Asymptotic invariance and the discretisation of nonautonomous forward attracting sets

The $\omega$-limit set $\omega_B$ of a nonautonomous dynamical system generated by a nonautonomous ODE with a positive invariant compact absorbing set $B$ is shown to be asymptotic positive invariant in general and asymptotic negative invariant if, in addition, the vector field is uniformly continuous 
in time on the absorbing set. This set has been called the forward attracting set of the nonautonomous dynamical system and is related to Vishik's concept of a uniform attractor. 
If $\omega_B$ is also assumed to be uniformly attracting, then its upper semi continuity in a parameter and the upper semi continuous convergence of its counterparts under discretisation by the implicit Euler scheme are established.

(Communicated by Wolf-Jürgen Beyn) Abstract. The ω-limit set ω B of a nonautonomous dynamical system generated by a nonautonomous ODE with a positive invariant compact absorbing set B is shown to be asymptotic positive invariant in general and asymptotic negative invariant if, in addition, the vector field is uniformly continuous in time on the absorbing set. This set has been called the forward attracting set of the nonautonomous dynamical system and is related to Vishik's concept of a uniform attractor. If ω B is also assumed to be uniformly attracting, then its upper semi continuity in a parameter and the upper semi continuous convergence of its counterparts under discretisation by the implicit Euler scheme are established.
1. Introduction. Invariant sets play a fundamental role in characterising the behaviour of dynamical systems, in particular their attractors. The weaker concepts of positive and negative invariance are also very useful in describing more easily determined absorbing sets that contain an attractor, in the first case, or in establishing the upper semi continuity of attractors in a parameter and the upper semi continuous convergence of discretised attractors in the second case [8,7,16,17]. This applies for autonomous dynamical systems and also for the more complicated situation of nonautonomous dynamical systems whose attractors consist of an invariant family of time-dependent subsets, A = {A t : t ∈ R}, which attracts all bounded subsets of the state space in either the pullback or forward sense [2,3,9,10,15].
In autonomous dynamical systems the attractor corresponds to the ω-limit set ω B of an absorbing set B. Although this set is defined as the limit as time goes to infinity, it actually exists for all time, since the time variable in an autonomous dynamical system π(t, x) is just the elapsed time from the start and not the actual time. The situation is rather different in a nonautonomous dynamical system φ(t, t 0 , x) which depends on both the actual time and starting time and not only their difference. ω-limit sets can be defined here, but depend on the starting time, i.e., ω B,t0 [14]. These are increasing in t 0 and the closure of their union ω B represents the set of all future limit points of the system. Vishik [18] introduced the concept of a uniform attractor as the smallest set that attracts all bounded sets uniformly in the initial time. It contains all forward ω-limit points, hence ω B , but can be a larger set as an example in the appendix of [2] shows. However neither this uniform attractor nor ω B need be invariant or even positive invariant, e.g., as the scalar ODEẋ = −x + e −t with ω B = {0} shows.
Bortolan et al [1] introduced the concept of lifted invariance of a set A, by which they mean, in terms of the processes or two-parameter semi-groups considered here, that for any point x 0 in A there is some time t 0 and an entire trajectory contained in A and taking the value x 0 at time t 0 . They show that the set ω B of a nonautonomous dynamical system formulated as a skew-product flow is lifted invariant, but their proof assumes that the state space of the driving system has a compact invariant subset, which does not apply for processes. Some related concepts for limiting equations and their associated skew-product flows are given in Chapter 4 and the appendix of LaSalle [14].
Recently, Kloeden & Yang [12] showed ω B is asymptotic positive invariant for nonautonomous difference equations. This concept has been long known in the differential equations literature [6,13]. In this note it will be shown that ω B is also asymptotic negative invariant provided the vector field of the nonautonomous ODE and its spatial gradient are uniformly continuous in time inside the compact absorbing set B.
These results are then used to show the upper semi continuity dependence of ω B on parameters and the upper semi continuous convergence of its counterparts for the implicit Euler scheme when, in addition, ω B is uniformly attracting.
The results here also hold for more general one-step schemes as well as more general dissipativity asumptions on the ODE. The main issue is to show that the numerical scheme inherits the dissipativity of the ODE and hence has a nonempty ω-limit set. This will be discussed briefly in the last section. This property is easily established for the implicit Euler scheme considered here, which allows the focus of the paper to be on the essentially new ideas of asymptotic invariance and the required uniformity assumptions.

Dissipative nonautonomous system. Consider a nonautonomous ODE in
Also assume that the nonautonomous system (2.1) is dissipative. In particular, assume that the nonautonomous system (2.1) is ultimately bounded in the closed and bounded (hence compact) subset B in R d .

Assumption 1.
There exists a φ-positive invariant compact subset B in R d such that for any bounded subset D of R d and every t 0 ≥ T * there exists a T D ≥ 0 for which It follows from this dissipativity assumption and the compactness of the set B that the ω-limit set is a nonempty compact set of R d for each t 0 ≥ T * . In fact, by the positive invariance is the Hausdorff semi-distance between nonempty compact subsets of R d ).
Hence, the set is nonempty and compact. It contains all of the future limit points of the process starting in the set B at some time The set ω B characterises the forward asymptotic behaviour of the nonautonomous system. It is closely related to what Vishik [18] (see also [1,2,3]) called the uniform attractor, but it may be smaller and does not require the generating ODE (2.1) to be defined for all time or the attraction to be uniform in the initial time [9,12].
Furthermore, the simple exampleẋ = −x + e −t with ω B = {0} shows that the set ω B need not be invariant or even positive invariant.
2.1. Asymptotic positive invariance. The set ω B = {0} in the previous example appears to become more and more invariant the later one starts in the future. This motivated the concept of asymptotic positive invariance in the literature for differential equations [6,13]. Positive invariance says that a set is mapped into itself at future times, while asymptotic positive invariance says it is mapped almost into itself and closer the later one starts. Definition 2.1. A set A is said to be asymptotic positively invariant if for any monotonic decreasing sequence ε p → 0 as p → ∞ there exists a monotonic increasing sequence T p → ∞ as p → ∞ such that The following result was proved in [12] for difference equations. The proof, essentially the same, is repeated here in the present context for the reader's convenience.
Theorem 2.2. Let the above assumptions hold. Then ω B is asymptotic positively invariant.
Suppose for an ε 1 > 0 that there are sequences t 0,j ≤ t j ≤ t 0,j + T 0 (t 0,j , ε 1 ) with t 0,j → ∞ as j → ∞ such that Define y j := φ(t j , t 0,j , b j ). Since the points y j ∈ B, which is compact, there exists a convergent subsequence y j k →ȳ ∈ B. Moreover,ȳ ∈ ω B by the definition. However, Hence for this ε 1 > 0 there exists The argument can be repeated inductively with ε p+1 < ε p and t p+1 (ε p+1 ) > t p (ε p ). It follows that ω B is asympotic positively invariant.
The set ω B was called the forward attracting set of the nonautonomous system in [12].
3. Asymptotic negative invariance. The concept of negative invariance of a set implies that any point in it can be reached in any even time from another point in it. The set ω B is generally not negatively invariant, but under an additional uniformity assumption it is asymptotic negatively invariant.
To show that this property holds the future uniform behaviour of the vector field f in time, the following condition is needed. . This assumption holds if, for example, f has the form f (x, t) = f (x, φ(t)), where φ : [T * , ∞) → R m is uniformly bounded and uniformly continuous, see [11], such as an almost periodic or recurrent function.
By Assumption 2, the vector field f of the ODE (2.1) is Lipschitz on the compact absorbing set B uniformly in time, i.e., Thus for any two solutions ϕ(t, τ, where 0 ≤ t − τ ≤ T . Note that the bound depends just on the length of the time interval and not on the starting point of the interval. Proof. To show this let ω ∈ ω B , ε > 0 and T > 0 be given. Then there exist sequences b n ∈ B and τ n < t n with τ n → ∞ and an integer N (ε) such that Define a n := ϕ (t n − T, τ n , b n ) ∈ B. Since B is compact, there exists a convergent subsequence a nj := ϕ t nj − T, τ nj , b nj → ω ε as n j → ∞. By definition, ω ε ∈ ω B .
From Assumption 2 the process ϕ is continuous in initial conditions uniformly on finite time intervals of the same length, i.e., (3.2). Hence By the 2-parameter semi-group property ϕ t nj , t nj − T, a nj = ϕ t nj , t nj − T, ϕ t nj , τ nj , b nj = ϕ t nj , τ nj , b nj . Then This is the desired result.

Upper semi continuity in a parameter. Now consider a parameterised family of nonautonomous ODEs in
Let φ ν (t, t 0 , x 0 ) denote the unique solution with the initial value x(t 0 ) = x 0 . Assumption 1 is strengthened so the ODEs (4.1) are equi-ultimately bounded in the compact absorbing B uniformly in ν ∈ [0, ν * ].
Assumption 3. The exists nonempty compact set B which is φ ν -positive invariant for each ν ∈ [0, ν * ] and for any bounded subset D of R d and t 0 ≥ T * there exists a T D ≥ 0 (independent of ν) such that This holds, for example, if the vector fields of the ODEs (4.1) satisfy a dissipative inequality such as: there exists an R * > 0 such that In this case, B is the closed and bounded ball B 0 [R * ] of radius R * about the origin in R d .
It will also be assumed that the vector fields of the ODEs (4.1) converge as the parameter ν → 0 uniformly in time on the set B.
The continuous convergence of the solutions of the ODEs (4.1) as the parameter ν → 0 on the set B uniformly in time then follows.
Proof. For the solutions ϕ 0 (t, τ, x 0 ) and ϕ ν (t, τ, x 0 ) of the ODEs (2.1) and (4.1) in the set B Finally, the uniform attraction of the set ω 0 B for the system ϕ 0 is needed for the following result.
Assumption 5. ω 0 B uniformly attracts the set B, i.e., for every ε > 0 there exists a T (ε), which is independent of t 0 ≥ T * , such that Proof. A proof by contradiction will be used. Suppose for some sequence of parameters ν j → 0 that the above limit is not true, i.e., there exists an ε 0 > 0 such that Then use Lemma 4.1 with this T to pick a ν j < δ(ε 0 /2, T ) to ensure that Fix such a ν j and use the asymptotic negative invariance of ω νj B to obtain the existence of an ω j,T ∈ ω νj B and a t j ε 0 so that Then, with t 0 taken as t j ε above, which contradicts the assumption (4.2).
Assumption 6. The vector field of the ODE (2.1) satisfies the dissipative inequality for some R * > 0.
Thus the compact ball B 0 [R * ] of radius R * centered at the origin in R d is positive definite absorbing set uniformly in the initial time t 0 for the ODE (2.1). Thus, the limit set ω B exists and is contained in It will also be assumed in this section that the vector field and its gradient are uniformly continuous in future time on B, i.e, satisfy Assumption 2, and that ω B is uniformly attracting, i.e, satisfies Assumption 6.
The ODE (2.1) will be discretised using variable time steps h n . Let H be the set of infinite sequence h = {h 0 , h 1 , h 2 , · · · } with values in (0, 1] such that ∞ n=0 h n = ∞ and let T(h, t 0 ) be set of times t n = t 0 + n−1 j=0 h j for a given (but otherwise arbitrary) t 0 ≥ T * .
The implicit Euler scheme for the ODE (2.1) and the initial value x 0 and t 0 ≥ T * is given by for a step size sequence h ∈ H with {t n } ∈ T(h, t 0 ). The implicit equation (5.1) is uniquely solvable for sufficiently small step sizes [17]. Thus the implicit Euler scheme n is the nth iterate of (5.1) at time t n starting at x 0 at time t 0 . The process Φ (h) is continuous in its initial value, see Lemma 5.2 below.
The Implicit Euler scheme (5.1) is also dissipative. Proof. By the inequality in Assumption 6 where L B be the Lipschitz constant of the vector field f on the set B, which gives Thus, the discrete time process Φ (h) corresponding to the implicit Euler scheme has a nonempty compact limit sets ω for all time step sequences h ∈ H with L B h ∞ < 1. See also [16].
The proofs of the following theorems are essentially the same as those of their continuous time counterparts, Theorems 2.2 and 3.2. is asymptotic negative invariant for the implicit Euler scheme (5.1).
The implicit Euler scheme (5.1) is a first order scheme. Under the above uniformity assumptions its global discretisation error depends only on the length of the time interval under consideration and not its starting time.
Lemma 5.5. Let Assumptions 2 and 6 hold and let L B h ∞ < 1. Then the global discretisation error of the implicit Euler scheme (5.1) on the set B satisfies the uniform estimate for all x 0 ∈ B, t 0 ≥ T * and t n ∈ [t 0 , t 0 + T ] for any finite T > 0, where h = h ∞ and K T does not depend on t 0 ≥ T * .
Proof. The estimate (5.2) is obtained [17] by iterating a difference inequality and using the local discretisation error for all x 0 ∈ B, t 0 ≥ T * and h ∈ (0, 1]. The local error (5. Proof. Suppose for some sequence of time step sequences h j ∈ H with h j ∞ → 0 that this limit is not true. Then there exists ε 0 > 0 such that Let t j n → ∞ as n → ∞ be the time sequence in T(h j , t 0 ) determined by adding the time steps starting at t 0 and let Φ (h j ) (n, m, b) be the value of the numerical scheme at time t j n starting at b at time t j m , where n ≥ m. First, by Assumption 5 pick T 0 = T (ε 0 /4) such that for any t 0 ≥ T * dist R d (ϕ (t 0 + T 0 , t 0 , B) , ω B ) < 1 4 ε 0 Next consider the global discretisation error of the implicit Euler scheme (5.1) on the interval of length 2T 0 , i.e., ϕ (h j ) (m, n, b) − ϕ t j m , t j n , b ≤ K 2T0 h j ∞ , t j n ≤ t j m ≤ t j n + 2T 0 , b ∈ B.
Recall that by Lemma 5.5 the constant K 2T0 is independent of the starting time t j n .
Under Assumption 6 the dissipativity of the inclusion (6.1) follow easily: Modifications of the arguments in Kloeden & Lorenz [8] with the Lyapunov function V (x) := max{ x 2 − (R * ) 2 , 0} can then be used to show that one-step scheme is also dissipative. More generally, when the ODE (2.1) satisfies Assumption 1, there are general existence theorems of Lyapunov function characterising this equi-ultimate boundedness of the system that can be used.