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

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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 nonau- tonomous di®erential equations and pullback attractors. This relationship is essential, e.g., for the numerical approximation of these manifolds. In the ¯rst step, we show that the unstable manifold is the pullback attractor of the di®er- ential 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 di®eren- tial equations) the stable and unstable manifold theorem|¯rst proved by Poincar¶e [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)f¸1; : : : ; ¸rg [ f¸r+1; : : : ; ¸ng consists of eigenvalues ¸1; : : : ; ¸r with negative real parts and ¸r+1; : : : ; ¸n 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 = fx0 : x(t; x0) ! 0 as t ! 1g and W = fx0 : x(t; x0) ! 0 as t ! ¡1g 2000 Mathematics Subject Classi¯cation. 34C30, 34D45, 37B55, 37D10, 65L20. Key words and phrases. Nonautonomous di®erential 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 ¸1 < ¸2 < 0; the unstable manifold therefore has dimension 2 whereas the strong stable manifold (consisting of solutions with decay rate ¸1) 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 di®erential 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 di®erent methods for computing (un)stable manifolds of vector ¯elds 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 mani- folds of a so-called hierarchy of invariant manifolds which are nonlinear analogues of the eigenspaces of the linearization, including strong stable manifolds and non- hyperbolic 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 de¯ne 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 := jxij for all (x1; : : : ; xN ) 2 R i=1 N N the R is a normed vector space. We write U²(x0)fx 2 R : kx ¡ x0k < ²g for N N the ²-neighborhood of a point x0 2 R . For arbitrary nonempty sets A; B ½ R and x 2 RN let d(x; A) : inffkx ¡ yk : y 2 Ag be the distance of x to A and d(AjB) := supfd(x; B): x 2 Ag be the Hausdor® semi-distance of A and B.A INVARIANT MANIFOLDS AS PULLBACK ATTRACTORS 581 function g : R ! RN is called γ+-quasibounded if + ¡γt kgk¿,γ := supfkg(t)ke : t ¸ ¿g < 1 for some ¿ 2 R . Accordingly, the function g is called γ¡-quasibounded if ¡ ¡γt kgk¿,γ := supfkg(t)ke : t · ¿g < 1 for some ¿ 2 R . In this article we are concerned with nonautonomous di®erential equations x_ = f(t; x) ; where f : R £ RN ! RN is a function which ful¯lls conditions ensuring global existence and uniqueness of solutions. This equation then gives rise to a general solution ¸(t; ¿; ») which is the solution satisfying the initial condition x(¿) = » with ¿ and » treated as additional parameters. For a nonautonomous linear di®erential equation x_ = B(t)x (1) with a continuous function B : R ! RN£N the general solution is linear with respect to », i.e., there exists a function © : R £ R ! RN£N , the so-called evolution operator of (1), such that ¸(t; ¿; ») = ©(t; ¿)» for all t; ¿ 2 R and » 2 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 2 R ; P (t)©(t; s) = ©(t; s)P (s) for all t; s 2 R : A subset A of R £ RN is called a nonautonomous set if for all t 2 R the so-called t-¯bers A(t) := fx 2 RN :(t; x) 2 Ag are nonempty. We call A closed or compact, if all t-¯bers are closed or compact, respectively. Finally, a nonautonomous set A is called invariant if ¸(t; ¿; A(¿)) = A(t) for all t; ¿ 2 R . In the literature pullback attractors are usually de¯ned to be compact (see e.g., Kloeden & Keller & Schmalfu¼ [11]), but the (global) nonautonomous manifolds under consideration here are always noncompact. If we want to establish connec- tions between pullback attractors and nonautonomous manifolds we need a more general notion of pullback attractor. This is prepared by the following de¯nition: an invariant nonautonomous set A is said to be compactly generated if there exists a compact set K ½ RN , a so-called generator of A, with the following property: for any compact set C ½ RN there exists a number T (K; C) > 0 such that for any ¿ 2 R we have ¸(¿; ¿ ¡ t; A(¿ ¡ t) \ K) ⊃ A(¿) \ 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 ½ X we have lim d(¸(¿; t; C)jA(¿)) = 0 for all ¿ 2 R ; t!¡1 where d is the Hausdor® semi-distance. A global pullback attractor is always unique. We have proved this result in [1], where we have also introduced algo- rithms to approximate pullback attractors. 582 B. AULBACH, M. RASMUSSEN AND S. SIEGMUND 3. Theory of nonautonomous manifolds. In the sequel we consider nonau- tonomous di®erential 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 2 R. Moreover we assume: (H1) Hypotheses on linear part: The evolution operators ©1 and ©2 of the linear equationsx _ 1 = B1(t)x1 andx _ 2 = B2(t)x2, respectively, satisfy the estimates ®(t¡s) k©1(t; s)k · Ke for all t ¸ s ; ¯(t¡s) k©2(t; s)k · Ke for all t · s with real constants K ¸ 1 and ® < ¯ .
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