ]), then system (1) has no periodic orbits near Γ. Theorem 3.7.
Assume conditions H1–H3 are
= 0, i = 1, 2, 0 < λ, |µ| 1 and satisfied, w112 w212 < 0. Then in addition to the homoclinic loop joining p2 , system has no periodic orbits as µ ∈ L12λ . M 1iµ
Corollary 3.4. Let all hypotheses of Theorem 3.7 be
fulfilled, then the following propositions are true. (1) For w212 > 0 and w112 < 0, if µ is situated in the neighborhood of L12λ and confined to the side pointed by M 11µ , then system (1) has a unique periodic orbit near Γ; if µ is situated in the neighborhood of L12λ and confined to the side pointed by −M 11µ , then system (1) has no periodic orbits near Γ. (2) For w212 < 0 and w112 > 0, if µ is situated in the neighborhood of L12λ and confined to the side pointed by M 11µ , then system (1) has no periodic orbits near Γ; if µ is situated in the neighborhood of L12λ and confined to the side pointed by −M 11µ , then system (1) has one periodic orbit near Γ.
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(a) λ = 0, µ ∈ L12
(b) λ = 0, µ ∈ L21
(c) λ = 0, µ ∈ L12
(d) λ > 0, µ ∈ Lλ 12
(e) λ > 0, µ ∈ L21λ
(f) λ > 0, µ ∈ L12λ
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Fig. 7.
To illustrate Theorems 3.1–3.7, we draw some of the figures, see Fig. 7. Remark 3.2. By virtue of Fig. 7, one may see that homoclinic loop connecting p01 and heteroclinic loop joining p01 , p2 cannot be bifurcated from Γ, which is exactly determined by the generic condition H1 .
In the following, −β ≤ x10 < λp (0 < β 1) will be considered. Due to Fig. 3 and (14), it follows from (18) that s2 = δ−1 (M 11λ λ + M 11µ µ) + h.o.t., x10 = (w212 )−1 δsβ2 2 + M 12λ λ + M 12µ µ + h.o.t.
(26)
Theorem 3.8. Assume the conditions H1–H3 are true, rank(M 11λ , M 11µ ) > 0 and rank(M 12λ , M 12µ ) > 0. Then
(1) there exists a surface Σ1 (µ, λ) = {µ(λ) : M 11λ λ + M 11µ µ + h.o.t. = 0, − β ≤ M 12λ λ + M 12µ µ + h.o.t. < λp , 0 < |µ|, λ 1}, (27) such that system (1) has two orbits heteroclinic to p11 , p2 , p01 as µ ∈ Σ1 (µ, λ), see Figs. 8(a)–8(c);
(2) there exists a region in the (λ, µ) space ∆ = {(λ, µ) : −β ≤ (w212 )−1 δ1−β2 (M 11λ λ + M 11µ µ)β2 + M 12λ λ + M 12µ µ + h.o.t. < λp , M 11λ λ + M 11µ µ > 0, 0 < |µ|, λ 1},
(28)
such that system (1) has a heteroclinic orbit connecting p11 and p01 for (λ, µ) ∈ ∆, see Figs. 8(d )–8(f ). Proof.
If s2 = 0 in (26), then 0 = δ−1 (M 11λ λ + M 11µ µ) + h.o.t., x10 = M 12λ λ + M 12µ µ + h.o.t.
(29)
(29) shows there exists a surface Σ1 (µ, λ) given by (27) such that (26) has a solution s2 = 0 and −β ≤ x10 < λp for µ ∈ Σ1 (µ, λ), which means system (1) has two heteroclinic orbits Γ1 and Γ2 , where Γ1 is heteroclinic to p11 and p2 , Γ2 is heteroclinic to p2 and p01 .
(2) If s2 > 0 in (26), one attains M 11λ λ + M 11µ µ > 0. Eliminating s2 in (26), we achieve x10 = (w212 )−1 δ1−β2 (M 11λ λ + M 11µ µ)β2 + M 12λ λ + M 12µ µ + h.o.t.,
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F. Geng et al.
(a) µ ∈ Σ11 (µ, λ)
(b) µ ∈ Σ21 (µ, λ)
(c) µ ∈ Σ31 (µ, λ)
(d) (λ, µ) ∈ ∆1
(e) (λ, µ) ∈ ∆2
(f) (λ, µ) ∈ ∆3
Fig. 8.
which shows that there exists a region ∆ given by (28) such that when (λ, µ) ∈ ∆, system (1) has one heteroclinic orbit Γ3 , where Γ3 is heteroclinic to p11 and p01 .
∆1 = {(λ, µ) : 0 < (w212 )−1 δ1−β2 × (M 11λ λ + M 11µ µ)β2 + M 12λ λ
Remark. The heteroclinic orbit Γ2 and Γ3 will go into p01 in different ways according to different fields of x10 , see Fig. 8, where we denote
∆2 = {(λ, µ) : (w212 )−1 δ1−β2 (M 11λ λ + M 11µ µ)β2
Σ11 (µ, λ) = {µ(λ) : M 11λ λ + M 11µ µ + h.o.t. = 0, 0 < M 12λ λ + M 12µ µ + h.o.t. < λp , 0 < |µ|, λ 1}, 2 Σ1 (µ, λ) = {µ(λ) : M 11λ λ + M 11µ µ + h.o.t. = 0, M 12λ λ + M 12µ µ + h.o.t. = 0, 0 < |µ|, λ 1}, 3 Σ1 (µ, λ) = {µ(λ) : M 11λ λ + M 11µ µ + h.o.t. = 0, −β ≤ M 12λ λ + M 12µ µ + h.o.t. < 0, 0 < |µ|, λ 1}.
+ M 12µ µ + h.o.t. < λp , M 11λ λ + M 11µ µ > 0, 0 < |µ|, λ 1}, + M 12λ λ + M 12µ µ + h.o.t. = 0, M 11λ λ + M 11µ µ > 0, 0 < |µ|, λ 1}, ∆3 = {(λ, µ) : −β ≤ (w212 )−1 δ1−β2 × (M 11λ λ + M 11µ µ)β2 + M 12λ λ + M 12µ µ + h.o.t. < 0, M 11λ λ + M 11µ µ > 0, 0 < |µ|, λ 1}.
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