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International Journal of Bifurcation and Chaos, Vol. 18, No. 3 (2008) 641–674 c World Scientiﬁc Publishing Company

HYPERBOLICITY AND INVARIANT MANIFOLDS FOR PLANAR NONAUTONOMOUS SYSTEMS ON FINITE TIME INTERVALS LUU HOANG DUC∗ and STEFAN SIEGMUND† Fachbereich Mathematik, J.W. Goethe Universit¨ at, Robert-Mayer-Str. 10, 60325 Frankfurt, Germany ∗[email protected] †[email protected] Received December 4, 2006; Revised July 31, 2007 The method of invariant manifolds was originally developed for hyperbolic rest points of autonomous equations. It was then extended from ﬁxed points to arbitrary solutions and from autonomous equations to nonautonomous dynamical systems by either the Lyapunov–Perron approach or Hadamard’s graph transformation. We go one step further and study meaningful notions of hyperbolicity and stable and unstable manifolds for equations which are deﬁned or known only for a ﬁnite time, together with matching notions of attraction and repulsion. As a consequence, hyperbolicity and invariant manifolds will describe the dynamics on the ﬁnite time interval. We prove an analog of the Theorem of Linearized Asymptotic Stability on ﬁnite time intervals, generalize the Okubo–Weiss criterion from ﬂuid dynamics and prove a theorem on the location of periodic orbits. Several examples are treated, including a double gyre ﬂow and symmetric vortex merger. Keywords: Hyperbolicity; invariant manifolds; nonautonomous dynamical systems on ﬁnite time intervals; Okubo–Weiss criterion.

The rest point 00 of (2) corresponds to the rest point π0 of (1). A Taylor expansion of (2) in 00 0 . The system yields x˙ = 0 1 x +

1. Introduction We start with an example of a stable manifold for the pendulum equation

y

y˙ = −sin x. (1) kπ The rest points are at 0 for k ∈ Z, for the phase portrait, see Fig. behavior in 1. To study the local the vicinity of π0 we transform π0 into the origin , i.e. we comwith the transformation xy → x+π y x˜ pute (1) in the new coordinates y˜ = x+π y , for notational convenience omit the ˜, and get x˙ = y,

x˙ = y, †

y˙ = −sin(x − π) = sin(x).

1 0

y

ξ˙ =

sin(x)−x

0 1 ξ 1 0

(3)

2 with 0ξ ∈ R is called the linearization of (2) at π 0 (or equivalently the linearization of (1) at +1 and −1 with corre0 ). Its eigenvalues are 1 sponding eigenvectors 1 and −11 . The system (or the rest point π0 ) is called hyperbolic, since

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Author for correspondence 641

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of (2) as well as with its exponential eAt = x(·) t −t et −e−t ∈ Cγ be a solu(1/2) eet +e y(·) −e−t et +e−t . Let s tion of (2). Applying π to the variation of x(t ) x(t) constants formula for (2) y(t) = eA(t−t0 ) y(t00 ) + t A(t−s) 0 − sin(π−x(s))−x(s) ds yields t0 e

3

2

1

0 −1

1 1 −(t−t0 ) [x(t0 ) − y(t0 )] = e π 2 −1

1 −t t s e [sin(π − x(s)) + e 2 t0 1 + x(s)] ds, −1 = (1/2)et−t0 [x(t0 ) + y(t0 )] 11 − similarly π u x(t) y(t) t (1/2)et t0 e−s [sin(π − x(s)) + x(s)] 11 ds and by letting t0 → ∞, we get

1 1 t ∞ −s u x(t) e [sin(π−x(s))+x(s)] ds = e π 1 2 y(t) t

−2

s

−3 −6

−4

Fig. 1.

−2

0

2

4

6

Phase portrait of pendulum equation.

no eigenvalues are on the imaginary axis. The subspaces E u = span 11 and E s = span −11 are invariant for the linearization (3) and are called the unstable and stable subspaces of (2) at 00 , respectively. They satisfy the dynamical characterization 0 u 2 and E = ξ(0) ∈ R : lim ξ(t) = t→−∞ 0 0 s 2 . E = ξ(0) ∈ R : lim ξ(t) = t→+∞ 0 In fact, solutions in E s decay exponentially with a negative rate γ which is larger than the negative eigenvalue −1, say γ = −(1/2), i.e. each solution ξ(·) of (3) with ξ(0) ∈ E s satisﬁes ξ(·) ∈ Cγ := ξ ∈ C 0 (R, R2 ) : ξγ −γt

= sup e t≥0

ξ(t) < ∞ .

The local stable manifold of the nonlinear system (2) at the rest point 00 is then deﬁned analogously as the set of starting values (close to the origin) of solutions which decay exponentially with rate γ, i.e. for some δ > 0 x(0) 0 x(·) s ∈ Bδ : ∈ Cγ W loc := y(0) 0 y(·) x(·) and is a solution to (2) . y(·) This characterization is the basis of the Lyapunov– Perron construction method for invariant mani- folds: Let π u = (1/2) 11 11 and π s = (1/2) −11 −11 u denote the projections to E u and E s . Then 0 π1 s and π commute with the linear part A = 1 0

x(t) y(t)

where we used the fact that et−t0 [x(t0 )+y(t0 )] → 0, x(t) s x(t) + π u x(t) since x(·) y(·) ∈ Cγ . Then y(t) = π y(t) y(t) satisﬁes 1 x(t) 1 = e−t [ξ1 − ξ2 ] 2 −1 y(t)

1 −t t s e [sin(π − x(s)) + e 2 0 1 + x(s)] ds −1

1 t ∞ −s e [sin(π − x(s)) + e 2 t 1 + x(s)] ds (4) 1 is a solution of (2) in Cγ , if and only if x(·) y(·) ∈ Bδ 00 such that i.e. W sloc is the set of x(0) y(0) x(·) ∈ Cγ satisﬁes the ﬁxed-point Eq. (4) with y(·) ξ1 = [x(0) − y(0)] −11 ∈ E s . Equaξ = ξ2 = π s x(0) y(0) tion (4) deﬁnes a contraction on Cγ with parameter ξ ∈ E s and unique ﬁxed point xyξξ ∈ Cγ . Then with x (0) ws (ξ) := y ξ(0) the stable manifold is the graph of ξ ws over E s W sloc := {ξ + ws (ξ) : ξ ∈ E s , ξ < δ}.

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Whereas the Lyapunov–Perron method results in the construction of W sloc as a graph, the Hadamard graph transformation method uses (5) as a starting point for the construction of W sloc utilizing its invariance in the following form (6) π u f (ξ + ws (ξ)) = Dws (ξ)π s f (ξ + ws (ξ)) y . Equation (6) is derived with f xy = sin(x) from the invariance relation π u xy (t, ξ + ws (ξ)) = ws (π s xy (t, ξ + ws (ξ))) by computing its time derivative at t = 0 where xy (t, ξ + ws (ξ)) denotes the solution of (2) at time t starting in ξ + ws (ξ) ∈ W sloc . ws is constructed as a ﬁxed point of (6) in the space of Lipschitz mappings w : E s → E u with w(0) = 0. Substituting a Taylor expansion of ws into the invariance equation (6) one can compute η ∈ Es an approximation for ws , e.g. if for ξ = −η we get s

ξ + w (ξ) =

η + 24η 3 −η + 24η 3

+ O(ξ4 ).

In the last decades, there have been many papers generalizing classical autonomous results on hyperbolicity and invariant manifolds to nonautonomous diﬀerential equations x˙ = f (t, x)

(7)

using exponential dichotomies and the Lyapunov– Perron method (see e.g. [Barreira & Valls, 2005; Henry, 1981; Sell & You, 2002] and the references therein). The construction is analogous, the meaning of the real parts of the eigenvalues as growth rates for the linearization is replaced by exponential dichotomy estimates, ws then depends on both t and x. However, a disadvantage of this approach is its insensitivity to changes of (7) on intervals of ﬁnite time. For example, if a linear system x˙ = A(t)x admits an exponential dichotomy on R and is then changed arbitrarily on an interval of the form I = [a, b], a < b, then it still admits an exponential dichotomy. For example, if the evolution operator Φ(t, s) of x˙ = A(t)x satisﬁes an exponential dichotomy of the special form Φ(t, s)x ≤ Ke−α(t−s) x for t ≥ s with K ≥ 1, α > 0, then the solutions are exponentially decaying for large t and ﬁxed s, however, the behavior of the solution, as well as a

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possible change of the system, on a ﬁnite time interval only results in a larger constant K. In other words, hyperbolicity and invariant manifolds based on exponential dichotomies and the Lyapunov– Perron method are asymptotic concepts (in t) and are in general not related to the behavior of (7) on ﬁnite intervals of time. At this point it is crucial to distinguish between two diﬀerent cases: (i) f is recurrent in the sense that for t in unbounded intervals f (t, ·) : Rn → Rn is related, or close in some sense, to f (t, ·) on a bounded interval t ∈ I = [a, b]. This is the case e.g. if f is periodic, quasi-periodic, almost-periodic or almost-automorphic in t (see e.g. [Berger & Siegmund, 2003]). (ii) f is only known on the interval I = [a, b] of ﬁnite time and little or nothing can be said about f (t, ·) : Rn → Rn for t in unbounded intervals, e.g. if f is a physically observed or numerically computed nonstationary velocity ﬁeld and the observation or computation starts at time a and ends at time b. In case (i) the behavior of (7) for t ∈ I = [a, b] is related to the asymptotic dynamics for t → ±∞. One common method (which plays no role in this paper) is to consider a corresponding skew product ﬂow over the Bebutov shift by encoding the recurrent properties of f into the topological properties of the hull (see e.g. [Berger & Siegmund, 2003; Sell, 1971]). However, in case (ii) diﬀerent approaches have been developed in many recent papers, often motivated by applications in ﬂuid mechanics (see e.g. [Haller, 2005; Haller & Poje, 1998; Haller & Yuan, 2000; Ide et al., 2002; Ju et al., 2002; Lekien et al., 2006; Malhotra & Wiggins, 1998; Mancho et al., 2003, 2004; Miller et al., 1997; Ottino, 1989; Sandstede et al., 2000; Shadden et al., 2005; Wiggins, 2005] and the references therein). In [Kloeden & Siegmund, 2005] the bifurcations and continuous transitions of attractors serve as prototypical nonautonomous bifurcations for t ∈ R. A main goal of this paper is to provide the tools for an extension of the approach in [Kloeden & Siegmund, 2005] to ﬁnite time intervals, see also Sec. 6.3. To this end, one of the central questions is whether attraction, hyperbolicity and invariant manifolds have any useful meaning on a ﬁnite time interval. In this paper, we extend the notions of ﬁnite time hyperbolicity and ellipticity in [Haller, 2001a] to systems that are not necessarily ﬂuid-mechanical and study their relation to

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a new notion of invariant manifolds on ﬁnite time intervals. Although some results are formulated for scalar equations and Eq. (7) in arbitrary dimension, we discuss hyperbolicity and invariant manifolds in this paper only for Eq. (7) in the plane, i.e. x ∈ R2 . The structure of the paper is as follows. In Sec. 2 we deﬁne attraction and repulsion on intervals of ﬁnite time by considering the rate of strain tensor of the linearized system. We prove an analog of the “Theorem of Linearized Asymptotic Stability” on ﬁnite time intervals and extend a result in [Ide et al., 2002] on the existence of attractors and repellors for scalar diﬀerential equations. Section 3 contains the main results on hyperbolicity on ﬁnite time intervals. Using the strain acceleration tensor of the linearized system we discuss a time-varying partition of the state space R2 into quasi-hyperbolic, hyperbolic, elliptic and degenerate points and show that this partition is preserved under time-varying orthogonal transformations. We generalize the Okubo–Weiss criterion. Stable and unstable manifolds are introduced in Sec. 4. We show that they always exist in the hyperbolic and quasi-hyperbolic case. We extend a result in [Haller, 2001a] on suﬃcient conditions ensuring nonexistence in the elliptic case. We prove that if the manifolds exist then they are “tangential” to the stable and unstable manifolds of the linearized system. In Sec. 5 we deal with Hamiltonian systems which correspond to the incompressible case in 2D ﬂuid mechanics. Our notion of hyperbolicity and ellipticity was introduced in this context ﬁrst in [Haller, 2001a]. Section 6 contains applications to three different problems. First we prove a theorem on the location of periodic orbits for two-dimensional autonomous diﬀerential equations based on the partition introduced in Sec. 3. Next, we compute stable manifolds on ﬁnite time intervals for a heteroclinic orbit of a double gyre ﬂow and ﬁnally we show that the merging of two-dimensional symmetric vortices can be understood as a transition of the time-varying partition from hyperbolic regions to elliptic regions.

Definition 1 (Finite-Time Attractor and Repellor).

Let I = [a, b], a < b, be an interval. A solution µ : I → Rn of Eq. (8) is called ﬁnite-time attractor or attractor on I if it is attractive on I, i.e. there exists a δ > 0 such that for t0 ∈ I, x0 ∈ Rn the function t → x(t, t0 , x0 ) − µ(t) is strictly decreasing for all t ∈ I with 0 < x(t, t0 , x0 ) − µ(t) < δ. Similarly µ is called ﬁnite-time repellor or repellor on I if it is repulsive on I, i.e. there exists a δ > 0 such that for t0 ∈ I, x0 ∈ Rn the function t → x(t, t0 , x0 ) − µ(t) is strictly increasing for all t ∈ I with 0 < x(t, t0 , x0 ) − µ(t) < δ. Remark 2. We often abbreviate ﬁnite-time by ft.

If (8) is deﬁned on I = R then common notions of attraction are pullback and forward attraction (see e.g. [Kloeden & Siegmund, 2005] and the references therein). A solution µ : R → Rn of (8) is called pullback attractor if for all x0 ∈ Rn lim x(t, t0 , x0 ) − µ(t) = 0 for all t ∈ R

t0 →−∞

and it is called forward attractor if lim x(t, t0 , x0 ) − µ(t) = 0

t→∞

We consider a system of nonautonomous diﬀerential equations (8)

for all t0 ∈ R.

The following example shows that pullback, forward and ﬁnite-time attraction are three independent concepts describing the “past”, “future” and “present” of the diﬀerential equation. Example 3. Consider the scalar nonautonomous

equations x˙ = −2tx

2. Attraction and Repulsion

x˙ = f (t, x)

where f : D → Rn is a continuous function with continuous derivatives Dx f , Dx2 f deﬁned on a set D ⊂ R×Rn. To not overburden notation we assume from now on that D = I × Rn for a nontrivial interval I ⊂ R and that each solution x(t, t0 , x0 ) starting at t0 ∈ I in x0 ∈ Rn exists for all t ∈ I. All statements can easily be adapted to the case D I ×Rn .

and

y˙ = 2ty 2

2

with solutions x(t, t0 , x0 ) = x0 et0 −t and y(t, t0 , 2 2 y0 ) = y0 et −t0 , see Fig. 2. For both equations µ(t) ≡ 0 is a solution. For x˙ = −2tx the zero solution µ is a forward attractor but not a

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(a) Fig. 2.

645

(b)

Solutions to (a) x˙ = −2tx, and (b) y˙ = 2ty.

pullback attractor, moreover, it is attractive on any interval I = [a, b] if 0 < a < b and repulsive if a < b < 0. For the equation y˙ = 2ty the zero solution µ is a pullback attractor but not a forward attractor and it is attractive on any interval I = [a, b] if a < b < 0 and repulsive if 0 < a < b. The system x˙ = 2|t|x combines the dynamics of x˙ = −2tx for t ≤ 0 and of x˙ = 2tx for t ≥ 0 and hence µ(t) ≡ 0 is neither a pullback nor forward attractor, see also [Kloeden & Siegmund, 2005]. The following example shows that an asymptotically stable solution, i.e. a solution which is stable and attractive in the sense of Lyapunov, does not need to be ft-attractive.

Fig. 3.

‚“ ‚“ ”‚ ”‚ ‚ x(t, 0, 1) ‚ ‚ x(t, 1, 0) ‚ ‚ y(t, 1, 0) ‚ is increasing and ‚ y(t, 0, 1) ‚ is

decreasing for small t.

To study attraction and repulsion in the vicinity of a ﬁxed solution µ : I → Rn of (8) we transform it into the zero solution of a new system by the transformation x → x − µ(t), i.e. we compute (8) in the new coordinates x ˜ = x − µ(t), for notational convenience omit the ˜, and get x˙ = f (t, x + µ(t)) − f (t, µ(t)).

(9)

Example 4. Consider the linear system

x˙ 2 −5 x = y y˙ 5 −4 with solution x(t, x0 , y0 ) = e−t [x0 (cos 4t + = (3/4) sin 4t) − (5/4)y0 sin 4t], y(t, x0 , y0 ) −t e [(5/4)x0 sin 4t + y0 (cos 4t − (3/4) sin 4t)] starting at time 0 in xy00 . The zero solution µ(t) ≡ 0 is asymptotically stable. To check

x(t, x0 , y0 ) whether y(t, x0 , y0 ) is increasing or decreasing, we compute 2 d x(t, x0 , y0 ) dt y(t, x0 , y0 ) |t=0 d = dt

x(t, x0 , y0 ) x(t, x0 , y0 ) , y(t, x0 , y0 ) y(t, x0 , y0 ) |t=0

= 4x20 − 8y02

and observe that the solution starting in 10 at time 0 is increasing on a small neighborhood of t = 0, whereas the solution starting in 01 is decreasing, see Fig. 3. Hence µ is neither a ft-attractor, nor a ft-repellor.

A Taylor expansion of (9) in x = 0 yields x˙ = Dx f (t, µ(t))x + g(t, x)

(10)

with g(t, x) := f (t, x + µ(t)) − f (t, µ(t)) − Dx f (t, µ(t))x. The system ξ˙ = Dx f (t, µ(t))ξ

(11)

is called the linearization of (8) along µ(t). Remark 5

(i) A solution µ of (8) is attractive on I if and only if the zero solution of (9) is attractive on I, since the transformation x → x − µ(t) respects ft-attraction. (ii) By (i) either all solutions or no solution of a linear equation x˙ = A(t)x are attractive on an interval I. We say that the system x˙ = A(t)x is attractive on I if the zero solution is attractive on I. Motivated by the rate of strain tensor, a notion from ﬂuid dynamics (see e.g. [Batchelor, 1967]), we

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compute for an arbitrary solution ξ : I → Rn of (11) 1 d ξ(t)2 2 dt 1 d

ξ(t), ξ(t) 2 dt 1 = ξ(t), [A(t, µ(t)) + A(t, µ(t))T ]ξ(t) 2

=

(12)

with A(t, x) = Dx f (t, x). Definition 6 (Rate of Strain Tensor). The symmet-

ric part S(t, x) :=

1 [A(t, x) + A(t, x)T ] 2

of A(t, x) = Dx f (t, x) is called the rate of strain tensor of Eq. (8). By formula (12) the rate of strain tensor describes growth and decay of solutions ξ(t) of the linearization (11). In fact the solutions of (11) are all strictly monotonically decreasing on I × Rn if S(t, µ(t)) is negative deﬁnite, i.e. if for all t ∈ I

ξ, S(t, µ(t))ξ < 0

for all ξ ∈ Rn ,

ξ = 0.

Using the fact that a matrix A is positive or negative deﬁnite if and only if its symmetric part S := (1/2)[A + AT ] is positive or negative deﬁnite, since

ξ, Aξ = (1/2) Aξ, ξ + (1/2) ξ, Aξ = ξ, Sξ, we deﬁne the attracting and repelling regions of Eq. (8) as follows. Definition 7 (Attracting and Repelling Region). For t ∈ I the set

A(t) := {x ∈ Rn : Dx f (t, x) is negative deﬁnite} is called the attracting region of Eq. (8) at t. Similarly the set R(t) := {x ∈ Rn : Dx f (t, x) is positive deﬁnite} is called the repelling region of Eq. (8) at t. Remark 8.

(i) By continuity of Dx f the sets A(t) and R(t) are open. (ii) The attracting and repelling region of (8) on the interval I is denoted by AI := {(t, x) : t ∈ I, x ∈ A(t)} and RI := {(t, x) : t ∈ I, x ∈ R(t)}. (iii) In the scalar case n = 1 the attracting and repelling regions are of the form A(t) = {x ∈ R : Dx f (t, x) < 0} and R(t) = {x ∈ R : Dx f (t, x) > 0}. For each t ∈ I they introduce

a partition of the state space R of x˙ = f (t, x) as the disjoint union R = A(t) ∪ R(t) ∪ {x ∈ R : Dx f (t, x) = 0}. Lemma 9 (Relation Between Attraction and Attracting Region A). Let I = [a, b], a < b, be an interval and µ : I → Rn a solution of Eq. (8). Then the following two statements are equivalent:

(i) µ(t) ∈ A(t) for t ∈ I. (ii) ξ˙ = Dx f (t, µ(t))ξ is attractive on I and every nonzero solution ξ has a positive rate of attraction, i.e. d 2 (13) min ξ(t) > 0. t∈I dt Proof

(i) ⇒ (ii): If µ(t) ∈ A(t) for all t ∈ I then Dx f (t, µ(t)) is negative deﬁnite by Deﬁnition 7 and d ξ(t)2 = 2 ξ(t), S(t, µ(t))ξ(t) dt = 2 ξ(t), Dx f (t, µ(t))ξ(t) < 0. Thus the linearization is ft-attractive. Moreover, by compactness of I and continuity (13) follows. (ii) ⇒ (i): By Deﬁnition 1 of ft-attraction (d/dt)ξ(t)2 ≤ 0. By (13) (d/dt)ξ(t)2 < 0 and hence µ(t) ∈ A(t) for t ∈ I. The classical theorem of linearized asymptotic stability states that a rest point of x˙ = f (x) is asymptotically stable if its linearization is asymptotically stable. The following result is an analog of this theorem for Eq. (8) on a nontrivial compact interval I. It states that a solution is attractive on I with positive rate of attraction if the linearization is attractive on I with positive rate of attraction. Theorem 10 (Linearized Finite-Time Attraction). Let I = [a, b], a < b, be an interval and µ : I → Rn a solution of Eq. (8). Then

µ(t) ∈ A(t)

for all t ∈ I ⇒ µ is attractive on I.

Moreover, there exists δ > 0 such that solutions which are δ-close to µ have a positive rate of attraction, i.e. for t0 ∈ I, x0 ∈ Rn and 0 < x0 − µ(t0 ) < δ d 2 (14) min x(t, t0 , x0 ) − µ(t) > 0 t∈[t0 ,b] dt where we take the min only for t ∈ [t0 , b] to ensure that the solution x(t, t0 , x0 ) of (8) is δ-close to µ(t).

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Assume that µ(t) ∈ A(t) for t ∈ I. Then Dx f (t, µ(t)) and hence S(t, µ(t)) are negative deﬁnite. As a consequence all eigenvalues of S(t, µ(t)) are negative. Let η(t) denote the largest eigenvalue, η(t) is continuous in t ∈ I by continuity of t → S(t, µ(t)). We deﬁne Proof.

ηmax := max η(t) < 0. t∈I

By Remark 5(i) µ is attractive on I if and only if the zero solution of (10) x˙ = Dx f (t, µ(t))x + g(t, x)

(15)

is attractive on I. Using the fact that g(t, 0) ≡ 0 and the mean value theorem, we get g(t, x) ≤ supt˜∈I,˜x≤δ Dx2 f (t˜, x ˜)x2 for t ∈ I, x ≤ δ. We choose δ > 0 such that for t ∈ I and x ≤ δ ηmax x g(t, x) ≤ − 2 and x = 0 is the only zero of g(t, x) for t ∈ I and x < δ. To prove that the zero solution of (15) is attractive on I, let ν : I → Rn be a solution of (15) with 0 < ν(t) ≤ δ for some t ∈ I. Then 1 d ν(t)2 = ν(t), Dx f (t, µ(t))ν(t) 2 dt + ν(t), g(t, ν(t)) = ν(t), S(t, µ(t))ν(t) + ν(t), g(t, ν(t)) ≤ η(t)ν(t)2 + g(t, ν(t)) ν(t) ≤ ηmax ν(t)2 −

ηmax ν(t)2 2

ηmax ν(t)2 < 0 = 2 which implies that µ is attractive on I for the nonlinear system (8) with positive rate of attraction (14). The converse of Theorem 10 does not hold. The zero solution µ(t) ≡ 0 of x˙ = −x3 is attractive with positive rate of attraction (14) on any interval I = [a, b], a < b, however, µ(t) ∈ A(t) for any t ∈ I, since every solution of the linearization ξ˙ = 0 is constant. The following example shows that the assumption of Theorem 10 cannot be weakened. A solution of (8) does not need to be ft-attractive even if its linearization is ft-attractive.

647

Example 11. Consider the damped pendulum

equation x˙ = y,

y˙ = −sin x − 2y.

(16)

Every nonzero solution of the linearization at the origin 0 1 ˙ξ = ξ (17) −1 −2 satisﬁes (1/2)(d/dt)ξ(t)2 = −2ξ2 (t)2 ≤ 0, i.e. (d/dt)ξ(t)2 = 0 if and only if ξ2 (t) = 0. Thus, ξ(t) is strictly decreasing for every nonzero solution ξ(t) and hence, by Remark 5(ii), the linearization (17) is attractive on any interval of the form I = [a, b], a < b, but not with positive rate of attraction (13). On the other hand, for δ ∈ Rthe solution δ satisﬁes ν(t) of (16) starting at time 0 in δ/100 1 d δ(49δ − sin δ) ν(t)2 = . 2 dt 100 Since α(0) = α (0) = 0 and α (0) = −(1/2500) < 0, we get (d/dt)ν(t)2t=0 > 0 for 0 < |δ| 1. Consequently the zero solution of the nonlinear system (16) is not attractive although its linearization (17) is attractive on I. α(δ) :=

The Wazewski principle is a topological tool to ensure the existence of solutions in given sets if only the dynamics across the boundary of that set is known [Conley, 1978; Hale, 1980; Mischaikow, 1999]. Using Wazewski type arguments and the ordering of solutions of scalar equations the following theorem provides conditions ensuring the existence of a ft-repellor, see Fig. 4. A similar statement yields a ft-attractor. Theorem 12 (Repellor for Scalar Equation). Consider Eq. (8) for n = 1 on an interval I = [a, b], a < b. Let w− , w+ : I → R be C 1 functions with graphs W− := {(t, w− (t)) : t ∈ I} and W+ := {(t, w+ (t)) : t ∈ I} and assume that the following conditions are satisﬁed:

(i) W− lies below W+ , i.e. w− (t) < w+ (t) for t ∈ I. (ii) Solutions leave the region between W− and W+ transversally, i.e. f (t, w− (t)) < w˙ − (t) and f (t, w+ (t)) > w˙ + (t).

(18)

Then there exists a solution µ : I → R of (8) with µ(t) ∈ (w− (t), w+ (t))

for all t ∈ I.

Moreover, if (w− (t), w+ (t)) ⊂ R(t) for t ∈ I, then µ is a repellor on I.

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nonempty closed interval and we can choose a solution µ : I → R with µ(t) ∈ (w− (t), w+ (t)) for t ∈ I. If additionally (w− (t), w+ (t)) ⊂ R(t) for t ∈ I then µ(t) ∈ R(t) and by an analog of Theorem 10 µ is a repellor on I and the proof of Theorem 12 is complete. Proof of Lemma 13. We divide the proof into three

steps.

Fig. 4. Solutions leave the region between W− and W+ transversally.

Let x(t, t0 , x0 ) denote the solution of (8) starting at time t0 in x0 . By continuity the set of starting values of solutions which stay between W− and W+ Proof.

Γ := {x0 ∈ [w− (a), w+ (a)] : x(t, a, x0 ) ∈ [w− (t), w+ (t)] for all t ∈ I} is either a nonempty closed interval or empty. Assume that it is empty. Deﬁne the exit time τ (x0 ) := sup{t ∈ I : x(s, a, x0 ) ∈ [w− (s), w+ (s)] for all s ∈ [a, t]} for x0 ∈ [w− (a), w+ (a)]. Then τ (w− (a)) τ (w+ (a)) = a and x(τ (x0 ), a, x0 ) = w− (τ (x0 )) or x(τ (x0 ), a, x0 ) = w+ (τ (x0 )).

=

(19)

To derive a contradiction, we need the following lemma. Lemma

13. τ (x0 )

is continuous in x0

∈

[w− (a), w+ (a)]. Assume that Lemma 13 is true, then by (19) the function e : [w− (a), w+ (a)] → W− ∪ W+ , e(x0 ) = (τ (x0 ), x(τ (x0 ), a, x0 )) is well-deﬁned and continuous. Consequently, e([w− (a), w+ (a)]) ⊂ W+ ∪ W− is connected. However, e(w− (a)) = (a, w− (a)) ∈ W− e(w+ (a)) = (a, w+ (a)) ∈ W+

and

and the sets W− , W+ are disconnected by assumption (i), which is a contradiction. Thus, Γ is a

Step 1. Solutions cross W− and W+ transversally, i.e. if t0 = τ (x0 ) < b for x0 ∈ [w− (a), w+ (a)] and x(t0 , a, x0 ) = w+ (t0 ) then there exists an ε0 > 0 such that x(t, a, x0 ) > w+ (t)

for t ∈ (t0 , t0 + ε0 )

and similarly if x(t0 , a, x0 ) = w− (t0 ). Because otherwise, there would exist a strictly decreasing sequence (tn )n∈N with tn → t0 and x(tn , a, x0 ) ≤ w+ (tn ), hence w+ (tn ) − w+ (t0 ) x(tn , a, x0 ) − x(t0 , a, x0 ) ≤ tn − t0 tn − t0 which by passing to the limit for n → ∞ leads to f (t0 , w+ (t0 )) = x(t ˙ 0 , a, x0 ) ≤ w˙ + (t0 ), contradicting (18). Step 2. τ (·) is upper semi-continuous at x0 , since for any ε ∈ (0, ε0 ) by Step 1 x(t0 + ε, a, x0 ) > w+ (t0 + ε) and there exists a neighborhood V of x(t0 +ε, a, x0 ) small enough such that v > w+ (t0 +ε) for v ∈ V and such that U := x(a, t0 + ε, V ) is a neighborhood of x0 in [w− (a), w+ (a)]. By construction x(t0 + ε, a, U ) = V and hence τ (u) < τ (x0 ) + ε for u ∈ U . Step 3. τ (·) is lower semi-continuous at x0 , since for any ε ∈ (0, t0 − a) there exists a neighborhood V of x(t0 − ε, a, x0 ) small enough such that v < w+ (t0 − ε) for v ∈ V and such that U := x(a, t0 − ε, V ) is a neighborhood of x0 in [w− (a), w+ (a)]. For u ∈ U Step 1 implies x(t, a, u) ∈ [w− (t), w+ (t)] for all t ∈ [a, t0 − ε] and hence τ (u) > τ (x0 ) − ε for u ∈ U . A direct application of Theorem 12 yields Theorem 3.1, part 2, in [Ide et al., 2002], about how to detect a distinguished ﬁnite time repeller for scalar equations (8) based on the location of instantaneous stagnation points (ISPs), i.e. points xsp (t) ∈ R with f (t, xsp (t)) = 0. In our situation, this theorem can be expressed as follows. Theorem 14 [Ide et al., 2002]. Consider Eq. (8) for

n = 1 on an interval I = [a, b], a < b, and assume

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that the following conditions are satisﬁed: (i) Existence of an ISP: For t ∈ I there exists an ISP, denoted by xsp (t), i.e. a function satisfying f (t, xsp (t)) = 0. (ii) Isolated ISP: For each t ∈ I xsp (t) is the unique ISP in the interval max [xmin sp , xsp ]

:= mint∈I xsp (t) and xmax := where xmin sp sp maxt∈I xsp (t). (iii) Repulsion of the box containing the ISP: For t∈I max [xmin sp , xsp ]

⊂ R(t).

Then there exists a solution µ : I → R of (8) with max µ(t) ∈ [xmin sp , xsp ] and µ is a repellor on I. max By (ii) f (t, xmin sp ), f (t, xsp ) = 0 for t ∈ min max I. By (iii) xsp , xsp ∈ R(t), i.e. Dx f (t, xmin sp ), max Dx f (t, xsp ) > 0 for t ∈ I. By continuity of f and Dx f there exists ε > 0 with

Proof.

|f (t, xmin sp )|,

|f (t, xmax sp )|, max Dx f (t, xsp ) > 2ε

Dx f (t, xmin sp ), for t ∈ I.

Since f and Dx f are uniformly continuous on compact sets there exists δ > 0 such that |f (t, x)|, Dx f (t, x) > ε min max max for t ∈ I, x ∈ [xmin sp − δ, xsp ] ∪ [xsp , xsp + δ], max i.e. [xmin sp − δ, xsp + δ] ⊂ R(t) and by (ii) the sign of max f (t, x), for t ∈ I ﬁxed and x ∈ [xmin sp − δ, xsp + δ], changes only at x = xsp (t). Since

f (t, x) = f (t, x) − f (t, xsp (t)) = Dx f (t, xsp (t))(x − xsp (t)) + O(|x − xsp (t)|2 ), the sign of f (t, x) is the sign x − xsp (t) for x close to xsp (t). As a consequence f (t, x) < 0

for x ∈ [xmin sp − δ, xsp (t)) and

f (t, x) > 0

for x ∈ (xsp (t), xmax sp − δ].

Then assumption (18) holds with w− (t) := xmin sp − δ + δ and by Theorem 12 and w+ (t) := xmax sp max Γδ := {x0 ∈ [xmin sp − δ, xsp + δ] : x(t, a, x0 ) max ∈ [xmin sp − δ, xsp + δ] for all t ∈ I}

649

is a nonempty closed interval. Using the fact that Γδ1 ⊂ Γδ2 for 0 < δ1 < δ2 < δ, the monotone intersection max Γδ = {x0 ∈ [xmin sp , xsp ] : x(t, a, x0 ) δ>0

max ∈ [xmin sp , xsp ] for all t ∈ I}

is nonempty and contains a solution µ : I → max [xmin sp , xsp ] which is a repellor on I. In the following example we give an application of Theorems 12 and 14. Example 15. For ε ≥ 0 consider the scalar

equation x˙ = [x − ν(t)] − [x − ν(t)]3

with ν(t) = ε sin t.

A direct computation yields the ISP xsp (t) = ν(t) in the repelling region xsp (t) ∈ R(t) = (ω− (t), ω+ (t)) for t ∈ R √ (t) := ν(t) − (1/ 3) and w+ (t) := ν(t) + with w − √ √ (1/ 3). As long as ε < 1/2 3 max [xmin sp , xsp ] = [−ε, ε] ⊂ R(t)

for t ∈ R

and the assumptions of Theorem 14 are satisﬁed on any interval I = [a, b], a < b. Hence there exists a deﬁned ft-repellor. However, Theorem 12 with w± √ as above yields a repellor on I for ε < 2/3 3, i.e. Theorem 12 is stronger than Theorem 14.

3. Hyperbolicity and Ellipticity In this section we extend the elliptic-parabolichyperbolic (EPH) partition in [Haller, 2001a] to systems which are not necessarily ﬂuid mechanical (cp. also Sec. 5 on Hamiltonian systems). We consider a planar nonautonomous diﬀerential equation x˙ = f (t, x)

(20)

where f : I × R2 → R2 is a continuous function with continuous derivatives Dx f , Dx2 f , and I ⊂ R is a nontrivial interval. As in Sec. 2 we study the behavior of solutions in the vicinity of a ﬁxed solution µ : I → R2 . By x → x − µ(t) the solution µ is mapped to the zero solution of (10) x˙ = Dx f (t, µ(t))x + g(t, x)

(21)

and we denote the linearization of (21) along µ by ξ˙ = A(t)ξ a(t) b(t) where A(t) = = Dx f (t, µ(t)) (22) c(t) d(t)

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with C 1 functions a, b, c, d : I → R. We recall that by (12) the rate of strain tensor S(t) := S(t, µ(t)) describes the growth and decay of solutions ξ(t). Next we deﬁne the set of zero strain by (d/dt)ξ(t) = 0 as in [Haller, 2001a].

Since det S(t) depends continuously on the coeﬃcients of (22), (i) and (iii) are open conditions whereas (ii) and (iv) do not persist under typical perturbations of (22).

Remark 19.

A solution ξ(t) of (22) can cross a line in Z(t) only perpendicular to that line. The sign of the second order derivative of ξ(t) characterizes whether ξ(t) crosses from a region with increasing norm to a region with decreasing norm or vice versa. With ˙ S(t) = S(t, µ(t)) and S(t) = (d/dt)S(t, µ(t)) we get

Definition 16 (Zero Strain Set). For t ∈ I the set

Z(t) := {ξ ∈ R2 : ξ, S(t)ξ = 0} is called the zero strain set of the linearization (22). The next example illustrates the geometric meaning of the zero strain set.

d 1 d2 ξ(t)2 = ξ(t), S(t)ξ(t) 2 2 dt dt ˙ ˙ = ξ(t), S(t)ξ(t) + ξ(t), S(t)ξ(t)

Example 17. Consider the system

x˙ 1 0 x = . y˙ 0 −4 y

(23)

˙ + S(t)ξ(t)

Its zero strain set Z(t) is independent of t∈ R and 1 − + consists of two lines ξ = 2 and ξ = −12 , see Fig. 5(a).

˙ = A(t)ξ(t), S(t)ξ(t) + ξ(t), S(t)ξ(t) + S(t)A(t)ξ(t) ˙ + S(t)A(t) = ξ(t), [S(t)

Proposition 18 (Characterization of Zero Strain Set). Consider system (22) with symmetric part S(t) = (1/2)[A(t) + A(t)T ]. Then the zero strain set Z(t) = Z(t, µ(t)) satisﬁes exactly one of the following four cases:

+ A(t)T S(t)]ξ(t).

The following notion was introduced in this context in [Haller, 2001a], it coincides with the Cotter– Rivlin rate of S in continuum mechanics.

(i) Z(t) is the origin if and only if det S(t) > 0. (ii) Z(t) consists of one line if and only if det S(t) = 0 and a(t)2 + d(t)2 = 0. (iii) Z(t) consists of two lines if and only if det S(t) < 0. (iv) Z(t) is the plane if and only if a(t) = b(t) + c(t) = d(t) = 0.

Definition 20 (Strain Acceleration Tensor). The

matrix ˙ x) + S(t, x)A(t, x) + A(t, x)TS(t, x) M (t, x) := S(t, is called the strain acceleration tensor of Eq. (22), ˙ x) := Dt S(t, x) + Dx S(t, x)f (t, x) is the where S(t, derivative of the symmetric S(t, x) part of A(t, x) = Dx f (t, x) along the solution of (20) starting at time t in x.

The number of lines in Z(t) is the number of linearly independent solutions of the quadratic form a(t)ξ12 + (b(t) + c(t))ξ1 ξ2 + d(t)ξ22 = 0.

Proof.

(a) Fig. 5.

(24)

(b) 1

Dynamics (a) across S and (b) across Z for system (23).

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Example 21. Consider again Example 17. A = 1 0 0 −4

is independent of t and x and we get 2 0 M= . 0 32 For the two lines ξ− = 12 and ξ + = −12 in the zero strain set Z we have ξ − , M ξ − > 0 and

ξ + , M ξ + > 0. By (24) solutions cross Z from regions of decreasing norm to regions of increasing norm, see Fig. 5(b). If S(t, µ(t)) is indeﬁnite then by Proposition 18 the zero strain set Z(t) consists of two lines ξ − and ξ + . By (24) the signs of ξ − , M (t, µ(t))ξ − and ξ + , M (t, µ(t))ξ + characterize the directions in which solutions of (22) cross ξ − and ξ + . The restriction MZ (t, µ(t)) of M (t, µ(t)) to Z(t) is positive/ negative deﬁnite if both signs are positive/negative and it is indeﬁnite if the signs are diﬀerent. The following deﬁnition extends the EPH partition in [Haller, 2001a]. Definition 22 (Dynamic Partition of R2 ). Consider

Eq. (20). For t ∈ I we deﬁne the following sets: (i) Attracting region:

A(t) := {x ∈ R2 : S(t, x) is negative deﬁnite} (ii) Repelling region: R(t) := {x ∈ R2 : S(t, x) is positive deﬁnite} (iii) Elliptic region: E(t) := {x ∈ R2 : S(t, x) is indeﬁnite and MZ (t, x) is indeﬁnite} (iv) Hyperbolic region: H(t) := {x ∈ R2 : S(t, x) is indeﬁnite and MZ (t, x) is positive deﬁnite} (v) Quasi-hyperbolic region: Q(t) := {x ∈ R2 : S(t, x) is indeﬁnite and MZ (t, x) is negative deﬁnite} (vi) Degenerate region: D(t) := R2 \[A(t) ∪ R(t) ∪ E(t) ∪ H(t) ∪ Q(t)] Remark 23

(i) If µ(t) ∈ H(t) for t ∈ I then the cone ﬁeld {(t, ξ) ∈ I × R2 : ξ, S(t, µ(t))ξ > 0} is forward invariant under the dynamics of the linearization (22), i.e. (d/dt)ξ(t1 ) > 0 implies

651

(d/dt)ξ(t2 ) > 0 for t1 , t2 ∈ I, t2 > t1 . Similarly, if µ(t) ∈ Q(t) for t ∈ I then {(t, ξ) ∈ I × R2 : ξ, S(t, µ(t))ξ < 0} is forward invariant, see also Fig. 6. (ii) A nonempty degenerate region D(t) may or may not be persistent under small pertur bations of the system, e.g. if x˙ = −b0 bd x, b, d ∈ R, with D = R2 , is perturbed in R2×2 then generically D = ∅ for the perturbed system. On the other hand, in Example 55 below, degenerate regions always exist as boundaries between other regions. (iii) If x ∈ D(t) then either both eigenvalues of S(t, x) are zero (S(t, x) = 0) or one eigenvalue of S(t, x) is zero (S(t, x) is semi-deﬁnite) or both eigenvalues of S(t, x) are nonzero of different sign (S(t, x) is indeﬁnite, and MZ (t, x) is either semi-deﬁnite or zero). Example

24. Consider

again

the

pendulum

equation x˙ = y,

y˙ = − sin x.

In order to get the dynamic partition we compute 0 1 A= and −cos x 0   1 (1 − cos x) 0  2  S=  1 (1 − cos x) 0 2 as well as M = S˙ + SA + AT S   y sin x −cos x(1 − cos x)   2 =  y sin x 1 − cos x 2 and hence ξ, Sξ = ξ1 ξ2 (1 − cos x), i.e. Z = span{ 10 , 01 } =: {ξ − , ξ + } if x = 2kπ and Z = R2 if x = 2kπ, i.e. the set { xy ∈ R2 : x = 2kπ, k ∈ Z} consists of degenerate points. Moreover, since + +

ξ − , M ξ − = − cos x(1 − xcos x) 2and ξ , M ξ = 1 − cos x ≥ 0, the set { y ∈ R : x ∈ (−(π/2) + 2kπ, (π/2) + 2kπ), k ∈ Z} consists of hyperbolic points, whereas { xy ∈ R2 : x ∈ ((π/2) + 2kπ, (3π/2) + 2kπ), k ∈ Z} consists of elliptic points, see Fig. 7. The transformation x → x − µ(t) transforms Eq. (20) into Eq. (21). The corresponding dynamical partitions of (20) and (21) are mapped onto

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Fig. 6.

Dynamics in the vicinity of µ(t) ∈ R2 .

3

2

1

0 −1 −2 −3 −6

−4

Fig. 7.

−2

0

2

4

Blue hyperbolic and red elliptic points for the pendulum equation.

6

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each other by this transformation. Let Q : I → R2×2 be a continuously diﬀerentiable function of orthogonal matrices Q(t). We transform solutions of (21) into rotated solutions of a new system by the transformation x → Q(t)T x, i.e. we compute (21) in the new coordinates x ˜ = Q(t)T x, for notational convenience omit the ˜, and get ˙ x˙ = [Q(t)TDx f (t, µ(t))Q(t) − Q(t)T Q(t)]x + h(t, x)

(25)

with h(t, x) = Q(t)T [f (t, Q(t)x+ µ(t))− f (t, µ(t))− Dx f (t, µ(t))Q(t)x]. The same transformation x → Q(t)T x transforms the linearization (22) of (20) into the linearization ξ˙ = B(t)ξ where ˙ (26) B(t) = Q(t)TDx f (t, µ(t))Q(t) − Q(t)T Q(t) of (25) along the zero solution. Since x → Q(t)T x is a norm-preserving rotation, the corresponding dynamical partitions of (21) and (25) are mapped onto each other by this transformation. In other words, the time-dependent shift and rotation x → Q(t)T [x − µ(t)] transforms system (20) and its dynamic partition into system (25) and its dynamic partition, respectively. In fact, any time-dependent shift and rotation x = Q(t)˜ x + u(t) preserves the dynamic partition as a direct computation shows. Proposition 25 (Dynamic Partition under Shift and Rotation). Let Q : I → R2×2 be a C 1 function of orthogonal matrices Q(t) and u : I → R2 be a C 1 function. Then the transformation x → Q(t)T [x − u(t)] transforms Eq. (20) to

x˙ = Q(t)T f (t, Q(t)x + u(t)) ˙ ˙ − Q(t)T u(t). − Q(t)T Q(t)x

(27)

Let x(t) be a solution of (22), then x ˜(t) = Q(t)T [x(t) − u(t)] is a solution of (27 ) with linearization ˙ ξ˙ = [Q(t)TDx f (t, x(t))Q(t) − Q(t)T Q(t)]ξ and x(t) is in one of the regions A(t), R(t), E(t), H(t), Q(t) or D(t) of Eq. (22) if and only if x ˜(t) is in the corresponding region of Eq. (27 ). Dresselhaus and Tabor [1992] suggested to study the linearization (22) in the eigenbasis of SA (t) = (1/2)[A(t) + A(t)T ] thus factoring out that part of time dependence of A(t) that comes from the rotation of the eigenvectors of the rate of strain tensor SA (t). We therefore transform Eq. (20) by

653

a smooth shift and rotation x → Q(t)T [x − µ(t)] into (25). The symmetric part SB (t) of B(t) is the rate of strain tensor of the linearization (26) of (25). T = −Q(t)T Q(t), ˙ ˙ Using the fact that [Q(t)T Q(t)] it satisﬁes 1 SB (t) = [B(t) + B(t)T ] = Q(t)T SA (t)Q(t). 2 The transformed rate of strain tensor SB (t) is diagonal if and only if the orthogonal matrix Q(t) consists of the eigenvectors of SA (t). In this case the linearization (26) of (25) is of the form (t) β(t) λ 1 ξ˙ = B(t)ξ where B(t) = (28) −β(t) −λ2 (t) with continuously diﬀerentiable functions λ1 , λ2 , β : I → R. Definition

26

(Strain Coordinates). Consider Eq. (20) and a solution µ : I → R2 . Suppose that Q : I → R2×2 is a continuously diﬀerentiable func- tion of orthogonal matrices Q(t) = v1 (t)|v2 (t) consisting of eigenvectors v1 (t), v2 (t) of S(t, µ(t)) for t ∈ I. Then x → Q(t)T [x − µ(t)] is called strain coordinate transformation. The transformed system (25) is said to be in strain coordinates. Remark 27 (Strain Coordinates for Indeﬁnite Linearization). If the linear part A(t) of the linearization (22) is indeﬁnite for t ∈ I then the eigenvalues of the symmetric part S(t) = (1/2)[A(t) + A(t)T ] are given by 1 λ1 (t) = [a(t) + d(t) + ∆(t)] and 2 (29) 1 −λ2 (t) = [a(t) + d(t) − ∆(t)], 2

with ∆(t) = [b(t) + c(t)]2 + [a(t) − d(t)]2 . They satisfy λ1 (t) > 0 > −λ2 (t) and there exists a strain coordinate transformation in a neighborhood of each t0 ∈ I. This follows from the implicit function theorem applied to F (v, λ, t) = ([S(t) − λI]v, v0T v − 1) where S(t0 )v0 = λ0 v0 . The invertibility of (∂F /∂(v, λ))(v0 , λ0 , t0 ) follows from Kellers ABCD Lemma [Keller, 1977] and the fact that the eigenvalues are distinct. To get an explicit strain coordinate transformation Q(t), consider the diﬀerentiable curve z : I → S 1 deﬁned by a(t) − d(t) b(t) + c(t) , , z(t) = ∆(t) ∆(t)

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and for an arbitrary t0 ∈ I deﬁne ω(t0 ) ∈ [0, 2π) with (cos ω(t0 ), sin ω(t0 )) = z(t0 ) and for each t ∈ I

ω(t) := ω(t0 ) +

t t0

˙ ) − a(τ ˙ ) + c(τ [d(τ ˙ )][b(τ ) + c(τ )] − [d(τ ) − a(τ )][b(τ ˙ )] dτ. ∆(τ )

Then the C 1 strain coordinate transformation Q(t) is of the form  1 1 cos ω(t) sin ω(t)   2 2 . Q(t) =    1 1 −sin ω(t) cos ω(t) 2 2 

1

Indeed, if we denote by n(t) = (∆(t))− 2 (−b(t) − c(t), a(t) − d(t)) the normalized normal vector of z(t) then the tangent vector u(t) of z(t) satisﬁes u(t) =

˙ − a(t)][b(t) ˙ + c(t)] [d(t) ˙ + c(t)] − [d(t) − a(t)][b(t) ˙ d z(t) = n(t). dt ∆(t)

Using the fact that the angle ρ(t) of z(t) ∈ S 1 satisﬁes (d/dt)ρ(t) = u(t), n(t) and ρ(t0 ) = ω(t0 ) + 2kπ for some k ∈ Z we get

ρ(t) = ρ(t0 ) +

t t0

u(τ ), n(τ )dτ = ω(t0 ) +

t

u(τ ), n(τ )dτ + 2kπ = ω(t) + 2kπ.

t0

As a consequence cos ω(t) = (a(t) − d(t))/ ∆(t) and sin ω(t) = −(b(t) + c(t))/ ∆(t), and we get (omitting the t-dependence) 

1 cos ω  2 QTSA Q =   1 sin ω 2

 1 a −sin ω  2   1  b + c cos ω 2 2

 1 b+c cos ω  2  2   1 −sin ω d 2



b+c a+d a−d + cos ω − sin ω  2 2 2  = b+c a−d sin ω + cos ω 2 2 √ 1  2 [a + d + ∆] =  0

a−d b+c sin ω + cos ω 2 2 a+d a−d b+c − cos ω + sin ω 2 2 2





0 √ 1 [a + d − ∆] 2

 1 sin ω 2   1  cos ω 2

 = 

λ1 0

   

0 , −λ2

i.e. the columns of Q(t) consist of eigenvectors of SA (t). A similar computation shows that the transformed linearization is of the form (28) with (29) and β(t) =

˙ − a(t)][b(t) ˙ + c(t)] b(t) − c(t) [d(t) ˙ + c(t)] − [d(t) − a(t)][b(t) ˙ + . 2∆(t) 2

(30)

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The following example shows that strain coordinates do not always exist. Example 28. The zero solution µ(t) ≡ 0 of x˙ =

A(t)x with 

 1 2 t 2    for t < 0 and  −1

 −1  A(t) =  1 2 t 2  1 −1 + t2  2 A(t) =   0

If S(t) is indeﬁnite then the zero strain set is of the form √ √ − λ2 λ2 √ √ Z(t) = , (32) λ1 λ1 and for ξ ∈ Z(t)

ξ, M (t)ξ = 2 ×

0

satisﬁes µ(t) ∈ A(t) for t ∈ [−1, 1], since the eigenvalues −1 ± (1/2)t2 of S(t) = A(t) are negative. However, the unique (up to exchanging columns) orthogonal matrix Q(t) consisting of the eigenvectors of A(t) satisﬁes 1  1 √ √  2 2  for t < 0 and Q(t) =   1 1  √ −√ 2 2 1 0 Q(t) = for t ≥ 0 0 1 

and therefore is not even continuous. As a consequence the system cannot be transformed into strain coordinates on [−1, 1]. For a system in strain coordinates we summarize explicit formulas for the rate of strain tensor, zero strain set and strain acceleration tensor in the following proposition. Proposition 29 (Linearization in Strain Coordinates). Consider the linearization (28) in strain coordinates. Then (omitting t-dependencies) 0 λ1 and S(t) = 0 −λ2 2λ1 β λ˙ 1 + 2λ21 M (t) = −λ˙ 2 + 2λ2 2λ2 β 2

and for ξ = (ξ1 , ξ2 ) ∈ R2

ξ, M (t)ξ = (λ˙ 1 + 2λ21 )ξ12 + 2(λ1 + λ2 )βξ1 ξ2 + (−λ˙ 2 + 2λ2 )ξ 2 . (31) 2

2

   for t ≥ 0 1 2 −1 + t 2

655

λ˙ 1 λ2 − λ1 λ˙ 2 ±β λ1 λ2 + √ 2 λ1 λ2 (λ1 + λ2 )

λ2 (λ1 + λ2 )ξ22 . λ1

(33)

For system (20) with trace Dx f (t, x) = 0 (e.g. incompressible ﬂuids) Okubo [1970] and Weiss [1991] suggested to study the sign of det Dx f (t, x) for each t to identify hyperbolic regions {x : det Dx f (t, x) < 0} and elliptic regions {x : det Dx f (t, x) > 0}. This so-called Okubo–Weiss criterion provides an easily testable condition to distinguish between hyperbolic and elliptic behavior provided the system is slowly varying, i.e. Dx f (t, x(t)), along a solution x(t), is “almost” independent of t [Haller, 2001a, Sec. V.A]. Hua and Klein computed a second order correction of the Okubo–Weiss criterion in [Hua & Klein, 1998]. Tabor and Klapper applied the Okubo–Weiss criterion in strain coordinates and thus factored out the dependence of the Okubo–Weiss criterion on the reference frame [Tabor & Klapper, 1994]. The Okubo– Weiss criterion in strain coordinates is equivalent to the EPH partition [Haller, 2001a, Sec. V.B]. The following theorem is a generalization of the Okubo–Weiss criterion in strain coordinates to general Eq. (20). Theorem 30 (Okubo–Weiss Criterion). Consider

(20) with linearization (22) and assume that Dx f (t, µ(t)) is indeﬁnite for t ∈ I. With (29) and (30) deﬁne α(t) :=

λ˙ 1 (t)λ2 (t) − λ1 (t)λ˙ 2 (t) . λ1 (t)λ2 (t)+ 2 λ1 (t)λ2 (t)(λ1 (t) + λ2 (t))

Then the following characterizations hold: (i) (ii) (iii) (iv)

µ(t) ∈ E(t) if and only if |α(t)| < |β(t)|. µ(t) ∈ H(t) if and only if α(t) > |β(t)|. µ(t) ∈ Q(t) if and only if −α(t) > |β(t)|. µ(t) ∈ D(t) if and only if |α(t)| = |β(t)|.

Follows from (33) in Proposition 29 and the fact that by Proposition 25 the strain coordinate transformation respects the dynamic partition.

Proof.

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Remark 31 (Autonomous Rest Point is not QuasiHyperbolic). Consider an autonomous equation x˙ = f (x) with rest point x0 ∈ R2 , i.e. f (x0 ) = 0. Then the linearization ξ˙ = Aξ with A = ac db = Df (x0 ) does not depend on t ∈ R. If A is indeﬁnite then det S < 0 and a strain coordinate transformation yields linearization ξ˙ = Bξ λ1theβ transformed with B = −β −λ2 and √ √ a+d+ ∆ a+d− ∆ λ1 = , λ2 = − 2 2

and β =

b−c , 2

2 2 ∆ √ = (b + c) + (a − d) . Theorem 30 implies α(t) = −det S > 0 and as a consequence an autonomous rest point x0 cannot be quasi-hyperbolic.

Remark 32 (Direction of Rotation in Elliptic Case). If µ(t) ∈ E(t) in Theorem 30 then rotation of solutions of (28) in strain coordinates is

clockwise, if |α| < β and counter-clockwise, if |α| < −β. This follows from the explicit formulas (32) and (33) in strain coordinates. In particular, |α| < β corresponds to ξ − , M (t)ξ − > 0, ξ + , M (t)ξ + < 0 and |α| < −β corresponds to ξ −√, M (t)ξ − < 0,

ξ + , M (t)ξ + > 0 with ξ − = √λλ21 and ξ + = −√λ2 √ , see Fig. 8. λ1 Next, we prove that zero strain set and dynamic partition of the linearization (22) at t0 describe the behavior of solutions of the nonlinear equation (20) locally in some Bδ (µ(t0 )) = {x : x − µ(t0 ) < δ}, δ > 0.

(a) Fig. 8.

Theorem 33 (Nonlinear Zero Strain Set and Strain Acceleration). Consider Eq. (20) and its linearization (22) along µ : I → R2 . Suppose that Dx f (t0 , µ(t0 )) is indeﬁnite for a ﬁxed t0 ∈ I and let ξ − , ξ + denote the two lines in the zero strain set Z(t0 ) = {ξ − , ξ + } of (22). Then there exists a δ > 0 and two C 1 curves γ − , γ + : (−ε, ε) → Bδ (µ(t0 )) for some ε > 0 with the following properties:

(i) “Tangency” to linear zero strain set: γ − (0) = γ + (0) = µ(t0 ) and the curves γ − , γ + are tangential to the lines ξ − , ξ + , i.e. T γ − (0) = ξ −

and

T γ + (0) = ξ +

where T denotes the tangent space. (ii) Nonlinear zero strain set: For x0 ∈ Bδ (µ(t0 )) d x(t, t0 , x0 ) − µ(t)|t=t0 = 0 dt ⇔ x0 ∈ graph γ − ∪ graph γ + where x(t, t0 , x0 ) denotes the solution of (20) starting at t0 in x0 . (iii) Nonlinear strain acceleration: Solutions of (20) close to µ(t0 ) cross the curves γ − , γ + in the same direction as solutions to the linearization (22) cross the lines ξ − , ξ + , i.e. if

ξ ± , M (t0 , µ(t0 ))ξ ± > 0 (< 0) then for x0 ∈ graph γ ± d2 x(t, t0 , x0 ) − µ(t)|t=t0 > 0 dt2

(< 0).

Proof. We divide the proof into three steps.

Step 1. By Proposition 25 it is enough to show the claim for the zero solution of Eq. (20) in strain coordinates. That is, we consider Eq. (25) x˙ = B(t)x + h(t, x), denote its solution by x(t, t0 , x0 )

(b)

Rotation in the elliptic case is (a) clockwise if |α| < β and (b) counter-clockwise if |α| < −β.

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and assume that it is transformed in strain coordinates, i.e. the linearization is of the form (28) β(t) ˙ξ = B(t)ξ where B(t) = λ1 (t) . −β(t) −λ2 (t) For x = 0 deﬁne

k(t, x) :=

x h(t, x) , x x

λ2 (t0 ) − k(t0 , x) > 0. (34)

Step 2. To prove (i) and (ii) for (25) in strain coordinates we compute the nonlinear analogue {x0 ∈ Bδ (0) : (d/dt)x(t, t0 , x0 )|t=t0 = 0} of the zero strain set Z(t) consisting of those x0 ∈ Bδ (0) such that instantaneously at t0 the norm of the solution x(t, t0 , x0 ) does not change. Deﬁne C(t0 , x0 ) := (1/2)(d/dt)x(t, t0 , x0 )2|t=t0 . Then C(t, x) = x, B(t)x + h(t, x) = λ1 (t)x21 − λ2 (t)x22 + (x21 + x22 )k(t, x) = [ λ1 (t) + k(t, x)x1 ]2 − [ λ2 (t) − k(t, x)x2 ]2 . With the two functions F − , F + : Bδ (0) → R deﬁned by F ± (x) := λ1 (t0 ) + k(t0 , x)x1 ± λ2 (t0 ) − k(t0 , x)x2 = 0,

(35)

we get the equivalence C(t0 , x) = 0

⇔

F − (x) = 0 or

F + (x) = 0.

From now on, we consider only F − , the arguments for F + are similar. F − (x) is C 1 in x = 0 and F − (0) = 0. To prove that F − is diﬀerentiable in x = 0 with (36) DF − (0)x = λ1 (t0 )x1 − λ2 (t0 )x2 we compute

F − (x1 , x2 ) − F − (0, 0) − [ λ1 (t0 )x1 − λ2 (t0 )x2 ] = [ λ1 (t0 ) + k(t0 , x) − λ1 (t0 )]x1 + [ λ2 (t0 ) − λ2 (t0 ) − k(t0 , x)]x2

k(t0 , x)x1 λ1 (t0 ) + k(t0 , x) +

+ ≤

and since h(t0 , x) = O(x2 ), limx→0 k(t0 , x) = 0 and with k(t0 , 0) := 0 the function k(t0 , ·) is continuous. By Remark 27, λ1 (t0 ), λ2 (t0 ) > 0. Then there exists δ0 > 0 such that for all 0 < δ < δ0 and x < δ λ1 (t0 ) + k(t0 , x) > 0 and

=

λ1 (t0 )

k(t0 , x)x2 λ2 (t0 ) − k(t0 , x) + λ2 (t0 ) |k(t0 , x)|x

λ1 (t0 ) + k(t0 , x) +

+

657

λ1 (t0 )

|k(t0 , x)|x . λ2 (t0 ) − k(t0 , x) + λ2 (t0 )

Since k(t0 , x) → 0 as x → 0, and λ1 (t0 ), λ2 (t0 ) > 0, we have − F (x1 , x2 ) − F − (0, 0) − [ λ1 (t0 )x1 − λ2 (t0 )x2 ] = 0, lim x x→0 proving (36). A direct computation shows that limx→0 Dx F − (x) = Dx F − (0), i.e. F − is a C 1 function. In particular, Dx2 F − (0) = − λ2 (t0 ) = 0. Thus, by applying the Implicit Function Theorem, we get an ε > 0 and a C 1 function γ − : (−ε, ε) → Bδ (0), γ − (0) = 0, such that for x < ε F − (x1 , x2 ) = 0 if and only if x2 = γ − (x1 ) and (ii) is proved. To show (i) we compute γ − (ξ1 ) ξ1 →0 ξ1 λ1 (t0 ) + k(t0 , ξ1 , γ − (ξ1 )) = lim ξ1 →0 λ2 (t0 ) − k(t0 , ξ1 , γ − (ξ1 )) λ1 (t0 ) (37) = λ2 (t0 )

Dγ − (0) = lim

and by (32) we get (i). Step 3. To prove (iii) let x(t) ∈ graph γ − ∪ graph γ + be a solution of (25) in strain coordinates. Then, omitting t-dependencies, d 1 d2 x2 = C(t, x) 2 dt2 dt = x, M (t)x + x, m(t, x) with m(t, x) := B(t)T h(t, x) + (∂h/∂t)(t, x) + (∂h/∂x)(t, x)B(t)x. Assume x(t0 ) ∈ graph γ − with x(t0 ) = 0. Then F − (x(t0 )) = 0 and by (34)

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x2 (t0 ) = 0. Using (31) we get at t = t0 1 d2 x2t=t0 = x, M (t)x + x, m(t, x) 2x22 dt2 2 x1 x1 x21 + x22 x m(t0 , x) 2 2 ˙ ˙ + 2(λ1 + λ2 )β − λ2 + 2λ2 + = (λ1 + 2λ1 ) , x2 x2 x x x22 2 x1 λ2 λ2 2 λ2 2 2 ˙ ˙ ˙ − λ2 + 2λ2 + (λ1 + 2λ1 ) − = (λ1 + 2λ1 ) + 2(λ1 + λ2 )β λ1 λ1 x2 λ1 ! "# $ ! "# $ (I)

x1 + 2(λ1 + λ2 )β − x2 "# !

(II)

λ2 λ1

(III)

Now let x ∈ graph γ −tend to 0. Then by (37) the expression (x1 /x2 ) − λ2 (t0 )/λ1 (t0 ) converges to 0 and hence also (II) and (III) converge to 0. Since m(t0 , x) = O(x2 ) also (IV) converges to 0. Using the fact that λ˙ 1 λ2 − λ1 λ˙ 2 +β λ1 λ2 + √ (I) = 2 2 λ1 λ2 (λ1 + λ2 ) λ2 (λ1 + λ2 ), × λ1 formula (33) implies that we can choose δ > 0 small enough such that for 0 < x(t0 ) < δ the sign of (1/2x22 )(d2 /dt2 )x(t)2|t=t0 is the sign of

ξ − , M (t)ξ − and the proof is complete. Remark 34. If Dx f (t, µ(t)) in Theorem 33 is indeﬁ-

nite for all t ∈ I then the curves γ − , γ + are C 1 also in t ∈ I. This follows by extending F − , F + in (35) as a function also of t ∈ I.

4. Stable and Unstable Manifolds The classical stable and unstable manifolds of a hyperbolic rest point x0 of an autonomous equation x˙ = f (x) is deﬁned asymptotically as starting points x of solutions which tend to x0 as t → ∞ or t → −∞, respectively, see also introduction. Haller and Poje [1998] constructed ﬁnite-time stable and unstable manifolds for planar equations (20) which are incompressible, by applying the Lyapunov– Perron approach to a modiﬁed ﬂow which is cut-oﬀ in space and time outside of the interval I = [a, b], a < b. Since the modiﬁcation of the equation outside of I can be chosen in diﬀerent ways, the ﬁnite-time

+ $

!

x1 x2

2

x m(t0 , x) , +1 . x x "# $

(IV)

invariant manifolds are only unique up to an error of O(e−c(b−a) ) with a constant c > 0. By construction each stable manifold of a solution µ is contained in the set {(t0 , x0 ) ∈ I × R2 | x(t, x0 , x0 ) − µ(t) ≤ x(a, t0 , x0 ) − µ(a) for all t ∈ I}

(38)

consisting of all solutions x(t, t0 , x0 ) of (20) whose distance from the solution µ(t) does not exceed the initial distance x(a, t0 , x0 ) − µ(a) for all t ∈ I. Similarly for unstable manifolds. Whereas classical stable manifolds typically have lower dimension, the set (38) has the dimension of the extended state space I × R2 . Haller [2000] extended the existence proof from [Haller & Poje, 1998] to not necessarily incompressible equations (20) and showed the existence of ﬁnite time stable and unstable manifolds of solutions µ which are uniformly hyperbolic on I in the sense of [Haller, 2000, Def. 1]. For such solutions there exists λ > 0 and a continuous time-dependent and invariant splitting R2 = E s (t) ⊕ E u (t), t ∈ I, such that solutions ξ s (t) ∈ E s (t) and ξ u (t) ∈ E u (t) of the linearization (26) satisfy for t1 , t2 ∈ I, t1 ≤ t2 ξ s (t2 ) ≤ e−λ(t2 −t1 ) ξ s (t1 ) and ξ u (t1 ) ≤ e−λ(t2 −t1 ) ξ u (t2 ), in particular, solutions in E s are strictly decreasing on I, whereas solutions in E u are strictly increasing. Taking Haller’s work as a motivation, a natural extension of the notions in Sec. 3 leads us to the following Definition 35 (Finite-Time Stable and Unstable Manifold). Let µ : I → R2 be a solution of (20).

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Then the set

659

Proof

W sI = W sI [µ] 2 d := (t0 , x0 ) ∈ I × R x(t, t0 , x0 ) dt − µ(t) < 0 for all t ∈ I is called the stable manifold of µ on I and W Iu = W Iu [µ] d := (t0 , x0 ) ∈ I × R2 x(t, t0 , x0 ) dt − µ(t) > 0 for all t ∈ I is called the unstable manifold of µ on I. In contrast to classical stable and unstable manifolds the ﬁnite-time stable and unstable manifolds in Deﬁnition 35 are not of lower dimension but have the dimension of the extended state space I × R2 . Remark 36

(i) For each t0 ∈ I the set W Is (t0 ) = W sI [µ](t0 ) = {x0 ∈ R2 : (t0 , x0 ) ∈ W sI } is called the t0 -ﬁber of W Is and similarly for W Iu . The ﬁbers W sI (t0 ) and W Iu (t0 ) are open subsets of R2 , since if d x(t, t0 , x0 ) − µ(t) = 0 for all t ∈ I dt then by continuity of the solution x, this holds also for small perturbations of (t0 , x0 ). (ii) We say that W Is exists if W Is (t0 ) = ∅ for all t0 ∈ I. By Deﬁnition 35 this is equivalent to W Is = ∅ and similarly for W uI . Theorem 37 (Properties of Finite-Time Stable and Unstable Manifolds). Let µ : I → R2 be a solution of (20). Then the ft-stable manifold (and similarly for the ft-unstable manifold) of µ satisﬁes the following statements:

(i) Invariance: x0 ∈ W Is (t0 ) implies x(t, t0 , x0 ) ∈ W sI (t) for all t ∈ I. (ii) Embedding Property: For each nontrivial interval J ⊂ I W Is (t) ⊂ W Js (t)

for t ∈ J.

(i) To prove invariance we choose t ∈ I and show x(t, t0 , x0 ) ∈ W sI (t), i.e. d x(s, t, x(t, t0 , x0 ))−µ(s) > 0 for all s ∈ I. ds Using the fact that x(s, t, x(t, t0 , x0 )) = x(s, t0 , x0 ) this follows from the assumption x0 ∈ W Is (t0 ). (ii) The embedding property is a direct consequence of Deﬁnition 35. The following example shows ft-stable and ft-unstable manifolds of a hyperbolic autonomous linear system. Example 38. We consider again Example 17. The

system x˙ 1 0 x = y˙ 0 −4 y

(39)

if hyperbolic on the interval I = [0, 2]. The stable and unstable manifolds W sI and W Iu both exist, see Fig. 9. By Remark 31 an autonomous linear system cannot be quasi-hyperbolic. The following nonautonomous example is quasi-hyperbolic and has stable and unstable manifolds. Example 39. The system

−4t x˙ e = 0 y˙

0 x −1 y

(40) 1

−4t

−4t

has the solution x(t, t0 , x0 , y0 ) = x0 e− 4 (e −e 0 ) , y(t, t0 , x0 , y0 ) = y0 e−(t−t0 ) and is quasi-hyperbolic on the interval I = [0, 1]. The stable and unstable manifolds W Is and W Iu both exist, see Fig. 10. The next example is elliptic with stable and unstable manifolds on intervals which are not too large. Example 40. We consider again Example 4. The

system

x˙ 2 −5 x = y˙ 5 −4 y

(41)

is elliptic on any interval of the form I = [a, b], a < b. If b − a is small enough then W Is and W Iu exist. However, due to the rotation around the origin, if b − a is too large then both manifolds are empty, sincethen solution crosses the zero any −2 }, see Fig. 11. strain set Z = { √22 , √ 2

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Fig. 9.

Part of the stable manifold W sI and unstable manifold W u I of the hyperbolic system (39) on I = [0, 2].

Fig. 10.

Part of the stable and unstable manifolds of the quasi-hyperbolic system (40) on I = [0, 1].

Fig. 11.

Part of the stable and unstable manifolds of the elliptic system (41) on I = [0, 0.2].

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661

Due to the invariance properties described in Remark 23(i) in the hyperbolic and quasihyperbolic cases (cp. also Fig. 6) an argument similar to the Wazewski principle can be utilized to prove the existence of stable and unstable manifolds.

is continuous in xa . Then also the mapping which maps the arc to the nonlinear strain set % C(t), Θ : Ψ(a) ∩ {x : x = δ} →

Theorem 41 (Existence of Stable and Unstable

is continuous. By Theorem 33 C(a) consists of two curves γ − , γ + . Choose x− ∈ graph γ − , x+ ∈ graph γ + in the upper half plane with x− = x+ = δ. Then Θ maps the arc Ψ(a) ∩ {x : x = δ} to a connected set, however, Θ(x− ) = x− and Θ(x+ ) = x+ lie in diﬀerent components of & t∈I C(t), which is a contradiction. The proof for W Iu is similar. An analog argument yields the claim if µ(t) ∈ Q(t) for all t ∈ I.

Manifolds for Hyperbolic and Quasi-Hyperbolic Solutions). Let µ : I → R2 be a ft-hyperbolic or quasi-hyperbolic solution of (20), i.e. either µ(t) ∈ H(t) for t ∈ I or µ(t) ∈ Q(t) for t ∈ I. Then both the stable and unstable manifolds of µ exist on I, i.e. W sI (t) = ∅

and

W Iu (t) = ∅

for all t ∈ I.

Suppose that µ(t) ∈ H(t) for all t ∈ I. We prove that W Is (t) = ∅ for all t ∈ I. By Proposition 25, it is enough to show the claim for the zero solution of Eq. (20) in strain coordinates. That is, we consider Eq. (25) x˙ = B(t)x + h(t, x), denote its solution by x(t, t0 , x0 ) and assume that it is transformed into strain coordinates. By Theorem 33 and Remark 34 there exists a δ > 0 such that the nonlinear zero strain set on I in Bδ (0) C := (t0 , x0 ) ∈ I × Bδ (0) :

t∈I

xa → x(τ (xa ), a, xa )

Proof.

d x(t, t0 , x0 )|t=t0 = 0 dt has ﬁbers C(t0 ) = {x0 : (t0 , x0 ) ∈ C} which intersect the x1 -axis and x2 -axis only at the origin. Deﬁne the subset of the upper half plane consisting of nonincreasing solutions d x(t, t0 , x0 )|t=t0 Ψ := (t0 , x0 ) ∈ I × Bδ (0) : dt ≤ 0 and (x0 )2 ≥ 0 . We prove by contradiction that there exists a solution on I in Ψ which then lies in W sI . Thereto assume that for every xa ∈ Ψ(a) there exists a t∗ ∈ I such that x(t∗ , a, xa ) ∈ C(t∗ ), i.e. every solution starting in Ψ(a) leaves Ψ through the nonlinear zero strain set C. By an argument which is similar to the proof of Lemma 13 one can show that for xa in the arc Ψ(a) ∩ {x : x = δ} the exit time τ (xa ) := sup{t ∈ I : x(s, a, xa ) ∈ Ψ(s) for all s ∈ [a, t]}

The following theorem gives a suﬃcient condition for both stable and unstable manifolds to be empty in the ft-elliptic case. Theorem 42 (Stable and Unstable Manifolds for Elliptic Solutions). Let µ : I → R2 be a ft-elliptic solution of (20), i.e. µ(t) ∈ E(t) for t ∈ I. Suppose

b ∆(t) dt > π (42) |β(t)| − 2 a

with β and ∆ deﬁned as in Remark 3. Then there exists a δ > 0 such that both W sI and W uI are empty in a δ-neighborhood of µ, i.e. W sI (t) ∩ Bδ (µ(t)) = W Iu (t) ∩ Bδ (µ(t)) = ∅ for all t ∈ I. W.l.o.g. consider system (20) in strain coordinates (25), i.e. with linearization (28). Rewriting the system in polar coordinates x1 = r cos θ, x2 = r sin θ, we get ( ' λ1 (t) − λ2 (t) λ1 (t) + λ2 (t) + cos 2θ(t) r r˙ = 2 2 Proof.

(43) + k1 (t, θ, r), √ ˙θ = −β(t) − ∆ sin 2θ(t) + k2 (t, θ, r). 2 By (32) and Theorem 33 on the structure of the zero strain set there exists a δ0 > 0 such that for any solution x(t) which is contained in W sI or W Iu and satisﬁes x(t) = r(t) < δ0 for t ∈ I |θ(a) − θ(b)| < π.

(44) M r2

and k2 (t, There exists M ≥ 0 with k1 (t, θ, r) ≤ θ, r) ≤ M r for t ∈ I and r ≤ δ0 . By (42) either (i) √ √ b b a −β−( ∆/2) dt > π or (ii) a β−( ∆/2) dt > π.

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√ b In case (i) deﬁne ε := a −β−( ∆/2) dt−π > 0 and δ := min{δ0 , (ε/M (b − a))}. Assume that either W sI (t) ∩ Bδ (0) or W uI (t) ∩ Bδ (0) are nonempty for each t ∈ I. Then there exists a solution x(t) = r(t)(cos θ(t), sin θ(t))T which is contained either in W sI (t) or in W Iu (t) for all t ∈ I and satisﬁes supt∈I r(t) ≤ δ. Equation (43) and the estimates for k1 , k2 imply that √ √ ∆ ∆ − M r(t) ≤ θ˙ ≤ −β + + M r(t), −β − 2 2 or equivalently √

b

b ∆ −β − M r(t) dt dt − 2 a a ≤ θ(b) − θ(a) √

b

b ∆ dt + ≤ −β + M r(t) dt. 2 a a As a consequence, we get the estimate √

b ∆ dt − M r(t) dt −β − θ(b) − θ(a) ≥ 2 a I √

b ∆ dt − ε > π −β − ≥ 2 a which contradicts (44), proving both W Is and W Iu are empty in Bδ (0). If condition (42) in Theorem 42 is not satisﬁed then a ft-elliptic system might have stable and unstable manifolds on arbitrarily large intervals as the following example shows. Example 43. The linear system

1 2 x x˙ 1 . = 2 4(t + 1) −2 −1 y y˙

is elliptic on any interval [a, b], a > b, with 1I = −1 zero strain set Z(t) = { 1 , 1 }. For the system in polar coordinates 1 (sin2 θ − cos2 θ)r, r˙ = 2 4(t + 1) θ˙ =

4(t2

1 (2 + sin 2θ) + 1)

a direct computation shows that 1/4(t2 + 1) ≤ θ˙ ≤ 3/4(t2 + 1), i.e. for t ∈ I

t 1 dτ ≤ θ(t) − θ(a) 0≤ 2 a 4(τ + 1)

t 3π π 3 dτ ≤ < . ≤ 2 8 2 a 4(τ + 1)

Therefore in both cones between the zero strain set there exist solutions which do not cross Z(t) and hence are contained in W sI and W Iu , respectively. Assume that µ(t) is a solution of (20) which is contained in exactly one of the regions E(t), H(t) or Q(t) for all t ∈ I = [a, b]. If µ(t) ∈ H(t) or Q(t) then by Theorem 41 stable and unstable manifolds W Is , W Iu of (20) as well as stable and unstable manifolds EIs , EIu of the linearization (22) exist in a δ-neighborhood of µ. In the elliptic case, we have to assume that W Is , W Iu and E Is , E Iu exist. If they do exist we have the following Theorem 44 (Linear Approximation of Stable and Unstable Manifolds). Assume that µ(t) is contained in exactly one of the regions E(t), H(t) or Q(t) for all t ∈ I = [a, b]. Then the stable manifold EIs of the linearization approximates the stable manifold W Is [µ] locally. More precisely, for each t0 ∈ I let ζ − and ζ + be the two lines in the boundary of EIs (t0 ). Then there exists δ > 0 and two C 1 curves η − , η + : (−ε, ε) → Bδ (µ(t0 )) for some δ > 0 with the following properties:

(i) η − , η + are boundaries of W Is (t0 ): graph η − , graph η + ⊂ bd W Is (t0 ). (ii) “Tangency” to linear manifold: η − (0) = η + (0) = µ(t0 ) and the curves η − , η + are tangential to the lines ζ − , ζ + , i.e. T η − (0) = ζ −

and

T η + (0) = ζ +

where T denotes the tangent space. Similarly for E Iu (t0 ) and W Iu (t0 ). Assume that µ(t) ∈ H(t) for t ∈ I = [a, b]. As in Remark 23(i) we observe that ψ − lin := {(t, ξ) ∈ I × R2 : ξ, S(t, µ(t))ξ < 0} is backward invariant 2 and ψ + lin := {(t, ξ) ∈ I × R : ξ, S(t, µ(t))ξ > 0} is forward invariant under the solution Φ(t, t0 )ξ of the linearization (22) on I and Proof.

EIs (b) = ψ − lin (b)

and EIu (a) = ψ + lin (a).

(45)

2 : By Theorem 33 ψ − nl := {(t0 , x0 ) ∈ I × R (d/dt)x(t, t0 , x0 ) − µ(t)|t=t0 < 0} is backward 2 : invariant and ψ + nl := {(t0 , x0 ) ∈ I × R (d/dt)x(t, t0 , x0 )−µ(t)|t=t0 > 0} is forward invariant under the solution x(t, t0 , x0 ) of Eq. (20) on I and

W sI (b) = ψ − nl (b)

and W Iu (a) = ψ + nl (a).

(46)

Let ξ − , ξ + denote the lines in Z(b). By Theorem 33 the nonlinear zero strain set of (20) consists of two curves graph γ − , graph γ + which are tangential to

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ξ − , ξ + . In fact, by (45), ξ − , ξ + are in the boundary − + of ψ − lin (b) and by (46) graph γ , graph γ are in the − boundary of ψ nl (b) ξ − , ξ + ⊂ bd EIs (b) and graph γ − , graph γ + ⊂ bd W sI (b).

that even if a linear system is hyperbolic on [0, ∞) then solutions in the stable manifold do not necessarily tend to 0. Example 46. By Theorem 30 the system

  1 0 x˙ x˙   = 1 y˙ y˙ 0 − 2 t +1

Now ﬁx a t0 ∈ I and deﬁne ζ ± := Φ(t0 , b)ξ ±

and η ± := x(t0 , b, γ ± ).

Since boundaries are mapped on boundaries by Φ(t0 , b) and x(t0 , b, ·), and using the fact that Φ is the linearization of x along µ, the claim follows. Similarly one can show the statements for the unstable manifolds. If µ(t) ∈ Q(t) for t ∈ I then the proof + u is analogous using EIs (a) = ψ − lin (a), EI (b) = ψ lin (b). If µ(t) ∈ E(t) for t ∈ I then we have to use the fact that EIs (and similarly EIu ) intersects the zero strain set Z in exactly one line at time a and exactly one line at time b (cp. also Example 40). The following example shows the approximation of a stable manifold by the stable manifold of its linearization.

is hyperbolic on I = [0, ∞).√ The two√lines in the t2 +1 2 zero strain set Z(t) = 1/ 1t +1 , 1/ −1 cons verge to the y-axis for t → ∞. Each ﬁber W I (t) of the stable manifold for t ∈ I is the y-axis. However, each nonzero solution x(t) ≡ 0, y(t) = e− arctan t y(0) starting in W sI (0) converges to (0, e−π/2 y(0)) = (0, 0) for t → ∞. The following example shows that even if all solutions in the stable manifold of a linear hyperbolic system on [0, ∞) converge to 0 they do not necessarily decay exponentially. Example 47. By Theorem 30 the system

1 0 x x˙ 1 = t + 1 0 −1 y y˙

Example 45. Consider the nonlinear system

x˙ = x,

y˙ = −y − 4y 3 .

(47)

The zero strain set for the zero solution is independent of t ∈ R and of the form C = {(x, y) : x = y 1 + 4y 2 }. By Theorem 44 the ﬁbers of the stable manifold W Is are tangential to the ﬁbers of the stable manifold EIs of the linearization, see Fig. 12. Next we give some examples of stable manifolds on unbounded intervals. The ﬁrst example shows

is hyperbolic on I = [0, ∞) and W Is (t) is the y-axis for each t ∈ I. However, the solutions in W Is do not decay exponentially. Remark 48

(i) Example 47 is in strain coordinates with λ1 (t) = λ2 (t) = 1/(t + 1). Every solution in W Is converges to 0. In fact, one can prove that this is true for any linear system in strain

(a) Fig. 12.

(a) Fibers of

(b) W sI

and

663

EIs

at t = 1/2 and (b) part of

W sI

on I = [0, 1] for (47).

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coordinates is hyperbolic on [a, ∞) and ∞ which satisﬁes a ( λ1 (t)λ2 (t) − |β(t)|)dt = ∞ and

and a, b, c : I → R are C 1 functions. As a consequence of Proposition 18 we have

λ2 (t) < M and λ1 (t) d λ1 (t) − λ2 (t) 1 <M λ1 (t) + λ2 (t) dt λ1 (t) + λ2 (t)

Corollary 49 (Characterization of Incompressible

0<m
0, by using Barbalat’s Lemma [1959], see [Duc, 2006]. (ii) Example 39 is in strain coordinates with λ1 (t) = e−4t , λ2 (t) ≡ 1. The two lines in the zero strain set converge to the x-axis and every solution converges to a point in the x-axis. In fact, one can prove that this is true for any linear system in strain coordinates which is quasihyperbolic on [a, ∞) and satisﬁes

∞ λ2 (t)dt = ∞ and 0 ≤ λ2 (t) ≤ M a

for t ∈ [a, ∞)

for some M ≥ 0, by using Barbalat’s Lemma [1959], see [Duc, 2006].

5. Hamiltonian Systems In this section we consider the important special case of planar Hamiltonian systems x˙ = J∇H(t, x)

with standard symplectic matrix J = −10 10 and Hamiltonian H : I × R2 → R2 on a nontrivial interval I ⊂ R, i.e. we consider the system ∂H (t, x1 , x2 ), x˙ 1 = ∂x2

∂H x˙ 2 = − (t, x1 , x2 ) (48) ∂x1

where we assume that (∂ 3 H/∂x3 )(t, x) exists and is continuous. In ﬂuid dynamics in the plane, the Hamiltonian case corresponds to incompressibility of the ﬂuid and H is usually called stream function ψ. Let µ : I → R2 be a solution of (48). The linearization along µ is ξ˙ = A(t)ξ where

a(t) b(t) A(t) = c(t) −a(t)   2 ∂2H ∂ H  ∂x ∂x ∂x22    2 1 =  (t, µ(t)) (49) 2  ∂2H ∂ H  − 2 − ∂x1 ∂x2 ∂x1

Zero Strain Set). Consider system (49) with symmetric part S(t) = (1/2)[A(t) + A(t)T ]. Then the zero strain set Z(t) = Z(t, µ(t)) satisﬁes exactly one of the following two cases: (i) Z(t) consists of two lines if and only if det S(t) < 0. (ii) Z(t) is the plane if and only if det S(t) = 0. Proposition 18 with det S(t) = −a(t)2 − (1/4)(b(t) + c(t))2 . Proof.

By Corollary 49 the zero strain set cannot be the origin. This corresponds to the fact that A(t) is neither positive deﬁnite nor negative deﬁnite. As a consequence the attracting regions A(t) and repelling regions R(t) of system (49) are empty for all t ∈ I. If we assume that A(t) is indeﬁnite on I then by Remark 27, Eq. (48), and hence its linearization (49), can be transformed into strain coordinates λ(t) β(t) ξ˙ = B(t)ξ where B(t) = −β(t) −λ(t) with (omitting the t-dependence) 1 λ = a2 + (b + c)2 4 β=

and

a(b˙ + c) ˙ − a(b ˙ + c) b − c . + 2 4a + (b + c)2 2

(50)

By Proposition 29

1 −1 Z(t) = , and 1 1 2λβ λ˙ + 2λ2 . M (t) = 2λβ −λ˙ + 2λ2

Since 11 , M (t) 11 + −11 , M (t) −11 = 4λ(t)2 ≥ 0, MZ cannot be negative deﬁnite and as a consequence the quasi-hyperbolic region Q(t) is empty for all t ∈ I. By Proposition 25 the dynamic regions are preserved by the strain coordinate transformation and we have proved the following Lemma 50 (Empty A, R, Q in Hamiltonian Case).

Consider (48). Then A(t) = R(t) = Q(t) = ∅

for all t ∈ I.

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Note that for Hamiltonian systems (48) the dynamic partition in Deﬁnition 22 is Haller’s EPH partition [Haller, 2001a] with the minor diﬀerence that Haller’s elliptic points with vanishing rate of strain tensor as well as the parabolic region are contained in the degenerate region D(t). Theorem 30 applied to Hamiltonian systems (48) is the Okubo–Weiss criterion in strain coordinates which was originally derived for 2D incompressible ﬂuids (cp. the discussion before Theorem 30). Corollary 51 (Okubo–Weiss criterion for Hamilto-

665

By Corollary 51, λ < |β| and assumption (51) can be rewritten as

b π (52) (|β| − λ)dt > . 2 a The proof is now similar to the proof of Theorem 42 using the fact that (52) implies (42) with π replaced by π/2, and the lines in the zero strain set Z(t) = {ξ − , ξ + } are ﬁxed with angles π/4 and 3π/4.

6. Applications

nian Systems). Consider Eq. (48) with linearization (49) and assume that A(t) is indeﬁnite for t ∈ I. Then, using (50), the following characterizations hold:

We present applications to the problem of location of periodic orbits for autonomous equations, a double gyre ﬂow and symmetric vortex merger.

(i) µ(t) ∈ E(t) if and only if λ(t) < |β(t)|. (ii) µ(t) ∈ H(t) if and only if λ(t) > |β(t)|. (iii) µ(t) ∈ D(t) if and only if λ(t) = |β(t)|.

6.1. Location of periodic orbits

Of course, all results on the zero strain set and approximations of E(t) and H(t) for nonlinear systems (Theorem 33) as well as all results about stable and unstable manifolds (Theorems 37, 41 and 44) also hold for Hamiltonian systems (48). However, Theorem 42 on the existence of stable and unstable manifolds in the elliptic case can be strengthened. It was ﬁrst proved in [Haller, 2001a, Proposition 2 and Theorem 2]. In our situation, this theorem can be expressed as follows.

with a C 2 function f : R2 → R2 . Then the dynamic partition deﬁned in Deﬁnition 22 is independent of t ∈ R. The following theorem states that a periodic orbit is either in the elliptic region or contains a degenerate point.

Theorem 52 [Haller, 2001a]. Let µ : I → R2 be a

ft-elliptic solution of (48), i.e. µ(t) ∈ E(t) for t ∈ I. Let ξ − (t), ξ + (t) be nonzero vectors lying on the two diﬀerent lines in the zero strain set of the linearization (49). Suppose −

b − + + min ξ , M ξ , ξ , M ξ dt > π . − − + + ξ Sξ ξ Sξ 2 a (51) Then both W Is and W uI are empty in a neighborhood of µ. A direct computation shows that (51) does not change if we pass to strain coordinates and w.l.o.g. ξ − = 11 , ξ + = −11 and (omitting the t-dependence) Proof.

ξ − , M ξ − = λ + β, ξ − Sξ −

ξ + , M ξ + = λ − β. ξ + Sξ +

Consider an autonomous equation in the plane x˙ = f (x)

(53)

Theorem 53 (Location of Periodic Orbits). Assume

that (53) has a nontrivial periodic orbit Γ = {µ(t) : t ∈ [0, T ]} with period T > 0. Then exactly one of the following two alternatives holds: (i) Γ ⊂ E or (ii) Γ ∩ D = ∅.

Assume that Γ ∩ D = ∅. Then the closed set Γ is contained entirely in one of the open regions A, R, E, H or Q. Since µ ¨(t) = Df (µ(t))µ(t), ˙ the function µ˙ is a nontrivial T -periodic solution of the periodic linear equation ξ˙ = A(t)ξ where A(t) = Df (µ(t)) (54)

Proof.

with symmetric part S(t) and zero strain set Z(t). Assume that Γ ⊂ A. Then µ(t) ∈ A for t ∈ [0, T ] and by Lemma 9 we get the contradiction µ(T ˙ ) < µ(0). ˙ Similarly Γ ⊂ R yields a contradiction. Assume that Γ ⊂ H. Deﬁne the two cones Ψ− (t) := {ξ ∈ R2 : ξ, S(t)ξ < 0} and ˙ ∈ Ψ− (t) Ψ+ (t) := {ξ ∈ R2 : ξ, S(t)ξ > 0}. If µ(t) for all t ∈ [0, T ] then (12) implies the contradiction µ(T ˙ ) < µ(0). ˙ Also µ(t) ˙ ∈ Ψ+ (t) for all t ∈ [0, T ] is impossible. That is, there exists a t0 ∈ [0, T ] with µ(t ˙ 0 ) ∈ Z(t0 ). But then by Remark 23(i)

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µ(t) ˙ ∈ Ψ+ (t) for t > t0 and we get the contradiction ˙ 0 + T ) ∈ Ψ+ (t0 + T ) = Ψ+ (t0 ). Simµ(t ˙ 0 ) = µ(t ilarly Γ ⊂ Q yields a contradiction. Consequently Γ ⊂ E.

Example 55. Consider the system x˙ = λx − y + (ax − by)(x2 + y 2 ) y˙ = x + λy + (bx + ay)(x2 + y 2 )

(55)

for λ, a, b ∈ R with λ > 0 > a and periodic orbit Γ = {(x, y) : x2 + y 2 = −(λ/a)}. Figure 13 shows The following example has periodic orbits in the location of Γ for λ = 10, a = −1 and b = 0 on the the elliptic region. left for system (55) and on the right for (55) under 0 Example 54. The linear autonomous system . the transformation x → D−1 x with D = 10 (5/2) The unstable ﬁxed point at the origin is contained x˙ 3 5 x in the repelling region R, the quasi-hyperbolic region = y˙ −5 3 y Q is green and H is depicted blue, the attracting region A is unbounded. After the coordinate transforis elliptic with E = R2 and Hamiltonian H(x, y) = mation, a red elliptic region E emerges. In accordance 5x2 + 6xy + 5y 2 . Every solution is periodic and conwith Theorem 53, the periodic orbit Γ has nonempty tained in E. intersection with the degenerate region D, in fact, in the left ﬁgure Γ is contained entirely in the degenNext we study the location of periodic orbits erate region between A and H. To get the dynamic for a Hopf bifurcation in normal form [Chow et al., partition explicitly we compute the linearization ξ˙ = 1994, Sec. 3.2]. A(µ(t))ξ along a solution µ by λ + 3ax2 − 2bxy + ay 2 −1 − bx2 + 2axy − 3by 2 . A(x, y) = 1 + 3bx2 + 2axy + by 2 λ + ax2 + 2bxy + 3ay 2 Using Theorem 30, a direct computation yields the dynamic partition A = {(x, y) ∈ R2 R = {(x, y) ∈ R2 H = {(x, y) ∈ R2 Q = {(x, y) ∈ R2 E = {(x, y) ∈ R2 D = {(x, y) ∈ R2

: det S(x, y) > 0 : det S(x, y) > 0 : det S(x, y) < 0 : det S(x, y) < 0 : det S(x, y) < 0 : det S(x, y) = 0

and Σ1 (x, y) < 0} and Σ1 (x, y) > 0} and Σ2 (x, y) > 0 and Π(x, y) > 0} and Σ2 (x, y) < 0 and Π(x, y) > 0} and Π(x, y) < 0} or Π(x, y) = 0}

4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4 −4

−3

−2

−1

0

1

2

3

4

−4 −4

−3

−2

−1

0

1

2

3

4

Fig. 13. Periodic orbit Γ and dynamic partition of Eq. (55) for λ = 10, a = −1, b = 0 (left) and after the transformation x → D−1 x (right).

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Hyperbolicity and Invariant Manifolds for Planar Nonautonomous Systems on Finite Time Intervals 4

4

3

3

2

2

1

1

0

0

−1

−1

−2

−2

−3

−3

−4 −4

−2

−3

−1

0

1

2

3

4

−4 −4

−2

−3

−1

0

1

2

3

667

4

Fig. 14. Periodic orbit Γ and dynamic partition of Eq. (55) for λ = 10, a = −1, b = −0.5, case IV (left) and b = −0.8, case III (right).

withthe following explicit formulas depending on r = x2 + y 2 det S(x, y) = [λ − ( b2 + a2 − 2a)r 2 ] × [λ + ( b2 + a2 + 2a)r 2 ] √ Π(x, y) = (λ + ar 2 )2 [2λ + (3a − 3b)r 2 ] √ × [2λ + (3a + 3b)r 2 ] Σ1 (x, y) = λ + 2ar 2

separates two diﬀerent cases then the dynamic partition bifurcates in the sense that degenerate regions emerge or vanish. The same happens in case IV for b = 0, see Fig. 13 (left), since then dΓ = d6 = d7 and d2 = d3 = d5 . The periodic orbit Γ is the degenerate region DΓ which is, for example, located between the hyperbolic regions for the parameter values λ = 10, a = −1, b = −0.5, case IV, and between the elliptic regions for b = −0.8, case III, see Fig. 14.

Σ2 (x, y) = (b2 − 3a2 )r 4 − 5λar 2 − 2λ2 . A direct computation shows that the periodic orbit Γ forms a degenerate region DΓ = Γ. We distinguish four diﬀerent cases depending on thelocation of degenerate regions Di = {(x, y) : r = x2 + y 2 = di } with λ λ d1 = √ dΓ = 2 −a a + b2 − 2a 2λ d2 = √ 3b − 3a λ d4 = −2a

d3 = √

a2

4λ + 8b2 − 5a

2λ d5 = √ − 3b − 3a

λ 4λ d7 = √ 2 2 − + b − 2a − a + 8b2 − 5a and the regions in the annuli in between, see Fig. 15. For the location of cases I–IV in the (a, b)-plane, see Fig. 16. If the parameters (a, b) cross a line which d6 =

√

a2

6.2. Double gyre flow In this section we apply the previous results to a double gyre ﬂow [Shadden et al., 2005] given by the stream function ψ(x, y) = A sin(πx) sin(πy) for (x, y) ∈ Ω := [0, 2] × [0, 1] with amplitude A > 0. The vector ﬁeld is given by x˙ = −

∂ψ (x, y) = −πA sin(πx) cos(πy) ∂y

(56)

∂ψ (x, y) = πA cos(πx) sin(πy) y˙ = ∂x see Fig. 17. Since (56) is autonomous its dynamic partition is independent of t ∈ R. The dynamic partition is determined by the linearization ξ˙ = B(x(t), y(t))ξ of (56) along a solution

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Fig. 15.

Schematic location of R, Q, E , H, A, degenerate regions Di and periodic orbit Γ = DΓ .

Fig. 16.

Diﬀerent cases in dependence of a, b ∈ R, a < 0.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

Fig. 17.

0.8

1

1.2

1.4

Velocity ﬁeld of (56) for A = 0.1.

1.6

1.8

2

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(x(t), y(t)) where B(x, y) = Aπ

2

669

sin(πx) sin(πy) . cos(πx) cos(πy)

−cos(πx) cos(πy) −sin(πx) sin(πy)

By Corollary 51 the dynamic partition of (56) is given by H = (x, y) ∈ Ω : | cos(πx) cos(πy)| > | sin(πx) sin(πy)| E = (x, y) ∈ Ω : | cos(πx) cos(πy)| < | sin(πx) sin(πy)| D = (x, y) ∈ Ω : | cos(πx) cos(πy)| ∈ {0, | sin(πx) sin(πy)|} . a < b, with 0 ∈ I. By Theorem 44, W Is and W Iu are approximated locally by the stable and unstable manifolds EIs and EIu of the linearization along µ. See Fig. 19 for parts of the stable manifolds EIs and EJs with I = [−10, −1] and J = [1, 10]. At the degenerate point µ(0) = (1, 1/2)T the manifolds are “twisted”, in fact the ﬁbers of EIs contain the xaxis whereas the ﬁbers of EJs contain the y-axis. We conjecture that this corresponds to the fact that the linearization along the heteroclinic solution µ does not admit an exponential dichotomy on R but only exponential dichotomies on the half-lines R− and R+ .

A direct computation shows that 1 3 5 , , or D = (x, y) ∈ Ω : x + y ∈ 2 2 2 1 1 3 x−y ∈ − , , 2 2 2 see Fig. 18. The heteroclinic orbit µ(t) = (1, (1/π) arccos tanh(Aπ 2 t))T connects the remaining points (1, 1) and (1, 0) and satisﬁes µ(0) = (1, 1/2)T ∈ D, µ(t) ∈ H for t = 0. The linearization along µ is ξ˙ = B(µ(t))ξ with 1 0 B(µ(t)) = Aπ 2 tanh(Aπ 2 t) 0 −1

6.3. Vortex merger

and it is hyperbolic for t = 0, has the solution Φ(t, t0 )ξ =

2

2

2

2

eAπ t0 + e−Aπ t0 eAπ t + e−Aπ t ξ , ξ2 1 2 2 2 2 eAπ t0 + e−Aπ t0 eAπ t + e−Aπ t

The interaction of two like-signed vortices embedded in an otherwise quiescent ﬂuid has been the subject of intense research for the last decades (see [Velasco Fuentes, 2001] and the references therein). The motion of an ideal ﬂuid in two dimensions is governed by the vorticity and Poisson equation for the vorticity (ω) and the stream function (ψ):

T

and the time-independent zero strain set Z = { 11 , −11 } for t = 0. Since µ(t) ∈ H for t = 0, Theorem 41 implies that µ has stable and unstable manifolds W Is and W Iu on any interval I = [a, b],

ωt + ψy ωx − ψx ωy = 0,

∇2 ψ = −ω.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

Fig. 18.

0.6

0.8

1

1.2

1.4

1.6

Dynamic partition of (56), D consists of lines.

1.8

2

(57)

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Fig. 19. Parts of stable manifolds for the linearization of (56) along the heteroclinic orbit µ on the intervals I = [−10, −1] and J = [1, 10].

The level curves {(x, y) ∈ R2 : ψ(x, y, t) = c}, c ∈ R, are tangential to the velocity ﬁeld (x, ˙ y) ˙ of the particles (x, y) in the ﬂuid. Hence x˙ =

∂ψ (x, y, t), ∂y

y˙ = −

∂ψ (x, y, t). ∂x

(58)

Figure 20 shows numerical solutions of Eq. (57) for one and two vortices with no-slip boundary conditions. Based on the Okubo–Weiss criterion for incompressible ﬂuids, Corollary 51, one can compute the elliptic and hyperbolic region. Figure 21 shows the elliptic region (yellow for counter-clockwise, red for clockwise, blue for hyperbolic) and the dynamic partition for one vortex and for two vortices. The vortices are in yellow elliptic

Fig. 20.

(counter-clockwise) regions, the red elliptic (clockwise) regions for two vortices are caused by so-called ghost vortices. Since we consider symmetric vortex merger, the two vortices are of the same shape and size. As a consequence, the center is a rest point, i.e. (58) has the constant solution µ(t) ≡ (1/2, 1/2)T . Before the vortices merge the center is contained in the hyperbolic region and after the merging it is the center of the new merged elliptic vortex. In other words, there exists a time t0 ∈ R such that µ(t) ∈ H(t) for t < t0 and µ(t) ∈ E(t) for t > t0 , see Fig. 22. We say that µ bifurcates from hyperbolic to elliptic and call t0 merging time. By the Okubo–Weiss criterion Corollary 51, the merging is described by λ(t) and β(t), i.e. the linearization along µ in strain coordinates. Before the merging,

One vortex (left) and two vortices (right): level curves of stream function ψ, velocity ﬁeld and particles.

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Hyperbolicity and Invariant Manifolds for Planar Nonautonomous Systems on Finite Time Intervals

Fig. 21.

Fig. 22.

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One vortex (left) and two vortices (right): elliptic (yellow/red) and hyperbolic region (blue).

Before the vortices merge the origin is hyperbolic (left) and after the merging it is elliptic (right).

the eigenvalues of the linearization in strain coordinates are on the real axis, approach the origin as t → t0 . At merging time t0 they pass through the origin and after the merging the eigenvalues are on the imaginary axis. A detailed study is the topic of a forthcoming paper [Duc & Siegmund, 2008]. A snapshot of a three-dimensional velocity ﬁeld of two vortices is shown in Fig. 23. We thank T. Frisius and T. Hasselbeck for simulating these

two tropical storms with the Lokalmodell of the German weather agency (DWD), see also [Frisius et al., 2006]. The left picture shows the velocity ﬁelds and particles in two layers and the right picture displays the instantaneous two-dimensional dynamic partition in these layers. The threedimensional velocity ﬁeld is incompressible but the restricted dynamics in horizontal layers is not incompressible anymore and allows for attracting

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Fig. 23. storms.

Particles (left) and dynamic partition (right) on horizontal layers of three-dimensional simulation of two tropical

(white), repelling (black) and quasi-hyperbolic (green) regions. A detailed study is the topic of a forthcoming note.

ﬁnite time intervals. We mention that Haller introduced elliptic and hyperbolic regions for threedimensional incompressible ﬂows in [Haller, 2005], see also [Berger et al., 2008].

7. Conclusion We introduced attracting, repelling, elliptic, hyperbolic and quasi-hyperbolic regions as an extension of the EPH partition [Haller, 2001a] by studying the change of distance of neighboring solutions at a ﬁxed time. Solutions with decreasing and increasing distance over a whole time interval form the stable and unstable manifolds, respectively. As a test that these extensions for ﬁnite time intervals be coherent in the sense that they are analogs of the classical asymptotically deﬁned notions of attractor, repellor, hyperbolicity and invariant manifolds, we generalized results by Ide et al. [2002] on the existence of attractors and repellors, an existence result by Haller [2002] on invariant manifolds and the Okubo–Weiss criterion from ﬂuid mechanics on ﬁnite time intervals. Theorem 53 on the location of periodic orbits as well as the Vortex Merger discussion suggest to study nonautonomous bifurcation phenomena under the aspect of a bifurcation in time of the dynamic partition (or EPH partition in the incompressible case). So far nonautonomous bifurcation theory has focused on the bifurcation of asymptotically deﬁned objects like pullback attractors [Kloeden & Siegmund, 2005; Langa et al., 2002; Rasmussen, 2007]. The application of the dynamic partition to the problem of vortex merger in Sec. 6.3 indicates that the study of the qualitative change of the dynamic partition in dependence of time and/or parameters might be the right concept to develop a nonautonomous bifurcation theory on

Acknowledgments The second author thanks George Haller for introducing him to the ideas in [Haller, 2001a]. Abigail Wacher provided the ﬁnite-diﬀerence scheme solver for Eqs. (57) during her postdoc year in Frankfurt and coded computation and visualization of the dynamic partition in the incompressible case. Maria Terechtchenko assisted modifying and extending the code. This work was supported by the Emmy Noether Programme of the Deutsche Forschungsgemeinschaft (DFG).

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