INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS of NONAUTONOMOUS DIFFERENTIAL EQUATIONS Bernd Aulbach1 and Martin Rasmussen2 Stefan S

DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS Volume 15, Number 2, June 2006 pp. 579-596

INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS OF NONAUTONOMOUS DIFFERENTIAL EQUATIONS

Bernd Aulbach1 and Martin Rasmussen2 Department of Mathematics University of Augsburg D-86135 Augsburg, Germany Stefan Siegmund3 Department of Mathematics University of Frankfurt D-60325 Frankfurt am Main, Germany (Communicated by Glenn F. Webb)

Abstract. We discuss the relationship between invariant manifolds of nonautonomous differential equations and pullback attractors. This relationship is essential, e.g., for the numerical approximation of these manifolds. In the first step, we show that the unstable manifold is the pullback attractor of the differential equation. The main result says that every (hyperbolic or nonhyperbolic) invariant manifold is the pullback attractor of a related system which we con- struct explicitly using spectral transformations. To illustrate our theorem, we present an application to the Lorenz system and approximate numerically the stable as well as the strong stable manifold of the origin.

1. Introduction. In the theory of dynamical systems (e.g., autonomous differential equations) the stable and unstable manifold theorem—first proved by Poincaré [13] and Hadamard [10]—plays an important role in the study of the flow near a hyperbolic rest point or a periodic solution. The idea is to show that the rest point x = 0, say, locally inherits the dynamical structure of its hyperbolic linearization xAx˙ , i.e., the spectrum s u σ(A) = σ (A) ∪ σ (A){λ1, . . . , λr} ∪ {λr+1, . . . , λn} consists of eigenvalues λ1, . . . , λr with negative real parts and λr+1, . . . , λn with positive real parts. Let V s and V u denote the sum of all generalized eigenspaces corresponding to the eigenvalues in σs(A) and σu(A), respectively. Then V s and V u are the stable and unstable manifold of the linear system, respectively, and the stable manifold theorem provides nonlinear analogues s u W = {x0 : x(t, x0) → 0 as t → ∞} and W = {x0 : x(t, x0) → 0 as t → −∞}

2000 Mathematics Subject Classiﬁcation. 34C30, 34D45, 37B55, 37D10, 65L20. Key words and phrases. Nonautonomous diﬀerential equation, Pullback attractor, Invariant manifold, Numerical approximation. 1Died unexpectedly on January 14, 2005. His coauthors dedicate this paper to his memory. 2Supported by the Deutsche Forschungsgemeinschaft, grant GK 283. 3Supported by the Deutsche Forschungsgemeinschaft, grant Si801.

579 580 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND

s u which are tangential to V and V , respectively; x(t, x0) being the general solution of the nonlinear system. The fact that the linear manifolds V s and V u are direct sums of eigenspaces consisting of solutions with an exponential growth or decay rate given by the real parts of the corresponding eigenvalues also carries over to W s and W u. A submanifold of W s which corresponds to some (but not all) eigenvalues with smallest real part is called a strong stable manifold. As for an example, the Lorenz system at the origin (for the usual parameters) has two negative eigenvalues λ1 < λ2 < 0; the unstable manifold therefore has dimension 2 whereas the strong stable manifold (consisting of solutions with decay rate λ1) has dimension 1. The analytic computation of invariant manifolds is possible only in rare cases. It is reasonable, therefore, to develop numerical tools for the approximation of this kind of set. For stable and unstable manifolds of autonomous differential equations this topic is well examined. In Dellnitz and Hohmann [6, 7], a subdivision as well as a continuation algorithm is introduced to approximate the global attractor of the system; and since the global attractor contains all unstable manifolds, these algorithms are suitable for the approximation of unstable manifolds. For a survey of different methods for computing (un)stable manifolds of vector fields which also treats the subdivision technique, see the paper by Krauskopf, Osinga et al. [12]. In this paper, we go one step further and develop an approximation result for more general types of invariant manifolds. In fact, we consider all invariant manifolds of a so-called hierarchy of invariant manifolds which are nonlinear analogues of the eigenspaces of the linearization, including strong stable manifolds and nonhyperbolic manifolds such as center or center-stable manifolds. We make use of the observation that (under suitable global assumptions) the unstable manifold is globally attracting—and therefore amenable to a numerical approximation. Then we use spectral transformations to relate an arbitrary invariant manifold of the above-mentioned hierarchy to the unstable manifold of a transformed system. The price one has to pay for this wide range of applications is that the transformed systems are nonautonomous even if the original system is autonomous. This fact is the major reason for dealing with nonautonomous systems from the outset. This paper is organized as follows. In Section 2, we introduce some notation and define nonautonomous attractors. Section 3 is devoted to nonautonomous invariant manifolds. The relationship between hyperbolic invariant manifolds and attractors is established in Section 4. In Section 5, we describe the nonhyperbolic case leading to the general result in Section 6. To illustrate our theorem, in Section 7, we compute and visualize the stable as well as the strong stable manifold of the origin of the Lorenz system using a nonautonomous extension of the software package GAIO.

2. Preliminaries. As usual we denote by R the set containing all reals and by RN×N the set of all real N × N matrices. We use the symbol 1 for the identity matrix. With XN N k(x1, . . . , xN )k := |xi| for all (x1, . . . , xN ) ∈ R i=1 N N the R is a normed vector space. We write U²(x0){x ∈ R : kx − x0k < ²} for N N the ²-neighborhood of a point x0 ∈ R . For arbitrary nonempty sets A, B ⊂ R and x ∈ RN let d(x, A) : inf{kx − yk : y ∈ A} be the distance of x to A and d(A|B) := sup{d(x, B): x ∈ A} be the Hausdorff semi-distance of A and B.A INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS 581 function g : R → RN is called γ+-quasibounded if + −γt kgkτ,γ := sup{kg(t)ke : t ≥ τ} < ∞ for some τ ∈ R . Accordingly, the function g is called γ−-quasibounded if − −γt kgkτ,γ := sup{kg(t)ke : t ≤ τ} < ∞ for some τ ∈ R . In this article we are concerned with nonautonomous differential equations x˙ = f(t, x) , where f : R × RN → RN is a function which fulfills conditions ensuring global existence and uniqueness of solutions. This equation then gives rise to a general solution λ(t, τ, ξ) which is the solution satisfying the initial condition x(τ) = ξ with τ and ξ treated as additional parameters. For a nonautonomous linear differential equation x˙ = B(t)x (1) with a continuous function B : R → RN×N the general solution is linear with respect to ξ, i.e., there exists a function Φ : R × R → RN×N , the so-called evolution operator of (1), such that λ(t, τ, ξ) = Φ(t, τ)ξ for all t, τ ∈ R and ξ ∈ RN . We say that a function P : R → RN×N is an invariant projector of (1) if P (t)2 = P (t) for all t ∈ R , P (t)Φ(t, s) = Φ(t, s)P (s) for all t, s ∈ R . A subset A of R × RN is called a nonautonomous set if for all t ∈ R the so-called t-fibers A(t) := {x ∈ RN :(t, x) ∈ A} are nonempty. We call A closed or compact, if all t-fibers are closed or compact, respectively. Finally, a nonautonomous set A is called invariant if λ(t, τ, A(τ)) = A(t) for all t, τ ∈ R . In the literature pullback attractors are usually defined to be compact (see e.g., Kloeden & Keller & Schmalfuß [11]), but the (global) nonautonomous manifolds under consideration here are always noncompact. If we want to establish connections between pullback attractors and nonautonomous manifolds we need a more general notion of pullback attractor. This is prepared by the following definition: an invariant nonautonomous set A is said to be compactly generated if there exists a compact set K ⊂ RN , a so-called generator of A, with the following property: for any compact set C ⊂ RN there exists a number T (K,C) > 0 such that for any τ ∈ R we have λ(τ, τ − t, A(τ − t) ∩ K) ⊃ A(τ) ∩ C for all t > T (K,C) . An invariant, closed and compactly generated nonautonomous set A is called a global pullback attractor if for any compact set C ⊂ X we have lim d(λ(τ, t, C)|A(τ)) = 0 for all τ ∈ R , t→−∞ where d is the Hausdorff semi-distance. A global pullback attractor is always unique. We have proved this result in [1], where we have also introduced algorithms to approximate pullback attractors. 582 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND

3. Theory of nonautonomous manifolds. In the sequel we consider nonautonomous diﬀerential equations of the form

x˙ 1 = B1(t)x1 + F1(t, x1, x2) (2) x˙ 2 = B2(t)x2 + F2(t, x1, x2) , N×N M×M N M N where B1 : R → R , B2 : R → R , F1 : R × R × R → R and N M M F2 : R × R × R → R are continuous functions satisfying F1(t, 0, 0) = 0 and F2(t, 0, 0) = 0 for all t ∈ R. Moreover we assume:

(H1) Hypotheses on linear part: The evolution operators Φ1 and Φ2 of the linear equationsx ˙ 1 = B1(t)x1 andx ˙ 2 = B2(t)x2, respectively, satisfy the estimates α(t−s) kΦ1(t, s)k ≤ Ke for all t ≥ s , β(t−s) kΦ2(t, s)k ≤ Ke for all t ≤ s with real constants K ≥ 1 and α < β . N M (H2) Hypotheses on perturbation: For all (x1, x2), (¯x1, x¯2) ∈ R × R and t ∈ R we have

kF1(t, x1, x2) − F1(t, x¯1, x¯2)k ≤ Lkx1 − x¯1k + Lkx2 − x¯2k ,

kF2(t, x1, x2) − F2(t, x¯1, x¯2)k ≤ Lkx1 − x¯1k + Lkx2 − x¯2k , β−α where the constant L satisﬁes the estimate 0 ≤ L < 4K . We denote the general solution of this system by N M λ(t, τ, ξ, η) = (λ1(t, τ, ξ, η), λ2(t, τ, ξ, η)) ∈ R × R β−α and choose an arbitrary constant δ ∈ (2KL, 2 ]. The theorems in this section are slight modiﬁcations of the results obtained in Aulbach & Wanner [3, 4]. First we state the fundamental existence theorem on nonautonomous manifolds which says that system (2) gives rise to two nonautonomous manifolds, the pseudo-stable manifold S0 and the pseudo-unstable manifold R0. If system (2) is hyperbolic, i.e., α < 0 < β, then S0 and R0 are called stable manifold and unstable manifold, respectively.

Theorem 3.1. There exists a uniquely determined continuous mapping s0 : R × RN → RM whose graph N S0 := {(τ, ξ, s0(τ, ξ)) : τ ∈ R, ξ ∈ R } allows the representation + S0 = {(τ, ξ, η): λ(·, τ, ξ, η) is γ -quasibounded} for every choice of γ ∈ [α + δ, β − δ]. Moreover, there exists a uniquely determined M N continuous mapping r0 : R × R → R whose graph M R0 := {(τ, r0(τ, η), η): τ ∈ R, η ∈ R } allows the representation − R0 = {(τ, ξ, η): λ(·, τ, ξ, η) is γ -quasibounded} for every choice of γ ∈ [α + δ, β − δ]. The nonautonomous sets S0 and R0 are invariant and their intersection is the trivial solution of (2).

We say that S0 and R0 are the nonautonomous manifolds of the trivial solution. Not only the trivial solution but every solution of (2) admits nonautonomous manifolds. This is the central statement of the following theorem. INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS 583

Theorem 3.2. There exists a mapping s : R × RN × R × RN × RM → RM such N M that for every (τ0, ξ0, η0) ∈ R × R × R the graph N Sτ0,ξ0,η0 := {(τ, ξ, s(τ, ξ, τ0, ξ0, η0)) : τ ∈ R, ξ ∈ R } allows the representation + Sτ0,ξ0,η0 = {(τ, ξ, η): λ(·, τ, ξ, η) − λ(·, τ0, ξ0, η0) is γ -quasibounded} for every choice of γ ∈ [α + δ, β − δ] . The function s is continuous and has the following properties: N M (S1) For all τ, τ0 ∈ R, ξ0, ξ1, ξ2 ∈ R and η0 ∈ R we have K2L(δ − KL) ks(τ, ξ , τ , ξ , η ) − s(τ, ξ , τ , ξ , η )k ≤ kξ − ξ k . 1 0 0 0 2 0 0 0 δ(δ − 2KL) 1 2 N M (S2) For all τ, τ0 ∈ R, ξ, ξ0 ∈ R , η0 ∈ R and γ ∈ [α + δ, β − δ] we have + kλ1(·, τ, ξ, s(τ, ξ, τ0, ξ0, η0)) − λ1(·, τ0, ξ0, η0)k ³ ´ τ,γ K3L2 −γτ ≤ K + δ(δ−2KL) kξ − λ1(τ, τ0, ξ0, η0)ke , + kλ2(·, τ, ξ, s(τ, ξ, τ0, ξ0, η0)) − λ2(·, τ0, ξ0, η0)kτ,γ 2 K L(δ−KL) −γτ ≤ δ(δ−2KL) kξ − λ1(τ, τ0, ξ0, η0)ke . Moreover, there exists a mapping r : R × RM × R × RN × RM → RN such that for N M every (τ0, ξ0, η0) ∈ R × R × R the graph M Rτ0,ξ0,η0 := {(τ, r(τ, η, τ0, ξ0, η0), η): τ ∈ R, η ∈ R } allows the representation − Rτ0,ξ0,η0 = {(τ, ξ, η): λ(·, τ, ξ, η) − λ(·, τ0, ξ0, η0) is γ -quasibounded} for every choice of γ ∈ [α + δ, β − δ] . The function r is continuous and has the following properties: N M (R1) For all τ, τ0 ∈ R, ξ0 ∈ R and η0, η1, η2 ∈ R we have K2L(δ − KL) kr(τ, η , τ , ξ , η ) − r(τ, η , τ , ξ , η )k ≤ kη − η k . 1 0 0 0 2 0 0 0 δ(δ − 2KL) 1 2 N M (R2) For all τ, τ0 ∈ R, ξ0 ∈ R , η, η0 ∈ R and γ ∈ [α + δ, β − δ] we have − kλ1(·, τ, r(τ, η, τ0, ξ0, η0), η) − λ1(·, τ0, ξ0, η0)kτ,γ 2 K L(δ−KL) −γτ ≤ δ(δ−2KL) kη − λ2(τ, τ0, ξ0, η0)ke , + kλ2(·, τ, r(τ, η, τ0, ξ0, η0), η) − λ2(·, τ0, ξ0, η0)k ³ ´ τ,γ K3L2 −γτ ≤ K + δ(δ−2KL) kη − λ2(τ, τ0, ξ0, η0)ke .

Remark 3.1. The properties (R1) and (R2) hold also true for the function r0 of Theorem 3.1 instead of r, because of r(·, ·, 0, 0, 0) ≡ r0(·, ·). This fact is needed later in the proof of Theorem 4.1. The nonautonomous set

FH (τ0, η0) := Sτ0,r0(τ0,η0),η0 is called the horizontal ﬁber bundle through (τ0, η0). The next theorem says – provided the constant L is small enough – that every point of the extended phase space lies on exactly one horizontal ﬁber bundle. 584 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND

Theorem 3.3. Suppose that the constant L of system (2) satisfies β − α p 0 ≤ L < (2 + K − 4 + K2) . 4K2 Then there exists a continuous mapping F : R×RN ×RM → RM with the following N M property: every point (τ0, ξ0, η0) ∈ R × R × R lies exactly on one horizontal fiber N bundle, namely FH (τ0, η) with η = F(τ0, ξ0, η0). Furthermore, for all τ ∈ R, ξ ∈ R and η ∈ RM the estimate 1 kF(τ, ξ, η)k ≤ (kξk + kηk) (3) 1 − C(L)2 holds with the constant C(L) defined by

K2L(δ − KL) C(L) := . δ(δ − 2KL)

4. Hyperbolic manifolds and pullback attractors. We suppose now that system (2) is hyperbolic. For simplicity we refer to (2) also by writingx ˙ = f(t, x). In the following theorem we provide suﬃcient conditions in terms of the spectral gap and the constant L in order to establish connections between the two hyperbolic manifolds and pullback attractors.

Theorem 4.1. We suppose that

0 ∈ (α + 2KL, β − 2KL) , (4) and that the constant L satisﬁes β − α p 0 ≤ L < (2 + K − 4 + K2) . 4K2

Then the unstable manifold R0 is the global pullback attractor of (2). Furthermore, we obtain the following connection between the global pullback attractor A of the system under time-reversal

x˙ = −f(−t, x) (5) and the stable manifold S0 of system (2):

A(t) = S0(−t) for all t ∈ R .

Proof. First we show that R0 is the global pullback attractor of system (2). Ob- viously, R0 is an invariant nonautonomous set and graph of the continuous function r0; hence a closed nonautonomous set. Let us prove now that R0 is compactly generated with generator U1(0). For this we choose an arbitrary compact set C ⊂ RN . Since C is bounded, we have an M > 0 with kxk < M for all x ∈ C. β−α Due to (4) there exists a number δ ∈ (2KL, 2 ] with γ := β − δ > 0. Thus we get a T (U1(0),C) > 0 such that for all t > T (U1(0),C) we have µ ¶ K2L(δ − KL) K3L2 + K + Me−γt < 1 . (6) δ(δ − 2KL) δ(δ − 2KL) INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS 585

Choose τ0 ∈ R and (ξ, η) = (r0(τ0, η), η) ∈ C ∩ R0(τ0) arbitrarily. Then for all t > T (U1(0),C) we have

kλ(τ0 − t, τ0, r0(τ0, η), η)k

= kλ1(τ0 − t, τ0, r0(τ0, η), η)k + kλ2(τ0 − t, τ0, r0(τ0, η), η)k µ ¶ (R2) K2L(δ − KL) K3L2 (6) ≤ + K + kηk e−γt < 1 . δ(δ − 2KL) δ(δ − 2KL) |{z}

Thus, for all t > T (U1(0),C) the relation

{(ξ, η)} = λ(τ0, τ0 − t, {λ(τ0 − t, τ0, r0(τ0, η), η)})

⊂ λ(τ0, τ0 − t, U1(0)) is fulﬁlled. Since (ξ, η) ∈ C ∩ R0(τ0) has been chosen arbitrarily we have

C ∩ R0(τ0) ⊂ λ(τ0, τ0 − t, U1(0)) for all t > T (U1(0),C) .

This means that R0 is compactly generated with generator U1(0) . Finally we prove that R0 attracts every compact set in the sense of pullback attraction. Therefore N choose τ0 ∈ R and a compact set C ⊂ R arbitrarily. Due to (4) there exists a β−α number δ ∈ (2KL, 2 ] with γ := α + δ < 0. We see that for all (ξ, η) ∈ C and t ∈ R we have

kr0(t, F(t, ξ, η)) − ξk ≤ kr0(t, F(t, ξ, η))k + kξk (R1) K2L(δ − KL) ≤ kF(t, ξ, η)k + kξk δ(δ − 2KL) (3) K2L(δ − KL) 1 ≤ (kξk + kηk) + kξk ≤ M δ(δ − 2KL) 1 − C(L)2 1 with M1 > 0 since C is bounded. Moreover, for all (ξ, η) ∈ C and t ∈ R the relation

s(t, r0(t, F(t, ξ, η)), t, ξ, η) = F(t, ξ, η) is fulﬁlled since the identity

(t, r0(t, F(t, ξ, η)), F(t, ξ, η)) ∈ St,ξ,η = FH (t, F(t, ξ, η)) holds for all (ξ, η) ∈ C and t ∈ R. Hence, for all (ξ, η) ∈ C and t ≤ τ0

kλ(τ0, t, r0(t, F(t, ξ, η)), F(t, ξ, η)) − λ(τ0, t, ξ, η)k

= kλ1(τ0, t, r0(t, F(t, ξ, η)), s(t, r0(t, F(t, ξ, η)), t, ξ, η)) − λ1(τ0, t, ξ, η)k

+kλ2(τ0, t, r0(t, F(t, ξ, η)), s(t, r0(t, F(t, ξ, η)), t, ξ, η)) − λ2(τ0, t, ξ, η)k µ ¶ (S2) K3L2 ≤ K + kr (t, F(t, ξ, η)) − ξkeγ(τ0−t) δ(δ − 2KL) 0 K2L(δ − KL) + kr (t, F(t, ξ, η)) − ξkeγ(τ0−t) δ(δ − 2KL) 0 µ ¶ K3L2 K2L(δ − KL) ≤ K + + M eγ(τ0−t) =: M eγ(τ0−t) δ(δ − 2KL) δ(δ − 2KL) 1 2 586 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND

γt˜ with M2 > 0. We choose an ² > 0. Then there exists a t˜ > 0 with M2e < ². Thus, for all (ξ, η) ∈ C and t ≥ t˜

d(λ(τ0, τ0 − t, ξ, η),R0(τ0))

≤ kλ(τ0, τ0 − t, r0(τ0 − t, F(τ0 − t, ξ, η)), F(τ0 − t, ξ, η)) − λ(τ0, τ0 − t, ξ, η)k γt ≤ M2e < ² .

This leads immediately to d(λ(τ0, τ0 − t, C)|R0(τ0)) < ² for all t ≥ t˜ and finishes the proof that R0 is the global pullback attractor of (2). The verification of the second assertion of Theorem 4.1 follows from the first one since system (5) fulfills the hypotheses of Section 3, too, and the t-fiber of the unstable manifold of (5) is S0(−t).

The following example shows that not every unstable manifold is a pullback attractor.

Example 4.1. An easy calculation shows that the function F : R2 → R deﬁned by 1 2 1 2 1 3 F (ξ, η) := − 2 ξ + 2 η + 3 ξ is a Hamiltonian of the planar autonomous system x˙ = y y˙ = x − x2 . This implies that the relation W u F (λ(t, 0, ξ, η)) = F (ξ, η) holds for all (ξ, η) ∈ R2 and t ∈ R. Then the global unstable manifold ª W u := {x ∈ R2 : lim λ(t, 0, x) = 0 t→−∞ of the hyperbolic equilibrium (0, 0) is not an attractor of the system: ﬁrst, for all x ∈ W u F (x) = lim F (λ(t, 0, x)) t→−∞ Figure 1. Homoclinic orbit as a = F ( lim λ(t, 0, x)) part of the unstable manifold t→−∞ = F (0) = 0 holds. An analysis of the function F shows then that a homoclinic orbit is part of the unstable manifold W u (see Figure 1). Due to phase plane analysis, it is clear that for every point x in the interior of the homoclinic orbit we have ¡ lim inf d λ(t, 0, {x})|W u) > 0 . t→∞ This means that the set {x} is not attracted by W u.

We consider now a numerical application for Theorem 4.1.

Example 4.2. The nonautonomous diﬀerential equation

x˙ = −x + 0.38 cos(t)(sin(x) + sin(y)) y˙ = y + 0.38 sin(t) sin(x) satisﬁes the assumptions of Section 3 with the constants n = 2,K = 1, α = −1, β = 1 and L0.38. INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS 587

R0 R0

S0 S0

Figure 2. Fiber t = −2 Figure 3. Fiber t = 2 √ β−α 2 Due to L < 0.382 ≈ 4K2 (2+ K − 4 + K ) the conditions of Theorem 4.1 are fulfilled. Since we are able to approximate pullback attractors (see Aulbach & Rasmussen & Siegmund [1]), Theorem 4.1 enables us to compute the stable and unstable manifold of this system: Figure 2 and Figure 3 visualize the −2- and 2-fibers in the square [−2, 2] × [−2, 2] of the stable and unstable manifold. In Figure 4 we have plotted the nonautonomous manifolds in the extended phase space in the cuboid [−2, 2] × [−2, 2] × [−π, π]. As one can see, the −π- and the π- fibers of the manifolds are identical. This is no coincidence, since in Aulbach & Wanner Figure 4. Manifolds in [3, Corollary 4.4] it is proved that the pe- the extended phase space riodicity of the differential equation carries over to the manifolds.

5. Nonhyperbolic manifolds and pullback attractors. In this section we generalize the results of the previous one by omitting the assumption of hyperbolicity. We use spectral transformations to relate this situation to the hyperbolic case. Theorem 5.1. We suppose that there exists a real constant c with 0 ∈ (c + α + 2KL, c + β − 2KL) , and that the constant L satisﬁes β − α p 0 ≤ L < (2 + K − 4 + K2) . 4K2

c(t−t0) We use the spectral transformation y = e x, t0 ∈ R , to transform (2) into

y˙ = cy + ec(t−t0)f(t, e−c(t−t0)y) . (7)

Then the following connection holds between the global pullback attractor A1 of system (7) and the pseudo-unstable manifold R0 of system (2):

c(t−t0) A1(t) = e R0(t) for all t ∈ R . 588 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND

In addition, we have the following connection between the global pullback attractor A2 of the time-reversed version of (7), y˙ = −cy − e−c(t+t0)f(−t, ee(t+t0)y) , (8) and the pseudo-stable manifold S0 of system (2):

−c(t+t0) A2(t) = e S0(−t) for all t ∈ R . Proof. We can write system (7) in the following form:

c(t−t0) −c(t−t0) −c(t−t0) x˙1 = (B1(t) + c1)x1 + e F1(t, e x1, e x2)

c(t−t0) −c(t−t0) −c(t−t0) x˙2 = (B2(t) + c1)x2 + e F2(t, e x1, e x2) .

This system fulﬁlls the assumptions of Section 3, since the evolution operators Φ˜ 1 and Φ˜ 2 ofx ˙ 1 = (B1(t) + c1)x1 andx ˙ 2 = (B2(t) + c1)x2, respectively, satisfy the estimates c(t−s) (c+α)(t−s) kΦ˜ 1(t, s)k = ke Φ1(t, s)k ≤ Ke for all t ≥ s , c(t−s) (c+β)(t−s) kΦ˜ 2(t, s)k = ke Φ2(t, s)k ≤ Ke for all t ≤ s , N M and for all t ∈ R and (x1, x2), (¯x1, x¯2) ∈ R × R we have

c(t−t0) c(t0−t) c(t0−t) c(t0−t) c(t0−t) ke (F1(t, e x1, e x2) − F1(t, e x¯1, e x¯2))k

≤ Lkx1 − x¯1k + Lkx2 − x¯2k,

c(t−t0) c(t0−t) c(t0−t) c(t0−t) c(t0−t) ke (F2(t, e x1, e x2) − F2(t, e x¯1, e x¯2))k

≤ Lkx1 − x¯1k + Lkx2 − x¯2k, β−α where the constant L satisfies the estimate 0 ≤ L < 4K . We can apply Theorem 4.1 because of 0 ∈ (c+α+2KL, c+β −2KL). The unstable manifold R˜0 of system (7) is therefore identical with the global pullback attractor of system (7). To prove the first assertion of the theorem it is thus sufficient to show that

c(t−t0) R˜0(t) = e R0(t) for all t ∈ R . An easy calculation yields that t 7→ µ(t) ∈ RN+M is a solution of (2) if and only if t 7→ ec(t−t0)µ(t) ∈ RN+M is a solution of (7). Hence, for every t ∈ R we have

− N+M x ∈ R˜0(t) ⇔ There exists a 0 -quasibounded solutionµ ˜ : R → R of (7) withµ ˜(t) = x . ⇔ There exists a solutionµ ˜ : R → RN+M of (7) withµ ˜(t) = x, and ec(t0− ·)µ˜(·) is (−c)−-quasibounded . ⇔ There exists a (−c)−-quasibounded solution µ : R → RN+M of (2) with µ(t) = ec(t0−t)x .

c(t0−t) ⇔ e x ∈ R0(t) .

c(t−t0) ⇔ x ∈ e R0(t) .

Analogously one shows that for the stable manifold S˜0 of system (7) the relation

c(t−t0) S˜0(t) = e S0(t) for all t ∈ R is fulﬁlled. Theorem 4.1 implies that the pullback attractor A2 of system (8) satisﬁes

c(−t−t0) A2(t) = S˜0(−t) = e S0(−t) for all t ∈ R , INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS 589 if system (8) is the time-reversal of (7) as introduced in Theorem 4.1. An obvious calculation shows that this is indeed true, so all assertions of the theorem are proved. Remark 5.1. • Specializing c = 0 we see that Theorem 5.1 is a generalization of Theorem 4.1. • Systems (7) and (8) are only suitable for the approximation of the t0-ﬁber of the manifolds R0 and S0, respectively, because simple examples show that an exponential transformation of the numerical covering of the pullback attractors A1 or A2, respectively, leads to useless results (see Rasmussen [14]). • Even for the approximation of pseudo-stable or pseudo-unstable manifolds of autonomous systems the nonautonomous theory is essential. This is because the spectral transformation always yields a nonautonomous system.

6. Generalizations. In this section we generalize the situation of Section 3 by first considering systems with coupled linearization and then systems which give rise to a hierarchy of nonautonomous manifolds. 6.1. Systems with coupled linearization. We consider differential equations of the form µ ¶ µ ¶ µ ¶ x˙ x F (t, x , x ) 1 = B(t) 1 + 1 1 2 (9) x˙ 2 x2 F2(t, x1, x2) (N+M)×(N+M) N M N with continuous functions B : R → R , F1 : R × R × R → R and N M M F2 : R × R × R → R satisfying F1(t, 0, 0) = 0 and F2(t, 0, 0) = 0 for all t ∈ R. Moreover we assume: (H1)ˆ Hypotheses on linear part: The linear systemx ˙ = B(t)x admits a nonhyperbolic exponential dichotomy, i.e., there exists an invariant projector P : R → R(N+M)×(N+M) with rank P = N such that the evolution operator Φ ofx ˙ = B(t)x satisfies the estimates kΦ(t, s)P (s)k ≤ Keα(t−s) for all t ≥ s , kΦ(t, s)[1 − P (s)]k ≤ Keβ(t−s) for all t ≤ s with real constants K ≥ 1 and α < β . We emphasize here that we do not assume 0 to be between α and β. N M (H2)ˆ Hypotheses on perturbation: For all t ∈ R and (x1, x2), (¯x1, x¯2) ∈ R × R we have

kF1(t, x1, x2) − F1(t, x¯1, x¯2)k ≤ Lkx1 − x¯1k + Lkx2 − x¯2k ,

kF2(t, x1, x2) − F2(t, x¯1, x¯2)k ≤ Lkx1 − x¯1k + Lkx2 − x¯2k , β−α where the constant L satisﬁes the estimate 0 ≤ L < 16K3 . We denote the general solution of this system by λ. To decouple the linear equationx ˙ = B(t)x we extend a lemma from Siegmund [16] (see also Coppel [5]). Lemma 6.1. There exists a function S : R → R(N+M)×(N+M) of invertible matrices such that the linear system x˙ = B(t)x can be decoupled via the transformation y = S(t)x into µ ¶ B (t) 0 y˙ = (S˙(t)S−1(t) + S(t)B(t)S−1(t))y = 1 y 0 B2(t) 590 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND

N×N M×M with continuous functions B1 : R → R and B2 : R → R . Furthermore, the estimates √ √ kS(t)k ≤ 2 K and kS−1(t)k ≤ 2 for all t ∈ R hold. The evolution operators Ψ1 and Ψ2 of the linear systems y˙1 = B1(t)y1 and y˙2 = B2(t)y2, respectively, satisfy 2 α(t−s) kΨ1(t, s)k ≤ 2K e for all t ≥ s , 2 β(t−s) kΨ2(t, s)k ≤ 2K e for all t ≤ s . Proof. In Siegmund [16, Lemma 3.1] all assertions are proved, except for the estimates for the evolution operators. It is also shown that µ ¶ 1 0 S(t)P (t)S−1(t) = N1×N1 N1×N2 for all t ∈ R . 0N2×N1 0N2×N2 Thus, for all t ≥ s we have °µ ¶ ° ° ° Ψ1(t, s) 0 −1 kΨ1(t, s)k = ° S(s)P (s)S (s)° ° 0 Ψ2(t, s) ° ° µ ¶ ° ° Ψ (t, s) 0 ° = °S(t)S−1(t) 1 S(s)P (s)S−1(s)° ° 0 Ψ2(t, s) ° = kS(t)Φ(t, s)P (s)S−1(s)k ≤ kS(t)k kΦ(t, s)P (s)k kS−1(s)k ≤ 2K2eα(t−s) . 2 β(t−s) Analogously one shows kΨ2(t, s)k ≤ 2K e for all t ≤ s . This completes the proof of the lemma.

We now apply the transformation y = S(t)x to the nonlinear system (9) and get (1) −1 y˙1 = B1(t)y1 + S (t)F1(t, S (t)y) (10) (2) −1 y˙2 = B2(t)y2 + S (t)F2(t, S (t)y) , where we have denoted the ﬁrst N rows of S(t) by S(1)(t) and the last M rows by S(2)(t). We denote the general solution of (10) by λ˜. It is easy to show that this system fulﬁlls the assumptions of Section 3 with the constants 2K2, α, β and 2KL. Hence, the existence of nonautonomous manifolds S˜0 and R˜0 is guaranteed by The- 3 β−α orem 3.1. Fixing a constant δ ∈ (8K L, 2 ] these manifolds can be characterized for every γ ∈ [α + δ, β − δ] as follows: ˜ + S˜0 = {(τ, ξ): λ(·, τ, ξ) is γ -quasibounded} , ˜ − R˜0 = {(τ, ξ): λ(·, τ, ξ) is γ -quasibounded} .

An obvious calculation yields that the sets S0 and R0, deﬁned by −1 −1 S0(t) := S (t)S˜0(t) and R0(t) := S (t)R˜0(t) for all t ∈ R , are nonautonomous manifolds of (9) with + S0 = {(τ, ξ): λ(·, τ, ξ) is γ -quasibounded} , − ª R0 = {(τ, ξ): λ(·, τ, ξ) is γ -quasibounded for every γ ∈ [α + δ, β − δ]. We now aim at the relationship between pullback attractors and the manifolds S0 and R0. Therefore we study the transformation behavior of pullback attractors. INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS 591

Lemma 6.2. Let T : R → Rn×n be a differentiable function of invertible matrices −1 with supt∈R max{kT (t)k, kT (t)k} ≤ Γ for some Γ > 0. We consider a differential equation x˙ = g(t, x) (11) with a function g : R × Rn → Rn fulfilling conditions ensuring the global existence and uniqueness of solutions. The transformation y = T (t)x yields the differential equation y˙ = T˙ (t)T −1(t)y + T (t)g(t, T −1(t)y) . (12) Then (11) has a global pullback attractor if and only if (12) has a global pullback attractor. If A is the global pullback attractor of (11) and A˜ the global pullback attractor of (12), then A˜(t) = T (t)A(t) for all t ∈ R . Proof. It suffices to prove one of the two implications since (11) is the result of the transformation x = T −1(t)y of system (12). An obvious calculation shows that the general solution of (12) is given by µ˜(t, τ, ξ) = T (t)µ(t, τ, T −1(τ)ξ) for all t, τ ∈ R and ξ ∈ Rn , provided that µ is the general solution of (11). Next we prove that the nonautonomous set A˜ defined by A˜(t) := T (t)A(t) for all t ∈ R is the global pullback attractor of (12) if (11) has the global pullback attractor A. The invariance of A˜ with respect to system (12) is a consequence of the special transformation and the invariance of A with respect to (11). A˜ is also closed since the linear mapping T (t): Rn → Rn is a homeomorphism for all t ∈ R. Let K be a generator of A. Since K is bounded, there exists an M1 > 0 with kxk < M1 for ˜ all x ∈ K. We show now that the set UΓM1 (0) is a generator of A. Therefore, we choose a compact set C ⊂ Rn arbitrarily. Due to the boundedness of C there exists an M2 > 0 with kxk < M2 for all x ∈ C. Since A is compactly generated (with respect to (11)) there exists a number T (K, UΓM2 (0)) > 0 such that for all τ ∈ R and all t > T (K, UΓM2 (0)) we get

µ(τ, τ − t, A(τ − t) ∩ K) ⊃ A(τ) ∩ UΓM2 (0) .

Thus, for all τ ∈ R and all t > T (K, UΓM2 (0)) we have µ˜(τ, τ − t, A˜(τ − t) ∩ U (0)) ⊃ µ˜(τ, τ − t, T (τ − t)(A(τ − t) ∩ K)) ΓM1 ¡ ¢ = T (τ)µ τ, τ − t, A(τ − t) ∩ K

⊃ T (τ)(A(τ) ∩ UΓM2 (0)) ⊃ T (τ)A(τ) ∩ C = A˜(τ) ∩ C. This means that A˜ is compactly generated with respect to (12). Completing this proof we show that every compact set C ⊂ Rn is attracted by A˜. Since C is bounded there is an M3 > 0 such that kxk < M3 for all x ∈ C. Then, for all τ ∈ R we get d(˜µ(τ, τ − t, C)|A˜(τ)) ≤ Γd(T −1(τ)˜µ(τ, τ − t, C)|T −1(τ)A˜(τ)) ¡ = Γd µ(τ, τ − t, T −1(τ − t)C)|A(τ)) ¡ ≤ Γd µ(τ, τ − t, UΓM3 (0))|A(τ)) . 592 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND

Letting t tend to inﬁnity the assertion follows and the lemma is proved.

For simplicity we refer to system (9) by also writingx ˙ = f(t, x). The analogue of Theorem 5.1 is then given as follows. Theorem 6.1. We suppose that there exists a real constant c such that 0 ∈ (c + α + 8K3L, c + β − 8K3L) and that the constant L satisﬁes β − α¡ p ¢ L < 2 + 2K2 − 2 1 + K4 . (13) 32K5

c(t−t0) We use the spectral transformation y = e x, t0 ∈ R , to transform (9) into y˙ = cy + ec(t−t0)f(t, e−c(t−t0)y) . (14)

Then the following connection holds between the global pullback attractor A1 of system (14) and the pseudo-unstable manifold R0 of system (9):

c(t−t0) A1(t) = e R0(t) for all t ∈ R . In addition, we have the following connection between the global pullback attractor A2 of the time-reversed version of (14), y˙ = −cy − e−c(t+t0)f(−t, ec(t+t0)y) , and the pseudo-stable manifold S0 of system (9):

−c(t+t0) A2(t) = e S0(−t) for all t ∈ R . √ β−α 2 4 Proof. From (13) it follows that 2KL < 4(2K2)2 (2 + 2K − 4 + 4K ). Thus, Theorem 5.1 can be applied to system (10). Therefore, the global pullback attractor A˜ of x˙ = S˙(t)S−1(t)x + ec(t−t0)S(t)f(t, ec(t0−t)S−1(t)x) (15) satisﬁes

c(t−t0) A˜(t) = e R˜0(t) for all t ∈ R . Since system (14) is obtained by the transformation y = S−1(t)x from (15), Lemma 6.2 implies

−1 c(t−t0) −1 c(t−t0) A1(t) = S (t)A˜(t) = e S (t)R˜0(t) = e R0(t) for all t ∈ R .

The corresponding assertion for the pullback attractor A2 can be proved similarly.

6.2. Hierarchies of nonautonomous manifolds. In this section we generalize our situation by considering the system

x˙ 1 = B1(t)x1 + F1(t, x1, x2, . . . , xn)

x˙ 2 = B2(t)x2 + F2(t, x1, x2, . . . , xn) . (16) .

x˙ n = Bn(t)xn + Fn(t, x1, x2, . . . , xn) ,

Ni×Ni N Ni where Bi : R → R , Fi : R×R → R (i = 1, . . . , n) are continuous mappings with Fi(t, 0, 0) = 0 for all t ∈ R (N = N1 + ··· + Nn). Moreover we assume: INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS 593

(H1) Hypotheses on linear part: There exist real constants K ≥ 1 and αi < βi (i1, . . . , n − 1) with βi ≤ αi+1 (i = 1, . . . , n − 2) such that the evolution operators Φi of the linear equationsx ˙ i = Bi(t)xi (i = 1, . . . , n) satisfy the estimates

αi(t−s) kΦi(t, s)k ≤ Ke for all t ≥ s ,

βi(t−s) kΦi+1(t, s)k ≤ Ke for all t ≤ s for all i = 1, . . . , n − 1. N (H2) Hypotheses on perturbation: For all x = (x1, . . . , xn), x¯ = (¯x1,..., x¯n) ∈ R , t ∈ R and i ∈ {1, . . . , n} we have Xn kFi(t, x1, . . . , xn) − Fi(t, x¯1,..., x¯n)k ≤ L kxj − x¯jk = Lkx − x¯k , j=1 where the constant L satisﬁes the estimate ½ ¾ β − α 0 ≤ L < min i i : i = 1, . . . , n − 1 . 4K(n − 1) We denote the general solution of (16) by λ and choose for i = 1, . . . , n − 1 βi−αi constants δi ∈ (2KL(n − 1), 2 ].

Theorem 6.2. There exist nonautonomous manifolds Wi,j (1 ≤ i ≤ j ≤ n), the so- called hierarchy of nonautonomous manifolds, with the following characterizations: N • W1,n = R × R . • For all i ∈ {1, . . . , n − 1} and γ ∈ [αi + δi, βi − δi] we have + W1,i = {(τ, ξ): λ(·, τ, ξ) is γ -quasibounded} , − Wi+1,n = {(τ, ξ): λ(·, τ, ξ) is γ -quasibounded} .

• For all 1 < i ≤ j < n we have Wi,j = W1,j ∩ Wi,n. Hence, for every choice of γ1 ∈ [αj + δj, βj − δj] and γ2 ∈ [αi−1 + δi−1, βi−1 − δi−1] the relation + − Wi,j = {(τ, ξ): λ(·, τ, ξ) is γ1 - and γ2 -quasibounded} is fulﬁlled. Proof. See Aulbach & Wanner [3, Theorem 5.1].

The following diagram visualizes the relations between the manifolds Wi,j of the hierarchy:

W1,1 ⊂ W1,2 ⊂ · · · ⊂ W1,n−1 ⊂ W1,n ∪ ∪ ∪ W2,2 ⊂ · · · ⊂ W2,n−1 ⊂ W2,n ∪ ∪ ...... Wn−1,n−1 ⊂ Wn−1,n ∪ Wn,n We state now the version of Theorem 5.1 which applies to system (16). For simplicity we refer to (16) by also writingx ˙ = f(t, x). 594 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND

Theorem 6.3. We suppose that there exists an i ∈ {1, .., n − 1} and a c > 0 with

0 ∈ (c + αi + 2KL(n − 1), c + βi − 2KL(n − 1)) , and that the constant L satisﬁes β − α p 0 ≤ L < i i (2 + K − 4 + K2) . 4K2(n − 1)

c(t−t0) We use the spectral transformation y = e x, t0 ∈ R, to transform (16) into y˙ = cy + ec(t−t0)f(t, e−c(t−t0)y) . (17)

Then the following connection holds between the global pullback attractor A1 of system (17) and the manifold Wi+1,n of system (16):

c(t−t0) A1(t) = e Wi+1,n(t) for all t ∈ R . In addition, we have the following connection between the global pullback attractor A2 of the time-reversed version of (17), y˙ = −cy − e−c(t+t0)f(−t, ee(t+t0)y) , and the manifold W1,i of system (16):

−c(t+t0) A2(t) = e W1,i(−t) for all t ∈ R . Proof. This result follows directly from Theorem 5.1 by putting the components 1, . . . , i and i + 1, . . . , n together. Remark 6.1. Provided the constant L of system (16) is small enough, every manifold of the hierarchy can be approximated numerically. For the manifolds W1,i and Wi+1,n, i = 1, . . . , n − 1, this follows from Theorem 6.3. The manifolds Wi,j for 1 < i ≤ j < n can be obtained by intersecting W1,j and Wi,n (see Theorem 6.2). Remark 6.2. In Section 6.1 the coupled linear part of system (9) was “decoupled” by means of the assumed exponential dichotomy. Using the Sacker-Sell or dichotomy spectrum (see [15]) we can apply the results in [16] to transform x˙ = B(t)x+F (t, x) to (16) with decoupled linear part where each component corresponds to a spectral interval of x˙ = B(t)x.

7. Approximation of a strong stable manifold of the Lorenz system. In 1963, the meteorologist Lorenz considered the three dimensional autonomous system x˙ = σ(y − x) y˙ = ρx − y − xz z˙ = −βz + xy .

For valuesp β > 0 andp ρ > 1 this system has thep three equilibriap r1 := (0, 0, 0), r2 := ( β(ρ − 1), β(ρ − 1), ρ − 1) and r3 := (− β(ρ − 1), − β(ρ − 1), ρ − 1) (see Guckenheimer & Holmes [9, Chapter 2.3]). The linearization in the equilibrium r1 is given by   −σ σ 0 D =  ρ −1 0  . 0 0 −β 8 For the parameter values suggested by Lorenz (σ = 10, ρ28 and β = 3 ) the matrix D has the eigenvalues λ1 ≈ −22.8, λ2 ≈ −2.7 and λ3 ≈ 11.8. Thus, the stable INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS 595

Figure 5. Stable and strong stable manifold of the origin from two diﬀerent perspectives

manifold of the origin has dimension two. Because of λ1 < λ2, there also exists a strong stable manifold corresponding to the eigenvalue λ1. In order to compute the stable and the strong stable manifold in the cuboid [−25, 25]3 we used the subdivision algorithm for the approximation of the pullback attractor of the time- reversed as well as the time-reversed and spectral-transformed system (c = 3). Since the Lorenz system does not fulﬁll the global assumptions of our theorems it is not (rigorously) guaranteed that the outcome of our computations – shown in Figure 5 – precisely represent the manifolds under consideration. However, using the ideas elaborated in this paper local versions of our theorems can also be derived.

It is worth mentioning that—with our approach—the approximation of the strong stable manifold is not possible in a purely autonomous setting. This is due to the fact that the spectral transformation always yields a nonautonomous system. 596 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND

Acknowledgements. The computations have been carried out with an extended version of the computer program GAIO (Global Analysis of Invariant Objects). This software package was developed in 1995 by Michael Dellnitz and Oliver Junge. The most important commands as well as some implemented algorithms can be found in Dellnitz & Froyland & Junge [8]. The two dimensional graphics in this paper have been created with the computer program Matlab. For the three dimensional pictures we used Grape from the research group ”Nichtlineare Partielle Diﬀerentialgleichungen” of the University of Bonn.

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