Attraction rates, robustness and discretization of attractors

: We investigate necessary and su(cid:14)cient conditions for the convergence of attractors of discrete time dynamical systems induced by numerical one{step approximations of ordinary di(cid:11)erential equations (ODEs) to an attractor for the approximated ODE. We show that both the existence of uniform attraction rates (i


Introduction
The long time behavior of dissipative dynamical systems is essentially determined by the attractors of these systems, since for large times its trajectories will typically stay on or near an attractor.Even for moderately complex finite dimensional systems, however, it is rarely possible to determine attractors by analytic methods.Hence numerical approximations form a natural part of a systematic analysis.It is therefore important to know about the effects of discretization errors on attractors in order to give a reasonable interpretation to numerical experiments and to justify numerical findings.
For dynamical systems induced by ordinary differential equations (ODEs), numerical onestep approximations like Runge-Kutta or Taylor schemes form an important class of schemes.It follows from a result of Kloeden and Lorenz in 1986 [17] for attracting sets, that if the ODE possesses an attractor A, then the discrete time dynamical system induced by such a numerical scheme also has an attractor contained in a neighborhood N of A, where the size of N shrinks down to A as the time-step of the discretization approaches 0.
A number of examples (see, e.g., [7,Example (2.12)], [8,Example 1.1.1]or [11] for the case of finite dimensional approximations of infinite dimensional systems) shows that the LARS GR ÜNE limit set for a convergent sequence of numerical attractors for vanishing time-steps may be strictly smaller than A. This fact, however, imposes a major problem for the interpretation of numerical results, since it implies that in general one cannot conclude the existence of a real attractor close to a numerical attractor.Hence it is important to derive techniques or conditions which allow to conclude convergence of numerical attractors to a real attractor.
There are three main approaches for tackling this problem: The first is to impose suitable conditions on the approximated system, which ensure a faithful numerical approximation and exclude the appearance of numerical artifacts.Typical examples of this approach are, for instance, results on the numerical approximation of Morse-Smale systems by Garay [5,6], on the discretization of gradient systems by Stuart and Humphries [21,Section 7.7] and a result on hyperbolic attractors by the author [9,Remark 2.10(ii)].
The second approach is to design algorithms which can be shown to converge to the right objects under no or under very mild conditions on the approximated system.An example for this approach is the subdivision algorithm for the computation of attractors originally developed by Dellnitz and Hohmann [3] using a rigorous discretization as proposed by Junge [13,14], see also [8,Section 6.3] for a description and a quantitative convergence analysis of this method based on robust Lyapunov functions.
The idea of the third approach is the formulation of conditions on the behavior of the numerical systems under which we can ensure convergence of the respective sets or the existence of respective nearby sets for the approximated system.A typical example are the sufficient conditions for the convergence of numerical attracting sets in the Galerkin approximation to Navier-Stokes equations by Kloeden [15].For finite dimensional systems, in [9] a necessary and sufficient condition of this type was developed based on uniform attraction properties of the numerical attractors, which were characterized using (uniformly) shrinking families of neighborhoods which are mapped onto each other by suitable perturbations of the numerical flows.
The present paper follows this third approach.As in [9] we are going to formulate necessary and sufficient conditions for the convergence of numerical attractors to a real attractor in terms of uniform attraction properties.The difference to [9] lies in the type of uniformity properties used for this purpose, because (i) here we formulate the properties directly in terms of the numerical attractors instead of using auxiliary attracting sets, (ii) we use comparison functions for characterizing attraction properties instead of using geometrical characterizations by means of shrinking neighborhoods which are difficult to identify in a numerical simulation, and (iii) most importantly, instead of using a condition on the rate of attraction (i.e., the speed of convergence towards the attractor) for perturbed numerical systems, here we "decouple" the rate and the perturbation and give two different conditions, one based on the attraction rate for the unperturbed numerical systems and the other based on the robustness against perturbations.
More precisely, we prove that a sequence of numerical attractors converges to a real attractor • if and only if the numerical attractors are attracting with uniformly bounded attraction rates, cf.Theorem 6.2(iii) • if and only if the numerical attractors are robust against perturbation with uniformly bounded robustness gains, cf.Theorem 6.2(ii) and Theorem 6.4.
In addition, in Theorem 6.5 we give estimates for the discretization error based on the local error of the numerical scheme and the robustness gains of the respective attractors.
The tools we need in order to obtain these results are developed step by step in this paper, which is organized as follows.After defining the setup and stating some preliminary results in Section 2, in Section 3 we define a suitable robustness concept for attracting sets with respect to perturbations, describe the concept of embedding systems into each other and show how this applies to numerical one-step approximations.In Section 4 we study the relation between robustness of attracting sets and their rate of attraction.In Section 5 we prove some useful results on the relation between a continuous time system and its timeh map and finally, in Section 6 we state the main results on attractors under one-step discretization.

Setup and Preliminaries
We consider ordinary differential equations given by and, for some time-step h > 0, discrete time systems of the form where f and Φ h are maps from For simplicity of exposition (cf.Remark 2.1, below) we assume global Lipschitz properties of the respective systems, i.e., we assume that there exists a constant L > 0 such that the inequalities It will often be convenient to combine continuous and discrete time trajectories in one notation.For this we use the notation Φ(t, x) which either denotes ϕ(t, x) or Φ h (t, x), where the precise meaning will be clear from the context.Whenever we consider a discrete time system with time-step h > 0 the time t is implicitly assumed to be in the respective discrete time-scale hN 0 .
For sets C ⊂ R d we use the notation A special type of a discrete time system is the time-h map of (2.1) which is defined by the discrete time system x(t + h) = ϕ(h, x(t)). (2.5) The trajectories of (2.5) are denoted by ϕ h (t, x).Note that if (2.1) satisfies (2.3) for some L > 0 then Gronwall's Lemma implies that the time-h map (2.5) satisfies (2.4) for the Lipschitz constant L = Le hL > L.
Another special type of a discrete time system (2.2) is a numerical one-step approximation Φ h of (2.1) which is supposed to satisfy (2.4) and is such that there exist constants c, q > 0 with Φ h (x) − ϕ(h, x) ≤ ch q+1 for all x ∈ R d . (2.6) Here q is called the order of the scheme.Examples for such approximations are Taylor and Runge-Kutta schemes; for details we refer, e.g., to the textbooks [4,12,20].Our main objects of interest are attracting sets and attractors as given by the following definition.
Remark 2.5 The functions β ∈ KL are closely related to the usual ε-δ definition of asymptotic stability.More precisely, for any function a : R + 0 × R + 0 → R + 0 satisfying the two properties (i) for all ε > 0 there exists δ > 0 such that if r ≤ δ then a(r, t) < ε for all t ≥ 0 (ii) for all ε > 0 and for all R > 0 there exists T > 0 such that a(r, t) < ε for all 0 ≤ r ≤ R and for all t ≥ T there exists a function β ∈ KL with a(r, t) ≤ β(r, t) for all r, t ≥ 0.
This fact was already implicitly used in Hahn's book [10]; in this form it is stated (but not proved) in Albertini and Sontag [1, Lemma 4.1] and proved (but not explicitly stated) by Lin, Sontag and Wang [18, Section 3].
Using class KL functions we can define rates of attraction for attracting sets.
Definition 2.6 Let Φ denote the trajectories of a system of type (2.1) or (2.2) and let A be an attracting set with attracted neighborhood B. Then holds for each x ∈ B and each t ≥ 0.

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The following lemma shows that each attracting set possesses a rate of attraction.
Lemma 2.7 Let Φ denote the trajectories of a system of type (2.1) or (2.2) and let A be an attracting set with attracted neighborhood B. Then there exists a rate of attraction β ∈ KL for A.
Proof: Using the forward invariance and attractivity properties of A, and the (uniform) continuous dependence of a trajectory on the initial value (as induced by Gronwall's Lemma for (2.1) or by induction for (2.2)) one easily verifies that the function a : satisfies the properties (i) and (ii) of Remark 2.5.Hence there exists β ∈ KL with a ≤ β and consequently for all x ∈ B and all t ≥ 0. This shows the claim.
We end this section by stating some useful properties of attractors which we will need in what follows.
Lemma 2.8 Let Φ denote the trajectories of a system of type (2.1) or (2.2) and let A be an attracting set with attracted neighborhood B. Then A contains an attractor with attracted neighborhood B.
Proof: Verification of the desired properties shows that is the desired attractor, see [ Proof: Let A be an attractor with attracted neighborhood B. Then, in particular, A is invariant.Now assume that Ã ⊂ A, Ã = A, is a forward invariant attracting set.Then there exists a neighborhood N ⊃ Ã with A ⊆ N , such that Φ(t, B) ⊂ N for some t ≥ 0, i.e., in particular Φ(t, A) = A which contradicts the invariance of A.
Let conversely A be a minimal forward invariant attracting set.Then by Lemma 2.8 A contains an attractor which again is a forward invariant attracting set.Hence by minimality it coincides with A.
The next Lemma shows that the attractor is also the maximal compact invariant set contained in intB.
Lemma 2.10 Let Φ denote the trajectories of a system of type (2.1) or (2.2) and let A be an attractor with attracted neighborhood B for Φ.Then each compact invariant set D ⊂ B for Φ is contained in A.
Proof: Let D ⊂ B be an invariant set for Φ.Then D = Φ(t, D) ⊂ Φ(t, B) for all t ≥ 0. On the other hand, for each neighborhood N ⊃ A we know that Φ(t, B) ⊂ N for all t ≥ 0 sufficiently large.Hence D ⊂ N which implies the assertion.

Inflation, robustness and embedding
The main technique that we will use in this paper in order to obtain results on convergence of numerical attractors is the embedding of the numerical approximation into a perturbed continuous time system, and vice versa.In this section we define suitable perturbed systems, the corresponding attracting sets and a useful robustness concept for attracting sets.
In addition, we give a mathematically precise meaning for the embedding property.
The following definition gives the appropriate perturbed systems (see also [16] for an equivalent definition using differential inclusions).For a continuous time system (2.1) we define the α-inflated system by and for a discrete time system we define it by For each initial value x ∈ R d and each w ∈ W α we denote the corresponding trajectory by ϕ(t, x, w) or Φ h (t, x, w), respectively.It should be noted that the discrete time inflation 2) of the time-h map (2.5) of a continuous time system (2.1) differs from the time-h map of the continuous time inflation (3.1) of system (2.1) defined by Throughout this paper, the term inflated time-h map refers to system (3.3),whose trajectories will be denoted by ϕ h (t, x, w).
As for the unperturbed systems, we use Φ(t, x, w) to denote both discrete and continuous time trajectories, depending on the context.Furthermore, for α > 0, x ∈ R d and a subset C ⊆ R d we use the notations Next we define suitable attracting sets for inflated systems and a robustness property of attracting sets.Let α 0 > 0 and γ ∈ K. Then an attracting set (or attractor) A with attracted neighborhood B for an unperturbed system (2.1) or (2.2) is called γ-robust for γ and α 0 , if for each α ∈ (0, α 0 ] there exists an α-attracting set A α with attracted neighborhood B for the corresponding inflated system with A ⊆ A α and Here γ ∈ K is called robustness gain.
Remark 3.3 Analogous to Lemma 2.7 one sees that for each α-attracting set A α with attracted neighborhood B there exists β ∈ KL such that for all x ∈ B, t ≥ 0 and all w ∈ W α .
We now define what we mean by an embedded system.For our purpose it is sufficient to define this concept for discrete time systems.for all t ∈ hN 0 .
Here we call Φ h the embedding system and Ψ h the embedded system.
Lemma 3.5 Consider three discrete time inflated systems Φ h , Ψ h and Θ h of type (2.2) and assume that Ψ h is (α Proof: Straightforward using Definition 3.4.
The following proposition shows how the inflated numerical system with trajectories denoted by Φ h (t, x, w), can be embedded into the inflated time-h map (3.3), and vice versa.
Proposition 3.6 Consider the numerical approximation Φ h of system (2.1) for some h > 0. Let α 0 > 0 and consider the constants L and c from (2.3) and (2.6).
Proof: Consider the auxiliary system defined by for t ∈ hN 0 and denote the trajectories with initial value x ∈ R d at initial time t = 0 by φh (t, x, w).It is immediate from Definition 3.4 and inequality (2.6) that φh is (ch q , 1)embedded in Φ h and that Φ h is (ch q , 1)-embedded in φh .
We claim that the system φh is (0, 1 + Lh)-embedded in ϕ h and that ϕ h is (0, e Lh )embedded in φh .Then the assertion follows from Lemma 3.5.
In order to prove the embedding relation between ϕ h and φh fix some w ∈ W α 0 .It is sufficient to show the embedding for t = h since then we can continue by induction.We first construct w such that ϕ h (h, x, w) = φh (h, x, w).

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Consider the perturbation Then we obtain which by the uniqueness of the solution to this differential equation implies From the Lipschitz estimate (2.3) we obtain for almost all τ ∈ [0, h] the inequality and thus shows the claim.
In the following two propositions we show the relations between attracting sets of embedding and embedded systems.
Proposition 3.7 Consider a discrete time system with trajectories Ψ h , which is (α, C)embedded in some other discrete time system with trajectories denoted by Φ h for some α ≥ 0, C ≥ 1. Assume that the embedding system Φ h has an attracting set A which is γ-robust for some γ ∈ K and some α 0 ≥ α.Then the embedded system Ψ h has an attracting set A with attracted neighborhood B which satisfies Proof: By the embedding property we obtain Ψ h (t, B) ⊆ Φ α h (t, B).Hence the α-attracting set A α for the inflated embedding system Φ α h is an attracting set for the embedded system Ψ h .Hence A = A α is the desired set.
The next proposition shows that we can even conclude the existence of robust attracting sets for the embedded system, if we are willing to allow a larger distance between A and A.
Proposition 3.8 Consider a discrete time system with trajectories Ψ h , which is (α, C)embedded in some other discrete time system with trajectories denoted by Φ h for some α ≥ 0, C ≥ 1. Assume that the embedding system Φ h has an attracting set A which is γ-robust for some γ ∈ K and some α 0 > α.Then for each D > 1 with Dα ≤ α 0 the embedded system Ψ h has a strongly attracting set A, which is γ(CD • /(D − 1))-robust for α 1 = α 0 (D − 1)/(CD) and satisfies d H ( A, A) ≤ γ(Dα).This shows the claim.

Robustness and Attraction Rates
In this section we investigate the relation between the robustness gain γ and the attraction rate β.We start by showing that we can find an upper bound for the robustness gain of an attracting set A which is essentially determined by its rate of attraction.
Furthermore, for any w ∈ W α and any x ∈ B inequality (4.1) implies that the trajectory satisfies and hence hits D α in some uniformly bounded finite time.Now we set These sets are α-forward invariant by construction and by (4.2) and α-attracting by (4.4).Furthermore they satisfy A ⊆ A α for α ∈ (0, α 0 ], B(r(α), A) ⊆ A α and because of (4.This shows the desired robustness property.
In general, this construction of γ might not yield the best possible robustness gain for a given system and attracting set.However, the importance of this theorem is the uniformity that can be deduced from it: knowing the attraction rate, the distance between B and A and the Lipschitz constant of the system allows us to give an upper bound for the robustness gain, no matter how the geometric structure of A or the behavior of Φ around or on A look like.In particular, when uniform attraction holds for a family of systems with uniform Lipschitz properties and uniform distance between the attracting sets and their attracted neighborhoods, then also uniform robustness can be deduced.
Let us illustrate one special case, in which the proof of Theorem 4.1 yields an explicit expression for µ.
Another interesting consequence of Theorem 4.1 is the following corollary.
Corollary 4.3 Consider an attracting set A for a system of type (2.1) or (2.2).Then there exists α 0 > 0 and γ ∈ K such that A is γ-robust for γ and α 0 .
Proof: By Lemma 2.7 there exists an attraction rate β ∈ KL for A. Hence Theorem 4.1 immediately gives the assertion.
Knowing that any attracting set admits a robustness gain, we can easily find an upper bound for a robustness gain for nested attracting sets.

Lemma 4.4
Let A be an attracting set with attracted neighborhood B for a system of type (2.1) or (2.2) and let A ⊃ A be an attracting set which is contained in B and is γ-robust for the α 0 -inflated system for some γ ∈ K and some α 0 > 0. Let ρ := dist( A, A).
Proof: By Corollary 4.3 there exists α0 > 0 and γ ∈ K such that A is γ-robust for the α0inflated system.Without loss of generality we may assume α0 ≤ α 0 and γ( α0 ) ≥ γ( α0 ) +ρ.Now for each α ∈ (0, α 0 ] there exists an α-attracting set This γ is easily verified to be of class K, thus the assertion follows.
We end this section by proving a "uniform attraction" property of the α-attracting sets appearing in the definition of the γ-robustness property.

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Proposition 4.5 Consider an attracting set A with attracted neighborhood B for a system of type (2.1) or (2.2), which is γ-robust for some γ ∈ K and some α 0 > 0. Then for each ε > 0 there exists a function β ∈ KL such that the inequality holds for all t ≥ 0, all x ∈ B, all α ∈ [0, α 0 ], all w ∈ W α and the trajectories of the corresponding inflated system.
Proof: It is easily seen that there exists a monotone decreasing sequence α n → 0 such that α 0 is the inflation parameter from the assumption and γ(α ) and r 0 = d H (B, A).Now for each r ∈ (0, r 0 ] we define the functions It is immediate that for all r, t > 0 the sequences σ n (r, t) and µ n (r, t) + d n are monotone decreasing in n and monotone increasing in r.From Remark 3.3 we obtain the existence of functions This implies lim sup and thus for each n ∈ N there exists T > 0 such that µ k (r, t) = 0 for all k = 1, . . ., n all r ∈ (0, r 0 ] and all t ≥ T. Furthermore, since A ⊆ A α for all α ∈ (0, α 0 ], from (4.6) for each n ∈ N and all r > 0 sufficiently small (depending on n) we obtain Hence for each n ∈ N there exists R > 0 such that µ k (r, t) = 0 for all k = 1, . . ., n, all r ∈ [0, R] and all t ≥ 0. ( Now consider the function a(r, t) := sup n∈N 0 µ n (r, t).From the definition of the µ n we obtain for all t ≥ 0, all α ∈ [α n+1 , α n ] and all w ∈ W α .Furthermore, a(r, t) is monotone increasing in r and monotone decreasing in t.If we fix n ∈ N and choose T > 0 such that (4.7) holds, then (4.7) and the monotonicity of µ n (r, t) for all t ≥ T .Similarly, from (4.8) one sees that for each n ∈ N and each t ≥ 0 there exists R ≥ 0 such that a(r, t) ≤ d n for all r ≤ R. Thus, since d n → 0 as n → ∞ we obtain a(r, t) → 0 if either r → 0 or t → ∞.Hence by Remark 2.5 a can be bounded from above by some function β ∈ KL which shows the claim.
Remark 4.6 Inequality (4.5) describes a property which in nonlinear control theory is known as input-to-state stability (ISS), see, e.g., the survey [19].For a detailed comparative study of various robustness concepts for attracting sets including their characterization via Lyapunov functions we refer to [8].

Discrete and Continuous Time Systems
By its very nature, a numerical one-step approximation with time-step h > 0 only gives an approximation to the time-h map ϕ h (2.5) of the continuous time system ϕ (2.1).It is therefore necessary to obtain information on the dynamical behavior of ϕ from its time-h map ϕ h .In this section we give two results for this purpose.
Proposition 5.1 Consider a sequence of time-steps h n → 0, and a sequence of γ n -robust attracting sets A n for γ n ∈ K and α n > 0, each with the same attracted neighborhood B for the inflated time-h n maps (3.3),where α n → α 0 > 0 as n → ∞.Assume there exists γ ∈ K such that lim sup n→∞ γ n (α) ≤ γ(α) and a compact set Then A is a γ-robust attracting set for the continuous time system (2.1) for γ and each α0 ∈ (0, α 0 ).
Proof: We first show that A is an attracting set for ϕ.For this, fix ε > 0 and consider n ∈ N such that d H (A n , A) < ε/3 and h n M < ε/3, where M is a bound on f (x) for x in a sufficiently large neighborhood of B. Then it is easily seen that there exists T > 0 such that ϕ(ih n , B) ⊂ B(ε/3, A n ) for all i ∈ N with ih n ≥ T .Consequently, we obtain ϕ(t, B) ⊂ B(ε, A) for all t ≥ T , and since ε > 0 was arbitrary, this shows the desired convergence dist(ϕ(t, B), A) → 0 as t → ∞.
It remains to show the γ-robustness.To this end, fix some α ∈ (0, α 0 ) and consider the set where the A α n denote the α-attracting sets for the inflated time-h n maps (3.3).Using the fact that ), with the same argument as above one sees that this set is αattracting for the inflated system.Since for each ε > 0 we find N ∈ N such that for all n ≥ N the inequalities hold we can conclude d H (A α n , A) ≤ γ(α) + ε for all n ≥ N and thus d H (A α n , A) ≤ γ(α) + ε which shows the desired distance since ε > 0 was arbitrary.
While in general an attracting set for the time-h map is not an attracting set for the continuous time system, this property is always true for attractors, as the following Lemma shows.
Lemma 5.2 Let h > 0 and A h be an attractor with attracted neighborhood B for the time-h map ϕ h (2.5) of the continuous time system (2.1).Then A h is also an attractor with attracted neighborhood B for the continuous time system (2.1).
Proof: We first show forward invariance of A h for ϕ, i.e., ϕ(t, A h ) ⊆ A h for each t ≥ 0. By invariance of A h for ϕ h for each t ≥ 0 we know ϕ h (h, ϕ(t, A h )) = ϕ(t, ϕ h (h, A h )) = ϕ(t, A h ), hence ϕ(t, A h ) is a compact invariant set for ϕ h , and by Lemma 2.10 it is contained in A h .Now we show that A h is an attracting set for the continuous time system ϕ.Forward invariance of A h and continuous dependence on the initial value imply that for each δ > 0 there exists ε > 0 such that for all t ∈ [0, h] and arbitrary bounded sets D ⊂ R d .Since attractivity of A h for ϕ h implies lim i→∞, i∈N dist(ϕ(ih, B), A) = 0 we can conclude lim t→∞ dist(ϕ(t, B), A h ) = 0, i.e.A h is also an attracting set for ϕ with attracted neighbourhood B.
It remains to show that A h is an attractor for ϕ.By Lemma 2.8 there exists an attractor A ⊆ A h for ϕ.This, in turn, is also an attractor set for ϕ h , hence by Lemma 2.9 it must coincide with A h .Thus A h is an attractor for ϕ.

Numerical Discretization
In this section we combine the results from the previous sections in order to derive criteria under which one can conclude the existence of attracting sets and attractors from numerical approximations.We start with sufficient conditions for the existence of attracting sets.Proposition 6.1 Consider the continuous time system (2.1) and a numerical one-step approximation Φ h for h > 0 satisfying (2.6) for c q > 0. Let L denote a Lipschitz constant for both systems from (2.3) and (2.4), respectively.
(a) Let A be a γ-robust attracting set for (2.1) for γ ∈ K and α 0 ≥ e hL ch q .Then there exists an attracting set A h for the discrete time system induced by the numerical approximation Φ h satisfying d H (A h , A) ≤ γ(e hL ch q ).
(b) Let A h be a γ-robust attracting set for Φ h for γ ∈ K and α 0 ≥ ch q .Then there exists an attracting set A h for the time-h map (2.5) of the continuous time system satisfying Proof: This follows directly from Proposition 3.6 and Proposition 3.7.
Theorem 6.2 Consider the continuous time system (2.1) and a family of numerical onestep approximation Φ hn satisfying (2.6) for a sequence of time-steps h n → 0 as n → ∞.Let A n be attractors for the discrete time systems induced by these numerical approximations, each with the same attracted neighborhood B, and assume that there exists a compact set Then the following properties are equivalent (i) A is an attractor for (2.1) with attracted neighborhood B (ii) there exist N ∈ N, γ ∈ K, α 0 > 0 and sequences C n → 1 and ρ n → 0 such that for each n ≥ N the attractor A n is γ n -robust for the numerical system Φ hn for α 0 and γ n (r) ≤ γ(C n r) + ρ n (iii) there exist N N, β ∈ KL and a sequence ε n → 0 such that for each n ≥ N the attractor A n for the numerical system Φ hn has attraction rate In addition, if (ii) holds for γ ∈ K and α 0 > 0 then A is γ-robust for this γ and each α0 ∈ (0, α 0 ) and if (iii) holds for β ∈ KL then A is attracting with this rate β for this continuous time system (2.1).
Proof: (i)⇒(ii): Since A is an attracting set by Corollary 4.3 it is also γ-robust for some suitable γ ∈ K and α0 > 0. Since by Proposition 3.6 the α 0 -inflated numerical system (3.4) for α 0 = α0 /2 and n sufficiently large is embedded in the α0 -inflated time-h n map (3.3), Proposition 3.8 applied with α = ch q n , C = 1 + h n L and D = 1/ ch q n implies the existence of γn -robust attracting sets A n with d H (A, A n ) ≤ γ( ch q n ) and Since the attractors A n converge to A and-by minimality-are contained in the attracting sets A n we can conclude that ρ n = dist(A n , A n ) → 0 as n → ∞, hence by Lemma 4.4 the A n are γ n -robust with γ n (r) ≤ γn (r) + ρ n and α 0 which shows the claim.
(i) ⇒ (iii): Since A is an attracting set by Corollary 4.3 and Proposition 4.5 it satisfies inequality (4.5) for suitable β, γ and ε.Thus by Proposition 3.6 for all sufficiently large n ∈ N, all x ∈ B and all i ∈ N we find some w ∈ W e hnL ch q n such that This yields

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which shows the assertion for ε n = d H (A n , A) + (1 + ε)γ(e hnL ch q n ).(ii) ⇒ (i): Similar to the arguments in case "(i) ⇒ (ii)" we obtain that for sufficiently large n ∈ N there exist γ n (D n •)-robust attracting sets A n for the inflated time-h n map (3.3) of the α n -inflated system for suitable constants C n → 1 and α n → α 0 , such that d H ( A n , A) → 0. Hence by Proposition 5.1 we obtain that A is a γ-robust attracting set for each α0 ∈ (0, α 0 ) for (2.1).It remains to show that A is an attractor.If this was not the case, then by Lemma 2.8 and Lemma 2.9 there exists an attractor A for ϕ with A ⊂ A, A = A and attracted neighborhood B. Denote η := d H ( A, A) > 0. Again following the arguments from the case "(i) ⇒ (ii)" this implies that for all sufficiently large n ∈ N the attractors A n for the numerical systems Φ hn must satisfy dist(A n , A) < η/2.This implies d H (A n , A) ≥ η/2 and hence contradicts the convergence d H (A n , A) → 0 for n → ∞.
(iii) ⇒ (i): Fixing some T > 0 and some ε > 0, by Gronwall's Lemma for all n > 0 sufficiently large and all x ∈ B we obtain the inequality Hence from the convergence of A n to A and from the properties of β n and β we obtain ϕ(T, x) A ≤ β( x A + ε, T + ε) + ε and since ε > 0 was arbitrary by continuity of β this implies ϕ(T, x) A ≤ β( x A , T ), which implies that A is an attracting set since T > 0 was arbitrary.The fact that A is an attractor follows similar to the case "(ii) ⇒ (i)", above.
In other words, Theorem 6.2 states that a sequence of "numerical" attractors converges to a "real" attractor if and only if the elements of this sequence are either uniform robust or attracting with a uniform rate.Remark 6.3 Note that we have used the minimality of attractors only in the proof of the implication "(i) ⇒ (ii)".Hence the equivalence "(i) ⇔ (iii)" and the implication "(ii) ⇒ (i)" remain true for general attracting sets.
In the next theorem we shift our attention to a sequence of uniformly robust numerical attractors (in the sense of Theorem 6.2 (ii)) without the a priori assumption about convergence of these sets.It turns out that this sequence of numerical attractors converges to a set if and only if this set is an attractor.Theorem 6.4 Consider the continuous time system (2.1) and a family of numerical onestep approximation Φ hn satisfying (2.6) for a sequence of time-steps h n → 0 as n → ∞.Let A n be attractors for the discrete time systems induced by these numerical approximations, each with the same attracted neighborhood B, assume that they are γ n -robust for the numerical system Φ hn for some α 0 > 0 and γ n (r) ≤ γ(C n r) + ρ n for some suitable γ ∈ K and sequences C n → 1 and ρ n → 0, and let A ⊂ B be a compact set.
Then the following statements are equivalent.
Proof: (i) ⇒ (ii): Since by Lemma 2.7 and Theorem 4.1 the attractor A is γ-robust for some suitable γ ∈ K, by Proposition 6.1(a) and Lemma 2.8 we can conclude dist(A n , A) → 0 as n → ∞.For the converse "dist" estimate, by the assumption on the γ n -robustness of the A n , for each ε > 0 and all n ∈ N sufficiently large there exist attracting sets A n for the time-h n map of the continuous time system with dist( A n , A n ) ≤ ε.By Lemma 2.8 each of these sets contains an attractor for the time-h n map (2.5) and attracted neighborhood B, which by Lemma 5.2 coincides with A. This implies dist(A, A n ) ≤ ε and since ε > 0 was arbitrary we obtain dist(A, A n ) → 0 as n → ∞.This shows the desired convergence.
In other words, Theorem 6.4 states that a sequence of uniformly robust "numerical" attractors converges to some set A if and only if it is a "real" attractor.
Finally, we are going to investigate the rates of convergence of A n to A under the assumptions of Theorem 6.4.Theorem 6.5 Consider the continuous time system (2.1) and a family of numerical onestep approximation Φ hn satisfying (2.6) for a sequence of time-steps h n → 0 as n → ∞ and constants c, q > 0. Let A n be attractors for the discrete time systems induced by these numerical approximations, each with the same attracted neighborhood B, and assume that they are γ n -robust for the numerical system Φ hn for some α 0 > 0 and γ n (r) ≤ γ(C n r) + ρ n for some suitable γ ∈ K and sequences C n → 1 and ρ n → 0. Let A ⊂ B be a compact set and assume that one of the two following conditions is satisfied (i) A is an attractor for (2.1) with attracted neighborhood B (ii) d H (A n , A) → 0 as n → ∞.
Then for all sufficiently large n ∈ N we obtain the estimates dist(A, A n ) ≤ γ(C n e Lhn ch q n ) + ρ n and dist(A n , A) ≤ γ(ch q n ) for the rate of convergence of A n to A.
Proof: Under the assumptions Theorem 6.4 implies that A is γ-robust for (2.1).Hence by Proposition 6.1 we obtain the existence of attracting sets for the numerical systems and the time-h n maps, respectively, with the desired distances.By Lemma 2.8, Lemma 2.9 and Lemma 5.2 the attractors A n and A are contained in these attracting sets, hence the "dist" estimates remain valid.

x
D := min y∈D x − y , the nonsymmetric Hausdorff distance between two compact sets by dist(C, D) := max x∈C min y∈D x − y , and the Hausdorff metric for compact sets by d H (C, D) := max{dist(C, D), dist(D, C)}.We use these distances for arbitrary bounded sets C, D ⊂ R d by defining x D := x cl D , dist(C, D) := dist(cl C, cl D) and d H (C, D) := d H (cl C, cl D).For ε > 0 we denote the (open) ε-ball around C by B(ε, C) := {y ∈ R d | y C < ε}.If C = {x} we also write B(ε, x).Note that for all bounded sets C, D, E ⊂ R d the equivalences dist(C, D) = 0 ⇔ C ⊆ cl D and d H (C, D) = 0 ⇔ cl C = cl D and the implication C ⊆ E ⇒ dist(C, D) ≤ dist(E, D) hold.

Definition 3 . 1
For α ∈ R, α ≥ 0 we define the set of perturbation values W α := {w ∈ R d | w ≤ α} and the space of measurable functions with values in W α by W α := {w : R → R d | w measurable with w(t) ∈ W α for almost all t ∈ R}.For functions w ∈ W α and real values a < b we define w [a,b] := ess sup t∈[a,b] w(t) .
Let C, D ⊂ R d be nonempty compact sets, x ∈ R d , and let • be the Euclidean norm on R d .We define the distance from a point to a set by Definition 2.3 Let Φ denote the trajectories of a system of type (2.1) or (2.2).Consider a compact set A ⊂ R d and an open and bounded set B ⊂ R d with A ⊂ B. Then A is called an attracting set with attracted neighborhood B if it is forward invariant, i.e., We define the following classes of comparison functions.
Definition 3.4Consider two inflated discrete time systems of type (3.2) with perturbations from W α0 and W α 0 , respectively.Denote the trajectories of the systems by Φ h and Ψ h , respectively, and let α ≥ 0 and C ≥ 1.Then we say that the second system Ψ h is (α, C)-embedded in the first Φ h if for each x ∈ R d and each w ∈ W α 0 there exist w ∈ W α0 with w [t,t+h] ≤ α + C w [t,t+h] and Φ h (t, x, w) = Ψ h (t, x, w)