The inflation of attractors and their discretization: the autonomous case

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1 The inflation of attractors and their discretization: the autonomous case P. E. Kloeden Fachbereich Mathematik, Johann Wolfgang Goethe Universität, D-654 Frankfurt am Main, Germany V.S. Kozyakin, Institute for Information Transmission Problems, Russian Academy of Sciences 19 Bolshoi Karetny Lane, Moscow, Russia AMS Subject Classification: 34A5, 34D45, 54C6 Key words: maximal attractor, inflated attractor, setvalued differential equation, perturbations, discretization. Dedictated to Professor V. Lakshmikantham on the occasion of his 75th birthday 1 Introduction The effects of perturbation or discretization on a maximal attractor of autonomous dynamical systems is now well understood, with the perturbed or discretized system having a nearby maximal attractor [5, 7, 1]. In contrast, for nonautonomous systems the very concept of a nonautonomous attractor itself is still undergoing intensive development and perturbation or discretization results have been obtained This work was supported by the DFG Forschungschwerpunkt Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme. The author was partially supported by the Russian Foundation for Basic Researches Grant

2 only for specific cases or under very restricted assumptions, see for example [2, 8, 9] and the references therein. There are also some practical problems for specific autonomous dynamical systems and types of perturbation or discretization processes due to the fact that the convergence of the perturbed or discretized attractor to that of the original attractor is usually only upper semicontinuous convergence, which means, roughly speaking, that the attractor may collapse under pertubation or discretization. While this is not unexpected for systems with parameter values near a bifurcation parameter, it does raise concerns about the faithfulness of numerical approximations. Of course, this collapse may not occur for another numerical scheme or step size or under appropriate assumptions about the dynamics inside the original attractor, but such details may not be known or confirmable in practice. In this paper we introduce the concept of inflation of an attractor, which will give us a larger positively invariant set for the original system that contains the maximal attractors of all possible autonomous perturbations and discretizations of a certain size, as well as even the attracting objects of corresponding nonautonomous perturbations and discretizations as described in [8, 9]. It also includes the effects of roundoff error and converges continuously to the original attractor. The existence of such an inflated attractor follows as in [7] from the uniform asymptotic stability of the original maximal attractor. While the details and underlying ideas are not really new in the autonomous case considered here, the interpretation is new and is more transparent here than in the technically more difficult nonautonomous case, to be treated in another paper [6], where it provides a means for handling the nonuniformities that are intrinsic in nonautonomous systems. The paper is organised as follows. Preliminary and background results are presented in the next section. Inflated dynamics are introduced in Section 3 in terms of a differential inclusion expanding the vector field of the given autonomous dynamical system. The existence and continuous convergence of a maxmal attractor, the inflated attractor, for this inflated system is formulated as Theorem 2 in Section 4 and its interpretation discussed. Proofs of the Theorem and an ancilliary proposition are then presented in the remaining two sections of the paper. 2 Maximal attractors Under assumptions assuring global uniqueness and extendability the solution x(t) = x(t; x ) of an autonomous differential equation dx dt = f(x), (1) 2

3 with initial value x(; x ) = x generates a continuous time semigroup φ = {φ t } t R + on R d defined by φ t (x ) := x(t; x ) for each t and x R d. We will assume that f in (1) satisfies a uniform Lipschitz condition on R d, i.e. there exists a constant K > such that f(x) f(y) K x y for all x, y R d, (2) and that the semidynamical system φ generated by (1) has a maximal attractor, by which we mean a nonempty compact subset A of R d which is φ invariant, i.e. with φ t (A ) = A for all t, and attracting, i.e. with lim t H ( φ t (D), A ) = for any bounded subset D R d. Here the Hausdorff separation H (A, B) of nonempty compact subsets A and B of R d is defined as H (A, B) := max dist(a, B) = max a A min a A b B a b, and H(A, B) = max {H (A, B), H (B, A)} is a metric, called the Hausdorff metric, on the space of nonempty compact subsets of R d. The existence of a maximal attractor follows from that of more easily found absorbing sets. A positively invariant compact subset B of R d, i.e. with φ t (B) B for all t, is called an absorbing set for the semidynamical system φ on R d if for every bounded subset D of R d there exists a t D R + such that φ t (D) B for all t t D. The maximal attractor is then given uniquely by A = t φ t (B). A maximal attractor is uniformly asymptotically stable [1]. Yoshizawa [12] gives various necessary and sufficient conditions involving Lyapunov functions for an arbitrarily shaped compact set A to be uniformly asymptotically stable for a differential equation (1); e.g. see Theorem 3 below. Such Lyapunov functions are a very convenient tool that does not require explicit knowledge of the solutions of the differential equation. For example, in Kloeden and Lorenz [7] a Lyapunov function for the maximal attractor was used to construct an absorbing set, hence to establish the existence of a maximal numerical attractor, for a discrete time semidynamical system corresponding to a numerical discretization of the differential equation (1). A similar construction will be used in this paper. 3

4 3 Inflated dynamics We shall inflate of the differential equation (1) to a setvalued differential equation (see, e.g. [1]) defined in terms a setvalued mappng formed by a closed ball about the function f in (1) at each point. Specifically, for ɛ > we define the ɛ inflated setvalued mapping F ɛ of f by F ɛ (x) := {y R d : y f(x) ɛ}. Let N ɛ [A] = {x R d : dist(x, A) ɛ} denote the closed ɛ neighbourhood of a nonempty compact subset A of R d. Then F ɛ (x) = N ɛ [{f(x)}] = {f(x)} + N ɛ [{}]. The ɛ inflated system corresponding to the differential equation (1) is thus the setvalued differential equation dx dt F ɛ(x). (3) The set F ɛ (x) is obviously nonempty, compact and convex. The mapping x F ɛ (x) also satisfies a uniform Lipschitz condition on R d with the same Lipschitz constant as in (2) for the function f. To see this we use Propostion from [3] to obtain H (F ɛ (x), F ɛ (y)) = H ({f(x)} + N ɛ [{}], {f(y)} + N ɛ [{}]) = H ({f(x)}, {f(y)}) = f(x) f(y) K x y. These properties ensure the maximal existence (see [1]) of an absolutely continuous solutions with initial value x() = x satisfying x(t) {x } + F ɛ (x(s)) ds for all t. Moreover, the attainability set Φ ɛ t(x ) formed by all such solutions is a nonempty compact connected subset of R d and generates a setvalued semigroup [11] on R d, i.e. satisfying Φ ɛ (x ) = {x }, Φ ɛ s (Φ ɛ t(x )) = Φ ɛ s+t(x ) for all s, t, x R d, for which the setvalued mapping (t, x ) Φt(x ɛ ) is continuous with respect to the Hausdorff metric. In fact, here the setvalued mapping (ɛ, t, x ) Φt(x ɛ ) is also continuous with φ t (x ) = ɛ> Φ ɛ t(x ) for all t and x R d. The following estimate will be proved in Section 5. Proposition 1 H ({φ τ (x )}, Φ ɛ τ(x )) K 1 e Kτ τɛ for all τ. 4

5 4 Inflated attractors and their interpretation Our main result is to show that the setvalued inflated semidynamical system has a maximal attractor that contains and converges to the maximal attractor of the original singlevalued dynamical system. Theorem 2 Suppose that f : R d R d satisfies the uniformly Lipschitz condition (2) and that the semidynamical system φ has a maximal attractor A. Then for every ɛ > the ɛ inflated semidynamical system Φ ɛ has a maximal attractor A ɛ such that A A ɛ and lim H ( A ɛ, A ) =. ɛ We call A ɛ the ɛ inflated attractor of the original singlevalued semidynamical system φ generated by the differential equation (1). From proof of Theorem 2, to be given in Section 6, we will see that A ɛ is in fact a positively invariant absorbing set for the singlevalued semidynamical system φ. For the remainder of this section we wish to provide an interpretation for A ɛ and to discuss its usefulness in connection with perturbed or numerical approximated dynamics of the original system φ and its maximal attractor A. A pth order one-step numerical scheme with constant step size h > [1] x n+1 = x n + hf (h, x n ) (4) applied to the differential equation (1) generates a discrete time semidynamical system, which, as shown in [7], has a maximal attractor A h num that converges upper semicontinuously to A, i.e. H (A h num, A ) as h. In general (see [1]), the Hausdorff separation H here cannot be replaced by Hausdorff metric H, so in the near limit the numerical attractor A h num may approximate only a proper subset of the original attractor A. A similar situation occurs for the maximal attractor of a perturbed differential equation dx dt = f(x) + hg(x) (5) with uniformly bounded perturbations g(x) 1 for all x R d, where g need only be continuous here and the solutions of (5) may even be nonunique, i.e. setvalued. The local discretization error of the numerical scheme (4) satisfies an estimate of the form x n+1 φ h (x n ) C p h p+1 for some constant C p. From Proposition 1 with τ = h, we see that if we choose h (, τ ) small enough so that C p h p+1 K 1 e Kh hɛ, (6) 5

6 then x n+1 Φ ɛ h(x n ). The numerical dynamics will thus be contained within and carried along by the setvalued dynamics of the discrete time system Φ ɛ h and, in particular, the numerical attractor A h num will be contained in the ɛ inflated attractor A ɛ. This also holds for the maximal attractor of a perturbed differential equation (5), provided h ɛ. In fact, it also holds for the numerical scheme with variable time steps h n [h min, h max ] provided C p h p+1 max K 1 e Kh min h min ɛ and for nonautonomous perturbations g(t, x) in (6), but now in the sense that the component subsets of the resulting nonautonomous pullback attractors belong to the inflated attractor A ɛ ; see [8, 9] for appropriate definitions and details. Real computations of a numerical scheme involve roundoff error as well as a theoretical discretization error. The roundoff error usually varies from step to step, so the actual numerical dynamical system generated within the computer will be nonautonomous even if a constant time step h is used. If ρ > denotes the maximum roundoff error and if the (possibly) variable time steps h n [h min, h max ] with C p h p+1 max + ρ K 1 e Kh min h min ɛ, then the component sets of the numerical pullback attractor will be contained in the inflated attractor A ɛ. There is an apparent relationship between the inflated attractor and the set of chain connected points of the original autonomous dynamical system. The latter is defined essentially by inflating the flow or solution mapping rather than the vector field as we have done here. While inflating the solution mapping has a number of theoretical advantages, this mapping is in practice generally not known explicitly. Moreover, in the nonautonomous case to be considered in [6] we will allow the inflation parameter to vary with state and time, in which case it is more convenient to work directly with the vector field. Finally, we note that it is always possible to construct, albeit artificial, approximate systems contained in the inflated dynamics for which the maximal attractor or pullback attractor components contain any specified point of the inflated attractor (see [6] for details). Hence the inflated attractor A ɛ is the smallest set containing all possible limiting behaviour or approximate autonomous attractors or nonautonomous attractor components resulting from all possible perturbations and approximations of appropriate magnitude of the original semidynamical system φ. In particular, there is no loss of information in the inflated attractor about the original asymptotic dynamics as may occur in using some type of approximate system for which the approximate attractors converge only upper semicontinuously to the original maximal attractor A. 6

7 5 Proof of Proposition 1 Since solution of the differential equation (1) is also a solution of the ɛ inflated system (3), we have φ t (x ) Φt(x ɛ ), and hence H ({φ t (x )}, Φt(x ɛ )), for all t and x R d. Now let x(t) be a solution of (3) with initial condition x() = x, i.e. satisfying (3). Using Propositions and in [3] we obtain { }) x(t) φ t (x ) = H ({x } + F ɛ (x(s)) ds, {x } + f (φ s (x )) ds = H ( { }) F ɛ (x(s)) ds, f (φ s (x )) ds = H (F ɛ (x(s)), {f (φ s (x ))}) ds H ({f(x(s))} + N ɛ [{}], {f (φ s (x ))}) ds H ({f(x(s))} + N ɛ [{}], {f(x(s)}) ds + H ({f(x(s)}, {f (φ s (x ))}) ds ɛ ds + so x(t) φ t (x ) ɛτ + K Hence by the Gronwall inequality we have f(x(s)) f(φ s (x )) ds, x(s) φ s (x ) ds for t τ x(τ) φ τ (x ) K 1 e Kτ τɛ for any τ. Taking the supremum over x(τ) Φτ(x ɛ ), we obtain H (Φ ɛ τ(x ), {φ τ (x )}) K 1 e Kτ τɛ and hence the desired result. 6 Proof of Theorem A Lyapunov inequality for the inflated dynamics Let A be the maximal attractor of the differential equation (1) and let V be a Lyapunov function characterizing the maximal uniform asymptotic stability of A 7

8 as in the following theorem which is a reformulation of Theorem 22.5 of Yoshizawa [12] to the maximal context considered here. Theorem 3 Suppose that f in (1) satisfies a uniform Lipschitz condition (2) on R d. Then a nonempty compact subset A of R d is globally uniformly asymptotically stable for the semidynamical system φ generated by differential equation (1) if and only if there exists a function V : R d [, ) for which: 1. V is uniformly Lipschitz on R d, i.e. there exists a constant L > such that V (x) V (y) L x y for all x, y R d ; 2. there exist continuous strictly increasing functions α, β : R d [, ) with α() = β() = and < α(r) < β(r) for all r > such that α(dist(x, A )) V (x) β(dist(x, A )) for all x R d ; 3. there exists a constant c > such that V (φ t (x ) e ct V (x ) for all t, x R d. (7) Similarly as in [7], the key tool in our proof is provided by the following Lyapunov function inequality. Lemma 4 V (x(t)) e ct V (x ) + L K 1 e Kt tɛ (8) for any solution x = x(t) of the ɛ inflated equation (3) with initial condition x() = x and any t. Proof: From the Lipschitz property of V in Theorem 3 we have V (x(t)) V (φ t (x )) + L x(t) φ t (x ). Applying the exponential decay inequality (7) of V in Theorem 3 to the first term on the right side and the inequality of Proposition 1 to the second then gives V (x(t)) e ct V (x ) + L K 1 e Kt tɛ for all t. 8

9 6.2 Existence of an absorbing set for the inflated dynamics Fix ɛ > and τ > and define η = η(ɛ, τ) := 2LK 1 e Kτ τɛ 1 e cτ. Lemma 5 Λ(η) := { x R d : V (x) η } is a nonempty compact subset of R d with H ( Λ(η), A ) α 1 (η). (9) Proof: Λ(η) is nonempty because it contains A = { x R d : V (x) = } and is closed in view of the continuity of V. Its boundedness and the inequality (9) follow from the inequality α(dist(x, A )) V (x) in the second property of V in Theorem 3 and the definition of Λ(η), giving dist(x, A ) α 1 (V (x)) α 1 (η) for all x Λ(η). The set Λ(η) is positively invariant with respect to the discrete time setvalued semigroup formed by iterating Φ ɛ τ with τ fixed, i.e. Lemma 6 Φ ɛ τ (Λ(η)) Λ(η). Proof: It suffices to consider any single point x 1 Φ ɛ τ(x ) for an arbitrary x Λ(η), so V (x ) η = η(ɛ, τ). Then by the key inequality (8) with t = τ and the definition of η(ɛ, τ) we have V (x 1 ) e cτ V (x ) + LK 1 e Kτ τɛ e cτ η(ɛ, τ) + 1 ( ) 1 e cτ η(ɛ, τ) 2 = 1 ( ) 1 + e cτ η(ɛ, τ) η(ɛ, τ), 2 so x 1 Λ(η). The set Λ(η) is in fact an absorbing set for Φτ ɛ provided τ is chosen small enough. Lemma 7 There is a τ > such that for each compact subset D and τ (, τ ) there exists an integer N D,ɛ,τ 1 for which Φ ɛ τ (D) Λ(η) for all n N D,ɛ,τ. 9

10 Proof: As in Lemma 3.4 of [7] there exists a γ > such that 1 + e cγ = 2e cγ/2 and 1 + e cτ < 2e cτ/2 for all < τ < τ = γ. Let x 1 Φ ɛ τ(x ) with x / Λ(η). Then by the key inequality (8) and the definition of η = η(ɛ, τ) we have V (x 1 ) e cτ V (x ) + LK 1 e Kτ τɛ = e cτ V (x ) + 1 ( ) 1 e cτ η(ɛ, τ) 2 < 1 ( ) 1 + e cτ V (x ) 2 < e cτ/2 V (x ) since V (x ) > η(ɛ, τ) and < τ < τ. Repeating this argument, we have V (x n ) < e cnτ/2 V (x ) where x n Φ ɛ τ(x n 1 ), x n 1 Φ ɛ τ(x n 2 ),..., x 1 Φ ɛ τ(x ) as long as x, x 1,..., x n 1 / Λ(η). Now V (x ) β ( dist(x, A ) ) β ( H (D, A ) ) < for all x D, so V (x n ) < e cnτ/2 β ( H (D, A ) ) as long as x, x 1,..., x n 1 / Λ(η). Define N D,ɛ,τ to be the smallest integer n for which e cnτ/2 β ( H (D, A ) ) η < e c(n 1)τ/2 β ( H (D, A ) ). Thus for each x D there exists an integer n N D,ɛ,τ, possibly, such that Φ ɛ nτ(x ) Λ(η). By the positive invariance of Λ(η) proved in Lemma 6 all successive iterates of Φ ɛ τ then remain in Λ(η), so the proof of Lemma 7 is complete. However, we need a nonempty compact subset of R d that is positively invariant and absorbing for the continuous time setvalued system Φt. ɛ For this we fix a τ as in Lemma 7 and define B ɛ := Φs ɛ (Λ(η)). s τ with η = η(ɛ, τ) as above. The B ɛ is compact and contains Λ(η), so is nonempty. Lemma 8 Φ ɛ t (B ɛ ) B ɛ for all t. 1

11 Proof: have First consider ρ (.τ]. Then by the setvalued semigroup property we Φ ɛ ρ (B ɛ ) = Φ ɛ ρ Φs ɛ (Λ(η)) = s τ Φρ+s ɛ (Λ(η)) B ɛ, s τ since Φ ɛ ρ+s (Λ(η)) B ɛ if ρ + s (, τ], whereas Φ ɛ ρ+s (Λ(η)) Φ ɛ ρ+s τ (Φτ ɛ (Λ(η))) Φ ɛ ρ+s τ (Λ(η)) B ɛ if ρ + s (τ, 2τ] by the previous step and Lemma 7. Now consider t > τ and write t = nτ + ρ where ρ (, τ). Then ( Φt ɛ (B ɛ ) Φ ɛ nτ Φ ɛ ρ (B ɛ ) ) Φnτ ɛ (B ɛ ) B ɛ by the step above. Lemma 9 For each compact subset D of R d there exists an integer T D,ɛ such that Φ ɛ t (D) B ɛ for all t T D,ɛ. Proof: Define D = s τ Φ ɛ s (D) and let t = nτ + ρ where ρ (, τ). Then ( Φ ɛ t (D) Φ ɛ nτ Φ ɛ ρ (D) ) Φ ɛ nτ ( D ) and by Lemma 7 there exists an N D,ɛ,τ 1 such that Φ ɛ nτ ( D ) Λ(η) B ɛ for all n N D,ɛ,τ. The result then follows with T D,ɛ = τn D,ɛ,τ. 6.3 Existence and continuous convergence of the inflated attractors We apply standard dynamical systems theoretic methods [11, 1] to the continuous time setvalued semigroup Φ ɛ t and the absorbing set B ɛ defined in the previous subsection to obtain the existence of a maximal attractor A ɛ for Φ ɛ t, namely with A ɛ = t Φ ɛ t (B ɛ ). Note that A ɛ does not depend on the choice of the τ used to construct the absorbing set B ɛ. 11

12 Let A be the maximal attractor for φ. Then A = φ t ( A ) φ t (Λ(η)) Φ ɛ t (Λ(η)) Φ ɛ t (B ɛ ) for all t, so A Φ ɛ t (B ɛ ) = A ɛ t for each ɛ >. Thus H (A, A ɛ ) =. In addition, A ɛ Λ(η(ɛ, τ) B ɛ so H ( A ɛ, A ) H ( B ɛ, A ) = H s τ Φs ɛ (Λ(η(ɛ, τ))), A max ( s τ H Φ ɛ s (Λ(η(ɛ, τ))), A ) K 1 e Kτ ( τɛ + H ( Λ(η(ɛ, τ)), A )) K 1 e Kτ ( τɛ + α 1 (η(ɛ, τ)) ) as ɛ by (9) for the last inequality and by a generalization of Proposition 1 with different initial values (see also Lemma 3.4 of [4]) for the second last inequality. Combining the results, we have lim ɛ H ( A ɛ, A ) =, that is, the inflated attractor A ɛ converges continuously to A as ɛ converges to. This completes the proof of Theorem 2. References [1] J.P. Aubin and A. Cellina, Differential Inclusions. Springer Verlag, Berlin, [2] D.N. Cheban, P.E. Kloeden and B. Schmalfuß, Pullback attractors in dissipative nonautonomous differential equations under discretization, DANSE Preprint, FU Berlin, [3] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientific, Singapore, [4] L. Grüne, Convergence rates of perturbed attracting sets with vanishing perturbations, DANSE Preprint, FU Berlin,

13 [5] J. Hale, Asymptotic Behavior of Dissipative Dynamical Systems. Amer. Math. Soc., Providence, RI, [6] P.E. Kloeden and V.S. Kozyakin, Inflated nonautonomnous pullback attractors. in preparation [7] P.E. Kloeden and J. Lorenz, Stable attracting sets in dynamical systems and in their one-step discretizations, SIAM J. Numer. Analysis 23 (1986), [8] P.E. Kloeden and B. Schmalfuß, Lyapunov functions and attractors under variable time step discretization, Discrete & Conts. Dynamical Systems 2 (1996), [9] P.E. Kloeden and D.J. Stonier, Cocycle attractors in nonautonomously perturbed differential equations. Dynamics of Discrete, Continuous and Impulsive Systems. 4 (1998), [1] A.M. Stuart and A.R. Humphries, Numerical Analysis and Dynamical Systems, Cambridge University Press, Cambridge [11] G.P. Szegö and G. Treccani, Semigruppi di Trasformazioni Multivoche, Springer Lecture Notes in Mathematics, Vol. 11, [12] T. Yoshizawa, Stability Theory by Lyapunov s Second Method, The Mathematical Society of Japan, Tokyo,