Persistence of attractors for one-step discretization of ordinary di(cid:11)erential equations

: We consider numerical one-step approximations of ordinary di(cid:11)erential equations and present two results on the persistence of attractors appearing in the numerical system. First, we show that the upper limit of a sequence of numerical attractors for a sequence of vanishing time step is an attractor for the approximated system if and only if for all these time steps the numerical one-step schemes admit attracting sets which approximate this upper limit set and attract with a uniform rate. Second, we show that if these numerical attractors themselves attract with a uniformly rate, then they converge to some set if and only if this set is an attractor for the approximated system. In this case, we can also give an estimate for the rate of convergence depending on the rate of attraction and on the order of the numerical scheme.


Introduction
Attractors play an important role in the understanding of the behaviour of complex dynamical systems. 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, which often are the only way to gather knowledge about complicated systems as analytic solutions are rarely available.
For one-step discretizations of ordinary differential equations (ODEs) the basic result in that direction has been obtained by Kloeden and Lorenz in 1986 [11]. In this paper it is shown that if the ODE possesses an attractor then the numerical approximations possess absorbing sets nearby, which converge to the attractor as the time step tends to 0. Later, this result was established also for multi-step schemes by the same authors [12].
One of the main contribution of these papers is the observation that the right approximating set for the attractor of an ODE in general is not an attractor but an absorbing set, which in turn contains an attractor of the numerical approximation. This fact, however, implies a major problem when looking at attractors of the numerical approximation: Even the upper limit set for vanishing time step of these attractors can be strictly smaller than the attractor of the approximated system, i.e. we only obtain semi-continuous convergence (see e.g. [4,Example (2.12)] for an illustration and also [6] for related results for finitedimensional approximations of infinite dimensional systems). In other words, the existence of "numerical attractors" does not imply the existence of a nearby attractor for the approximated ODE; one can only conclude that if the unperturbed system has an attractor A then there exist attractors for the numerical approximations whose upper limit is contained in A. Unfortunately, also the knowledge about the existence of the absorbing sets close to the original attractor does not help in general, since in this case for each time step there exist infinitely many absorbing sets, and thus it is difficult to single out those sets approximating the attractor.
In Chapter 7 of the monograph [14] these results are discussed in detail, furthermore in Section 7.7 of this reference several conditions for convergence are given. For example, it is shown that convergence holds if both the continuous and the numerical attractors attract exponentially (in this case also an estimate for the rate of convergence is given), or if the continuous time attractor consists of the unstable manifolds of finitely many hyperbolic equilibrium points (which is shown to be true for gradient systems with a bounded set of hyperbolic equilibria). These conditions have in common that certain assumptions on the dynamics of the approximated ODE are made.
In the present paper we take the converse point of view. We ask whether it is possible to obtain criteria for convergence just by looking at the numerical approximations (this approach is also taken in [9] for attracting sets of Galerkin approximations to Navier-Stokes equations using, however, rather different techniques). And in fact we can give a positive answer, at least under the assumption that we have information about the behaviour of the numerical aproximations for arbitrarily small time steps. More precisely we prove that the upper limit of the numerical attractors for vanishing time step is a "true" attractor if and only if for arbitrarily small time steps the numerical one-step schemes admit attracting sets with uniform attraction rate approximating this upper limit set (the suitable concept of attraction rate is defined precisely in Section 2). Furthermore, we show that if the numerical attractors themselves are attracting with uniform rate, then they converge to some set A if and only if this set A is an attractor for the approximated ODE. In this case, we can also give an estimate for the rate of convergence.
It is clear that an assumption "for arbitrarily small time steps" will be hard to check rigorously in practice. Nevertheless, the results suggest the following procedure for numerical simulations: When an attractor is observed in a numerical system then redo the computation with different time steps and compare the rates of attraction. If these rates vary for different time steps then the observed attractor is likely to be a numerical artefact and the numerical results should be interpreted with care.
Concerning a rigorous verification, we expect that the relation of the assumed rates of attraction to Lyapunov functions (see Remark 2.6) might lead to practicable ways of checking the desired property. (The details are currently under investigation and will be addressed in a later paper.) Furthermore, the assumptions are always satisfied in the presence of hyperbolicity, with the additional nice outcome that in this case the order of convergence of the numerical attractors coincides with the order of convergence of the numerical scheme, see Remark 2.9. In any case, the results precisely show what is theoretically possible, and thus contribute to the understanding and interpretation of numerical results. This paper is organized as follows: In Section 2 we fix the setup and notation and state the main results. Section 3 provides some facts on attractors and attracting sets. In Section 4 two basic results on the stability of uniformly attracting sets are proved, and finally in Section 5 we prove the main results.

Setup and Main Results
We consider the ordinary differential equation in R ḋ for all x, y ∈ R d and constants M , L > 0. (These global assumptions can easily be weakened since we are only interested in the behaviour on compact subsets of the state space.) The solutions of (2.1) with initial value x 0 ∈ R d for initial time t 0 = 0 will be denoted by ϕ(t, x 0 ).
In order to provide a framework for numerical one-step approximations of (2.1) we fix some h 0 > 0 and consider difference equations for time steps h ∈ (0, h 0 ] where t ∈ hZ := {hk | k ∈ Z}, and Ψ h : A special case of (2.2) is the time-h map of (2.1) given by Note that L and M from (2.1) need to be slightly enlarged in order to meet the assumptions on (2.2).
Our main object of interest are the numerical one-step approximations of (2.1) (or, more precisely, of (2.3)) which we will denote by Here we assume that (2.4) is of type (2.2) and, in addition, for some p ∈ N and some K > 0. The value p is called the order of the scheme. Typical examples of these schemes are Runge-Kutta and Taylor schemes, which are described in any textbook on numerical methods for ordinary differential equations, see e.g. [2,8,13].
Each of these equations defines a (semi-)dynamical system either in continuous or discrete time, which we denote by ϕ(t, ·), Ψ h (t, ·), Φ h (t, ·) andΦ h (t, ·). In what follows we will sometimes write Φ(t, ·) which can be either a continuous or a discrete time system, and set T = R for continuous time and T = hZ for discrete time systems, the proper meaning being clear from the context. Furthermore, we abbreviate T + := {t ∈ T | t > 0}, and for subsets B ⊂ R d we use the convention Φ(t, B) = x∈B {Φ(t, x)}.
As we are going to derive estimates for distances between compact sets we briefly recall basic definitions for several of these distances. and, if C ⊆ D, the minimal distance by For ε > 0 we denote the ε-ball around C by B(ε, Now we can define our objects of interest.

Given two compact forward invariant sets
A is called a attractor (with attracted neighbourhood B) if it is invariant and attracting with attracted neigbourhood B.
Note that A is a local attractor here, i.e. it is not assumed that each compact set B ⊂ R d is an attracted neighbourhood.
The following definition, which is a slight variation of [5, Definition 2.8], is a tool to give some structure to the attraction property.
A family of compact, forward invariant sets B ϑ , ϑ ∈ R + 0 , which depend continuously on ϑ (w.r.t. the Hausdorff metric d H ) and satisfy B ⊆ B 0 is called a contracting family of neighbourhoods if The next definition introduces a measure for the rate of convergence of B ϑ to A as ϑ → ∞.

Definition 2.4 Consider a contracting family of neighbourhoods B ϑ and some class
Throughout this paper the β-shrinking property will be used to give a uniformity condition for contracting families.
Next we give the property which will turn out to be crucial for estimating rates of convergence. Again, this is a slight variation of a concept from [5], namely of Definition 4.1 in this reference.
Definition 2.5 Let Φ = ϕ or Φ = Ψ h for some h > 0, and let γ be a class K function. We say that an attracting set A with attracted neighbourhood B is γ-attracting if it admits a contracting family of neighbourhoods B ϑ , ϑ ≥ 0, which for all ϑ ≥ 0 and all t ∈ [0, T ϑ ] ∩ T for some T ϑ ∈ T + satisfies the inequality and we call it (γ, β)-attracting if this contracting family can be chosen to be β-shrinking.
If, in addition, A is an attractor then we call A a γ-attractor or a (γ, β)-attractor, respectively.
Remark 2.6 (i) Each attracting set is (γ, β)-contracting for suitable γ, β of class K, which can be seen e.g. by taking the B ϑ as suitably parameterized sublevel sets of Lyapunov functions V provided by [14, Theorem 1.7.6 and 2.7.6] (the construction in [14] actually goes back to [3] and [15,Theorem 22.1] and was also used in [11]): Then the bounds on V and the Lipschitz continuity allow an explicit computation of γ and β. It is, however, an open question whether we can always find a Lyapunov function whose sublevel sets realize the optimal rates γ and β.
(ii) In the case of exponential attraction, i.e. when there exists constants C, is easily verified as a Lipschitz continuous Lyapunov function (cf. [15,Theorem 19 The construction from (i) then gives the appropriate sublevel sets. (For a direct construction of a contracting family of neighbourhoods in the exponential case we refer to [5,Section 7].) As an illustration of this concept consider the simple two-dimensional systeṁ form a contracting family with β(1/ϑ) = e −δϑ . The function γ, however, has to be chosen as γ(s) ≥ s δ . Thus exponential shrinking of the B ϑ does not necessarily imply γ(s) = cs, however, a suitable choice of this family (i.e. δ = 1) guarantees linearity of γ. An interesting (but to the author's knowledge unsolved) question in this context is whether for a given system one can always obtain a relation between β and γ by chosing a suitable family B ϑ .
Using the concept of (γ, β)-attraction we can now formulate our main results.
Theorem 2.7 Consider a positive sequence of time steps h n → 0 as n → ∞. Let A hn be attractors for the numerical schemeΦ hn with attracted neighbourhood B and consider the upper limit set Then A is an attractor for the continuous time system (2.1) if and only if there exist functions γ, β of class K and n 0 ∈ N such that for all n ≥ n 0 there exist (γ, β)-attracting In this case A is a (2γ(5 ·), 2β(4 ·))-attractor.
The proof is given in Section 5.
In other words, the upper limit A is an attractor if and only if we find uniformly attracting sets (i.e. all with the same rates γ and β) for vanishing time step which approximate A.
Note that it is not necessary that the attractors A hn themselves are (γ, β)-attracting. In this case, however, further implications are possible, as formulated in the following theorem.
Theorem 2.8 Consider a positive sequence h n → 0 as n → ∞ and two functions γ, β of class K. Assume there exist (γ, β)-attractors A hn with attracted neighbourhood B for the numerical schemeΦ hn and let A ⊂ intB be a compact set. Then the following four statements are equivalent.
i) A is an attractor with attracted neighbourhood B for the continuous time system (2.1).
The proof can also be found in Section 5. (ii) If the system (2.1) is uniformly hyperbolic in a neighbourhood of A, then also the one-step approximations are uniformly hyperbolic for h > 0 sufficiently small. In this case the attractors attract exponentially (see [7]), i.e. we obtain γ(s) = cs from Remark 2.6 (ii). Thus hyperbolicity implies for some suitable C > 0 without any additional assumptions on A and A h .

Some facts on attractors and attracting sets
For the convenience of the reader in this section we provide some facts about attractors and attracting sets for the system and its time-h map.
It is well known that each forward invariant attracting set contains an attractor (see e.g. [ Proof: Let A be an attractor with attracted neighbourhood B. Then in particular A is invariant. Now assume thatÃ ⊂ A,Ã = A, is a forward invariant attracting set. Then there exists a neighbourhood N ⊃Ã with A ⊆ N , such that Φ(t, B) ⊂ N for some t ∈ T + , 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 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. In the next two lemmas we investigate the relation between attracting sets and attractors for the continuous time system and its time-h map.

1). Then
A h is also an attractor with attracted neighbourhood B for system (2.1).

and by Lemma 3.2 it is contained in
Thus we can conclude ϕ(t, A h ) ⊆ A h for each t ∈ R, hence also A h = ϕ(−t, ϕ(t, A h )) ⊆ ϕ(−t, A h ) for each t ∈ R and consequently A h is invariant for ϕ.
Finally, since A h is invariant for ϕ and an attracting set for Φ h , by Lemma 3.3 it is also an attracting set for ϕ with attracted neighbourhood B, thus an attractor.

Stability of attracting sets
In this section we provide two stability results for (γ, β)-attracting sets in the Propositions 4.2 and 4.6. The first considers stability under perturbation of a difference equation, the second stability when passing to a limit of time-h maps as h → 0.
Before formulating these result we introduce two auxiliary systems which will be useful for the proofs. Consider the differential equation (2.1) and the difference equation (2.2). For these equations we consider the α-perturbed systemṡ with solutions ϕ α (t, x, u(·)) and Ψ α h (t, x, u h (·)), where u(·) ∈ U and u h (·) ∈ U h with U := {u : R → B(1, 0) | u measurable} and U h := {u h : hZ → B(1, 0)}. (Recall that B(1, 0) is the ball with radius 1 around the origin in R d ). The set valued maps are called the α-inflated dynamics (they can alternatively be defined via differential or difference inclusions, cf. [10]  We now turn to a more specific perturbation of difference equations which covers our numerical one-step approximations. For the difference equation (2.2) given by Ψ h we consider the perturbed systemΨ 2h with doubled time step 2h In order to estimate the difference between Ψ h andΨ 2h we make the following assumption on the perturbed system for some α > 0.
Note that we do not make any regularity assumptions onΨ. The doubled time step 2h is necessary in order to make a connection between attracting sets of Ψ h andΨ 2h , which will become clear in the proof of the following proposition. This provides a stability result for (γ, β)-attracting sets under perturbations of type (4.1).
Proof: Consider the contracting family of neighbourhoods B ϑ realizing the assumed attraction rate and fix some α ∈ (0, α 0 ]. We choose ϑ α minimal such that and define a family of neighbourhoods B 2h ϑ of A α 2h by The continuity and β(2 ·)-shrinking w.r.t. A α 2h of this family follow immediately from the construction. Furthermore, which yields the desired estimate for the γ-contraction the B α ϑ . In addition, for each ϑ ≥ 0 with ϑ ≤ 2ϑ α − 2h the construction implies which shows that the B α ϑ form a contracting family and thus finishes the proof. We now turn to the question of (α, β)-attraction for the continuous time system (2.1) for limits of (α, β)-attracting sets for its time-h map (2.3) as h → 0. Before proving the desired result in Proposition 4.6 we provide a series of lemmas necessary for the proof.  ϕ(t, B), B) ≥ Ct for all t ∈ [0, T ] and some T > 0. Then for each t ∈ R + the inequality d min (ϕ(t, B), B) ≥ tC/(Lt + 1) holds for the Lipschitz constant L of (2.1).
We will make use of the following concept: Consider a family of compact sets B i ⊂ R d . Then we define the limes inferior of these sets by We refer to [1] for more information on this concept. The following Lemma summarizes those properties of this set which we will need in what follows. (ii) For all α ≥ 0 and all u(·) ∈ U we have ϕ α (t, B, u(·)) ⊆ Liminf i→∞ ϕ α (t, B i , u(·)).
In order to prove (iii) let x ∈ ϕ infl (t, B, α). Then there exist y ∈ B and u(·) ∈ U such that x = ϕ α (t, y, u(·)). Hence using first (ii) and then the assumption and (i) we obtain and hence the assertion.
For the proof of (iv) observe that by (i) the assumption implies B ⊆ C. Hence we obtain the equalities for all i ∈ N sufficiently large and for each point x ∈ R d and any sequence x i with x i → x as i → ∞. Now let x ∈ C. Then by the definition of Liminf there exists a sequence x i ∈ C i with x i → x as i → ∞. Thus Since x was arbitrary in C this implies the assertion.
Now we have all necessary tools for proving the second stability result.
Proposition 4.6 Consider a sequence h n → 0, and assume that there exist functions γ, β of class K and (γ, β)-attracting sets with attracted neighbourhood B for the time-h n map Φ hn of the continuous time system (2.1). Assume, furthermore, that there exists a set A with d H (A hn , A) → 0. Then for each C > 1 the set A is (Cγ(C ·), Cβ(2 ·))-attracting for the continuous time system (2.1).
Proof: Fix some C > 1 and letC ∈ (1, C). Assuming all h n to be sufficiently small and to be monotone decreasing (by taking an appropriate subsequence) by Lemma 4.4 we know that the A hn are (Cγ(C ·),Cβ(2 ·)) attracting for Φ 2hn , with contracting families B n ϑ satisfying We claim that the family of sets Then the assumption implies Φ 2h n (t, B n ϑ 1 ) ⊆ B n ϑ 2 for each t ∈ 2h n Z, t ≥ ϑ 2 − ϑ 1 , and consequently for each ε > 0 and each n with 2h n ≤ ε the inclusion ϕ(ϑ 2 −ϑ 1 +ε, B n ϑ 1 ) ⊆ B n ϑ 2 holds. Thus by Lemma 4.5 (ii) and (i) The same construction shows Property (iii) of Definition 2.3, and the desired distance from A follows from Lemma 4.5 (iv).

Proofs of Theorem 2.7 and Theorem 2.8
Proof of Theorem 2.7: Throughout the proof we fix the attracted neighbourhood B, i.e. "attracting" is to be understood as "attracting with attracted neighbourhood B".
Let A be an attractor. Then by Remark 2.6 there exist functionsγ,β such that A is (γ,β)-attracting for the continuous time system, thus also for each time-h n /2 map. Hence by Proposition 4.2 with h = h n /2, A h = A, Ψ h = Φ h ,Ψ 2h =Φ 2h and α = Kh p , for γ(r) =γ(4r), β(r) =β(2r) and all sufficiently small h n > 0 there exist (γ, β)-attracting setsÃ hn for the one-step scheme converging to A as h n → 0. By Lemma 3.1 these sets contain the attractors A hn .
By Lemma 3.1 there exists an attractorĀ ⊆ A. AssumeĀ = A, then there exists a compact neighbourhood N ⊂ R d withĀ ⊂ intN and A ⊂ N . As in the first part of the proof, by Remark 2.6 and Proposition 4.2 we can conclude the existence of attracting sets A h forΦ h converging toĀ. In particular this yieldsĀ h ⊂ N for h > 0 sufficiently small. Since the A hn are attractors forΦ hn , by Lemma 3.1 they must be contained in theĀ hn , which implies A hn ⊂ N , and hence Proof of Theorem 2.8: As in the proof of Theorem 2.7 we fix the attracted neighbourhood B. Now "(iv) ⇒ (iii)" and "(ii) ⇒ (i)" are obvious and "(iii) ⇒ (ii)" follows immediately from Theorem 2.7.
Hence by the assumption on the A hn and by Proposition 4.2 with h = h n , Ψ h =Φ h , Ψ 2h = Φ 2h and α = (1 + e Lhn )Kh p n we can conclude the existence of attracting setsĀ 2hn for the time-2h n map satisfying d h (Ā 2hn , A hn ) ≤ γ(10Kh p n ) for all h n sufficiently small. By Lemma 3.1 and Lemma 3.4 we know A ⊂Ā 2hn , hence dist(A, A hn ) ≤ γ(10Kh p n ) which shows (iv).
Finally, we show "(i) ⇒ (iii)", which finishes the proof: Observe by Remark 2.6 that there exist class K functionsγ,β such that A is a (γ,β)attractor. Without loss of generality we may assumeγ ≥ γ andβ ≥ β. Hence also the A hn are (γ,β)-attractors, and by the same arguments as for "(ii) ⇒ (iv)", above, we obtain d H (A, A hn ) ≤γ(Ch p n ) for some suitable C > 0 implying (iii).