Synchronization of Dissipative Dynamical Systems Driven by Non-Gaussian Lévy Noises

Abstract

Dynamical systems driven by Gaussian noises have been considered extensively in modeling, simulation, and theory. However, complex systems in engineering and science are often subject to non-Gaussian fluctuations or uncertainties. A coupled dynamical system under a class of Lévy noises is considered. After discussing cocycle property, stationary orbits, and random attractors, a synchronization phenomenon is shown to occur, when the drift terms of the coupled system satisfy certain dissipativity and integrability conditions. The synchronization result implies that coupled dynamical systems share a dynamical feature in some asymptotic sense.

1. Introduction

Synchronization of coupled dynamical systems is an ubiquitous phenomenon that has been observed in biology, physics, and other areas. It concerns coupled dynamical systems that share a dynamical feature in an asymptotic sense. A descriptive account of its diversity of occurrence can be found in the recent book [1]. Recently Caraballo and Kloeden [2, 3] proved that synchronization in coupled deterministic dissipative dynamical systems persists in the presence of various Gaussian noises (in terms of Brownian motion), provided that appropriate concepts of random attractors and stochastic stationary solutions are used instead of their deterministic counterparts.

In this paper we investigate a synchronization phenomenon for coupled dynamical systems driven by nonGaussian noises. We show that couple dissipative systems exhibit synchronization for a class of Lévy motions.

This paper is organized as follows. We first recall some basic facts about random dynamical systems (RDSs) as well as formulate the problem of synchronization of stochastic dynamical systems driven by Lévy noises in Section 2. The main result (Theorem 3.3) and an example are presented in Section 3.

Throughout this paper, the norm of a vector 𝑥 in Euclidean space ℝ^𝑑 is always denote by |𝑥|.

2. Dynamical Systems Driven by Lévy Noises

Dynamical systems driven by nonGaussian Lévy motions have attracted much attention recently [4, 5]. Under certain conditions, the SDEs driven by Lévy motion generate stochastic flows [4, 6], and also generate random dynamical systems (or cocycles) in the sense of Arnold [7]. Recently, exit time estimates have been investigated by Imkeller and Pavlyukevich [8], and Imkeller et al. [9], and Yang and Duan [10] for SDEs driven by Lévy motion. This shows some qualitatively different dynamical behavior between SDEs driven by Gaussian and nonGaussian noises.

2.1. Lévy Processes

A Lévy process or motion on ℝ^𝑑 is characterized by a drift parameter 𝛾∈ℝ^𝑑, a covariance 𝑑×𝑑 matrix A, and a nonnegative Borel measure 𝜈, defined on (ℝ^𝑑,ℬ(ℝ^𝑑)) and concentrated on ℝ^𝑑⧵{0}, which satisfies

_{ℝ𝑑⧵{0}}(|𝑦|²1)𝜈(𝑑𝑦)<∞, (2.1) or equivalently

_{ℝ𝑑⧵{0}}|𝑦|²1+|𝑦|²𝜈(𝑑𝑦)<∞. (2.2) This measure 𝜈 is the so-called the Lévy jump measure of the Lévy process 𝐿_𝑡. Moreover Lévy process 𝐿_𝑡 has the following Lévy-Itô decomposition:

𝐿_𝑡=𝛾𝑡+𝐵_𝑡+_|𝑥|<1𝑥N(𝑡,𝑑𝑥)+_|𝑥|≥1𝑥𝑁(𝑡,𝑑𝑥), (2.3) where 𝑁(𝑑𝑡,𝑑𝑥) is Poisson random measure,

N(𝑑𝑡,𝑑𝑥)=𝑁(𝑑𝑡,𝑑𝑥)−𝜈(𝑑𝑥)𝑑𝑡, (2.4) is the compensated Poisson random measure of 𝐿_𝑡, and 𝐵_𝑡 is an independent Brownian motion on ℝ^𝑑 with covariance matrix 𝐴 (see [4, 10–12]). We call (𝐴,𝜈,𝛾) the generating triplet.

General semimartingales, especially Lévy motions, are thought to be appropriate models for nonGaussian processes with jumps [11]. Let us recall that a Lévy motion 𝐿_𝑡 is a nonGaussian process with independent and stationary increments. Moreover, its sample paths are only continuous in probability, namely, ℙ(|𝐿_𝑡−𝐿_𝑡0|≥𝛿)→0 as 𝑡→𝑡₀ for any positive 𝛿. With a suitable modification [4], these paths may be taken as càdlàg, that is, paths are continuous on the right and have limits on the left. This continuity is weaker than the usual continuity in time. In fact, a càdlàg function has finite or at most countable discontinuities on any time interval (see, e.g., [4, page 118]). This generalizes the Brownian motion 𝐵_𝑡, since 𝐵_𝑡 satisfies all these three conditions, but additionally, (i) almost every sample path of the Brownian motion is continuous in time in the usual sense, and (ii) the increments of Brownian motion are Gaussian distributed.

The next useful lemma provides some important pathwise properties of 𝐿_𝑡 with two-sided time 𝑡∈ℝ.

Lemma 2.1 (pathwise boundedness and convergence). Let 𝐿_𝑡 be a two-sided Lévy motion on ℝ^𝑑 for which 𝔼|𝐿₁|<∞ and 𝔼𝐿₁=0. Then we have the following.(i)lim_𝑡→±∞(1/𝑡)𝐿_𝑡=0,𝑎.𝑠.(ii)The integrals ∫^𝑡_−∞𝑒^{−𝜆(𝑡−𝑠)}𝑑𝐿_𝑠(𝜔) are pathwisely uniformly bounded in 𝜆>1 on finite time intervals [𝑇₁,𝑇₂] in ℝ.(iii)The integrals ∫^𝑡_𝑇1𝑒^{−𝜆(𝑡−𝑠)}𝑑𝐿_𝑠(𝜔)→0 as 𝜆→∞, pathwise on finite time intervals [𝑇₁,𝑇₂] in ℝ.

Proof. (i) This convergence result comes from the law of large numbers, in [11, Theorem 36.5].
(ii) Since the function ℎ(𝑡)=𝑒^−𝜆𝑡 is continuous in t, integrating by parts we obtain
^𝑡_−∞𝑒^{−𝜆(𝑡−𝑠)}𝑑𝐿_𝑠(𝜔)=𝐿_𝑡(𝜔)−𝜆^𝑡_−∞𝑒^{−𝜆(𝑡−𝑠)}𝐿_𝑠(𝜔)𝑑𝑠. (2.5) Then by (i) and the fact that every càdlàg function is bounded on finite closed intervals, we conclude (ii).
(iii) Integrating again by parts, it follows that
^𝑡_𝑇1𝑒^{−𝜆(𝑡−𝑠)}𝑑𝐿_𝑠(𝜔)=(𝐿_𝑡−𝐿_𝑇1)𝑒^{−𝜆(𝑡−𝑇1)}+𝜆^𝑡_𝑇1𝑒^{−𝜆(𝑡−𝑠)}(𝐿_𝑡(𝜔)−𝐿_𝑠(𝜔))𝑑𝑠, (2.6) from which the result follows.

Remark. The assumptions on 𝐿_𝑡 in the above lemma are satisfied by a wide class of Lévy processes, for instance, the symmetric 𝛼-stable Lévy motion on ℝ^𝑑 with 1<𝛼<2. Indeed, in this case, we have ∫_|𝑥|>1|𝑥|𝜈(𝑑𝑥)<∞, and then 𝐸|𝐿₁|<∞, see [11, Theorem 25.3].

For the canonical sample space of Lévy processes, that is, Ω=𝐷(ℝ,ℝ^𝑑) of càdlàg functions which are defined on ℝ and taking values in ℝ^𝑑 is not separable, if we use the usual compact-open metric. However, it is complete and separable when endowed with the Skorohod metric (see, e.g., [13], [14, page 405]), in which case we call 𝐷(ℝ,ℝ^𝑑) a Skorohod space.

2.2. Random Dynamical Systems

Following Arnold [7], a random dynamical system (RDS) on a probability space (Ω,ℱ,ℙ) consists of two ingredients: a driving flow 𝜃_𝑡 on the probability space Ω, that is, 𝜃_𝑡 is a deterministic dynamical system, and a cocycle mapping 𝜑∶ℝ×Ω×ℝ^𝑑→ℝ^𝑑, namely, 𝜑 satisfies the conditions

𝜑(0,𝜔)=𝑖𝑑_ℝ^𝑑,𝜑(𝑡+𝑠,𝜔)=𝜑(𝑡,𝜃_𝑠𝜔)∘𝜑(𝑠,𝜔) (2.7) for all 𝜔∈Ω and all 𝑠,𝑡∈ℝ. This cocycle is required to be at least measurable from the 𝜎-field ℬ(ℝ)×ℱ×ℬ(ℝ^𝑑) to the 𝜎-field ℬ(ℝ^𝑑).

For random dynamical systems driven by Lévy noise we take Ω=𝐷(ℝ,ℝ^𝑑) with the Skorohod metric as the canonical sample space and denote by ℱ∶=ℬ(𝐷(ℝ,ℝ^𝑑)) the associated Borel 𝜎-field. Let 𝜇_𝐿 be the (Lévy) probability measure on ℱ which is given by the distribution of a two-sided Lévy process with paths in 𝐷(ℝ,ℝ^𝑑).

The driving system 𝜃=(𝜃_𝑡,𝑡∈ℝ) on Ω is defined by the shift

(𝜃_𝑡𝜔)(𝑠)∶=𝜔(𝑡+𝑠)−𝜔(𝑡). (2.8) The map (𝑡,𝜔)→𝜃_𝑡𝜔 is continuous, thus measurable ([7, page 545]), and the (Lévy) probability measure is 𝜃-invariant, that is,

𝜇_𝐿(𝜃_𝑡⁻¹(𝐴))=𝜇_𝐿(𝐴) (2.9) for all 𝐴∈ℱ, see [4, page 325].

We say that a family 𝐴={𝐴(𝜔),𝜔∈Ω} of nonempty measurable compact subsets 𝐴(𝜔) of ℝ^𝑑 is 𝑖𝑛𝑣𝑎𝑟𝑖𝑎𝑛𝑡 for a RDS (𝜃,𝜑), if 𝜑(𝑡,𝜔,𝐴(𝜔))=𝐴(𝜃_𝑡𝜔) for all 𝑡>0 and that it is a random attractor if in addition it is pathwise pullback attracting in the sense that

𝐻^∗_𝑑(𝜑(𝑡,𝜃_−𝑡𝜔,𝐷(𝜃_−𝑡𝜔)),𝐴(𝜔))→0as𝑡→∞ (2.10) for all suitable families (called the attracting universe) of 𝐷={𝐷(𝜔),𝜔∈Ω} of nonempty measurable bounded subsets 𝐷(𝜔) of ℝ^𝑑, where 𝐻^∗_𝑑(𝐴,𝐵)=sup_𝑥∈𝐴inf_𝑦∈𝐵|𝑥−𝑦| is the Hausdorff semi-distance on ℝ^𝑑.

The following result about the existence of a random attractor may be proved similarly as in [2, 15–18].

Lemma 2.3 (random attractor for càdlàg RDS). Let (𝜃,𝜑) be an RDS on Ω×ℝ^𝑑 and let 𝜑 be continuous in space, but càdlàg in time. If there exits a family 𝐵={𝐵(𝜔),𝜔∈Ω} of nonempty measurable compact subsets 𝐵(𝜔) of ℝ^𝑑 and a 𝑇𝐷^,𝜔≥0 such that 𝜑(𝑡,𝜃_−𝑡𝜔,𝐷(𝜃_−𝑡𝐷𝜔))⊂𝐵(𝜔),∀𝑡≥𝑇_,𝜔, (2.11) for all families 𝐷={𝐷(𝜔),𝜔∈Ω} in a given attracting universe, then the RDS (𝜃,𝜑) has a random attractor 𝐴={𝐴(𝜔),𝜔∈Ω} with the component subsets defined for each 𝜔∈Ω by 𝐴(𝜔)=_{𝑠>0𝑡≥𝑠}𝜑(𝑡,𝜃_−𝑡𝜔,𝐵(𝜃_−𝑡𝜔)). (2.12) Forevermore if the random attractor consists of singleton sets, that is, 𝐴(𝜔)={𝑋^∗(𝜔)} for some random variable 𝑋^∗, then 𝑋^∗_𝑡(𝜔)=𝑋^∗(𝜃_𝑡𝜔) is a stationary stochastic process.

We also need the following Gronwall's lemma from [19].

Lemma. Let 𝑥(𝑡) satisfy the differential inequality 𝑑𝑑𝑡₊𝑥≤𝑔(𝑡)𝑥+ℎ(𝑡), (2.13) where (𝑑/𝑑𝑡₊)𝑥∶=lim_ℎ↓0⁺((𝑥(𝑡+ℎ)−𝑥(𝑡))/ℎ) is right-hand derivative of 𝑥. Then 𝑥(𝑡)≤𝑥(0)exp[^𝑡₀𝑔(𝑟)𝑑𝑟]+^𝑡₀exp[^𝑡_𝑠𝑔(𝑟)𝑑𝑟]ℎ(𝑠)𝑑𝑠. (2.14)

2.3. Dissipative Synchronization

Suppose that we have two autonomous ordinary differential equations in ℝ^𝑑,

𝑑𝑥𝑑𝑡=𝑓(𝑥),𝑑𝑦𝑑𝑡=𝑔(𝑦), (2.15) where the vector fields 𝑓 and 𝑔 are sufficiently regular (e.g., differentiable) to ensure the existence and uniqueness of local solutions, and additionally satisfy one-sided dissipative Lipschitz conditions

max{⟨𝑥₁−𝑥₂,𝑓(𝑥₁)−𝑓(𝑥₂)⟩,⟨𝑥₁−𝑥₂,𝑔(𝑥₁)−𝑔(𝑥₂)⟩}≤−𝑙|𝑥₁−𝑥₂|² (2.16) on ℝ^𝑑 for some 𝑙>0. These dissipative Lipschitz conditions ensure existence and uniqueness of global solutions. Each of the systems has a unique globally asymptotically stable equilibria, 𝑥 and 𝑦, respectively [18]. Then, the coupled deterministic dynamical system in ℝ^2𝑑

𝑑𝑥𝑑𝑡=𝑓(𝑥)+𝜆(𝑦−𝑥),𝑑𝑦𝑑𝑡=𝑔(𝑥)+𝜆(𝑥−𝑦) (2.17) with parameter 𝜆>0 also satisfies a one-sided dissipative Lipschitz condition and, hence, also has a unique equilibrium (𝑥^𝜆,𝑦^𝜆), which is globally asymptotically stable [18]. Moreover, (𝑥^𝜆,𝑦^𝜆)→(𝑧,𝑧) as 𝜆→∞, where 𝑧 is the unique globally asymptotically stable equilibrium of the “averaged” system in ℝ^𝑑

𝑑𝑧=1𝑑𝑡2(𝑓(𝑧)+𝑔(𝑧)). (2.18) This phenomenon is known as synchronization for the coupled deterministic system (2.17). The parameter 𝜆 often appears naturally in the context of the problem under consideration. For example in control theory it is a control parameter which can be chosen by the engineer, whereas in chemical reactions in thin layers separated by a membrane it is the reciprocal of the thickness of the layers; see [20].

Caraballo and Kloeden [2], and Caraballo et al. [3] showed that this synchronization phenomenon persists under Gaussian Brownian noise, provided that asymptotically stable stochastic stationary solutions are considered rather than asymptotically stable steady state solutions. Recall that a stationary solution 𝑋^∗ of a SDE system may be characterized as a stationary orbit of the corresponding random dynamical system (𝜃,𝜑) (defined by the SDE system), namely, 𝜑(𝑡,𝜔,𝑋^∗(𝜔))=𝑋^∗(𝜃_𝑡𝜔).

The aim of this paper is to investigate synchronization under nonGaussian Lévy noise. In particular, we consider a coupled SDE system in ℝ^𝑑, driven by Lévy motion

𝑑𝑋_𝑡=(𝑓(𝑋_𝑡)+𝜆(𝑌_𝑡−𝑋_𝑡))𝑑𝑡+𝑎𝑑𝐿¹_𝑡,𝑑𝑌_𝑡=(𝑔(𝑌_𝑡)+𝜆(𝑋_𝑡−𝑌_𝑡))𝑑𝑡+𝑏𝑑𝐿²_𝑡, (2.19) where 𝑎,𝑏∈ℝ^𝑑 are constant vectors with no components equal to zero, 𝐿¹_𝑡, 𝐿²_𝑡 are independent two-sided scalar Lévy motion as in Lemma 2.1, and 𝑓,𝑔 satisfy the one-sided dissipative Lipschitz conditions (2.16).

In addition to the one-sided Lipschitz dissipative condition (2.16) on the functions 𝑓 and 𝑔, as in [2] we further assume the following integrability condition. There exists 𝑚₀>0 such that for any 𝑚∈(0,𝑚₀], and any càdlàg function 𝑢∶ℝ→ℝ^𝑑 with subexponential growth it follows

^𝑡_−∞𝑒^𝑚𝑠|𝑓(𝑢(𝑠))|²𝑑𝑠<∞,^t_−∞𝑒^𝑚𝑠|𝑔(𝑢(𝑠))|²𝑑𝑠<∞. (2.20) Without loss of generality, we assume that the one-sided dissipative Lipschitz constant 𝑙≤𝑚₀.

In the next section we will show that the coupled system (2.19) has a unique stationary solution (𝑋^𝜆_𝑡,𝑌^𝜆_𝑡) which is pathwise globally asymptotically stable with (𝑋^𝜆_𝑡,𝑌^𝜆_𝑡)→(𝑍^∞_𝑡,𝑍^∞_𝑡) as 𝜆→∞, pathwise on finite time intervals [𝑇₁,𝑇₂], where 𝑍^∞_𝑡 is the unique pathwise globally asymptotically stable stationary solution of the “averaged" SDE in ℝ^𝑑

𝑑𝑍_𝑡=12[𝑓(𝑍_𝑡)+𝑔(𝑍_𝑡1)]𝑑𝑡+2𝑎𝑑𝐿¹_𝑡+12𝑏𝑑𝐿²_𝑡. (2.21)

3. Systems Driven by Lévy Noise

For the coupled system (2.19), we have the following two lemmas about its stationary solutions.

Lemma 3.1 (existence of stationary solutions). If the Assumption (2.20) holds, 𝑓 and 𝑔 are continuous and satisfy the one-sided Lipschitz dissipative conditions (2.16) with Lipschitz constant 𝑙, then the coupled stochastic system (2.19) has a unique stationary solution.

Proof. First, the stationary solutions of the Langevin equations [4, 21] 𝑑𝑋_𝑡=−𝜆𝑋_𝑡𝑑𝑡+𝑎𝑑𝐿¹_𝑡,𝑑𝑌_𝑡=−𝜆𝑌_𝑡𝑑𝑡+𝑏𝑑𝐿²_𝑡 (3.1) are given by 𝑋^𝜆_𝑡=𝑎𝑒^{−𝜆𝑡𝑡}_−∞𝑒^𝜆𝑠𝑑𝐿¹_𝑡,𝑌^𝜆_𝑡=𝑏𝑒^{−𝜆𝑡𝑡}_−∞𝑒^𝜆𝑠𝑑𝐿²_𝑡. (3.2) The differences of the solutions of (2.19) and these stationary solutions satisfy a system of random ordinary differential equations, with right-hand derivative in time 𝑑𝑑𝑡₊(𝑋_𝑡−𝑋^𝜆_𝑡)=𝑓(𝑋_𝑡)+𝜆(𝑌_𝑡−𝑋_𝑡)+𝜆𝑋^𝜆_𝑡,𝑑𝑑𝑡₊(𝑌_𝑡−𝑌^𝜆_𝑡)=𝑔(𝑌_𝑡)+𝜆(𝑋_𝑡−𝑌_𝑡)+𝜆𝑌^𝜆_𝑡. (3.3)
The equations (3.3) are equivalent to 𝑑𝑑𝑡₊𝑈^𝜆_𝑡=𝑓(𝑋_𝑡)+𝜆(𝑉^𝜆_𝑡−𝑈^𝜆_𝑡)+𝜆𝑌^𝜆_𝑡,𝑑𝑑𝑡₊𝑉^𝜆_𝑡=𝑔(𝑌_𝑡)+𝜆(𝑈^𝜆_𝑡−𝑉^𝜆_𝑡)+𝜆𝑋^𝜆_𝑡, (3.4) where 𝑈^𝜆_𝑡=𝑋_𝑡−𝑋^𝜆_𝑡 and 𝑉^𝜆_𝑡=𝑌_𝑡−𝑌^𝜆_𝑡. Thus, 12𝑑𝑑𝑡₊(|𝑈^𝜆_𝑡|²+|𝑉^𝜆_𝑡|²)=(𝑈^𝜆_𝑡,𝑓(𝑈^𝜆_𝑡+𝑋^𝜆_𝑡)−𝑓(𝑋^𝜆_𝑡))+(𝑉^𝜆_𝑡,𝑔(𝑉^𝜆_𝑡+𝑌^𝜆_𝑡)−𝑔(𝑌^𝜆_𝑡))+(𝑈^𝜆_𝑡,𝑓(𝑋^𝜆_𝑡)+𝜆𝑌^𝜆_𝑡)+(𝑉^𝜆_𝑡,𝑔(𝑌^𝜆_𝑡)+𝜆𝑋^𝜆_𝑡)−𝜆|𝑈^𝜆_𝑡−𝑉^𝜆_𝑡|²𝑙≤−2(|𝑈^𝜆_𝑡|²+|𝑉^𝜆_𝑡|²2)+𝑙|𝑓(𝑋^𝜆_𝑡)+𝜆𝑌^𝜆_𝑡|²+2𝑙|𝑔(𝑌^𝜆_𝑡)+𝜆𝑋^𝜆_𝑡|². (3.5) Hence, by Lemma 2.4, |𝑈^𝜆_𝑡|²+|𝑉^𝜆_𝑡|²≤(|𝑈^𝜆_𝑡0|²+|𝑉^𝜆_𝑡0|²)𝑒^{𝑙(𝑡−𝑡0)}+4𝑒^−𝑙𝑡𝑙^𝑡_𝑡0𝑒^𝑙𝑠[|𝑓(𝑋^𝜆_𝑡)+𝜆𝑌^𝜆_𝑡|²+|𝑔(𝑌^𝜆_𝑡)+𝜆𝑋^𝜆_𝑡|²]𝑑𝑠. (3.6) Define |𝑅_𝜆(𝜔)|²4=1+𝑙⁰_−∞𝑒^𝑙𝑠[|𝑓(𝑋^𝜆(𝜃_𝑠𝜔))+𝜆𝑌^𝜆(𝜃_𝑠𝜔)|²+|𝑔(𝑌^𝜆(𝜃_𝑠𝜔))+𝜆𝑋^𝜆(𝜃_𝑠𝜔)|²]𝑑𝑠 (3.7) and let 𝐵^𝜆_2𝑑(𝜔) be a random closed ball in ℝ^2𝑑 centered on the origin and of radius 𝑅_𝜆(𝜔).
Now we can use pathwise pullback convergence (i.e., with 𝑡₀→−∞) to show that |𝑈^𝜆_𝑡|²+|𝑉^𝜆_𝑡|² is pathwise absorbed by the family 𝐵^𝜆_2𝑑={𝐵^𝜆_2𝑑(𝜔),𝜔∈Ω}, that is, for appropriate families 𝐷, there exists 𝑇𝐷^,𝜔≥0 such that 𝜑(𝑡,𝜃_−𝑡𝜔,𝐷(𝜃_−𝑡𝜔))⊂𝐵^𝜆_2𝑑𝐷(𝜔),∀𝑡≥𝑇_,𝜔. (3.8)
Hence, by Lemma 2.3, the coupled system has a random attractor 𝐴^𝜆={𝐴^𝜆(𝜔),𝜔∈Ω} with 𝐴^𝜆(𝜔)⊂𝐵^𝜆_2𝑑(𝜔).
Note that, by Lemma 2.1, it can be shown that the random compact absorbing balls 𝐵^𝜆_2𝑑(𝜔) are contained in the common compact ball for 𝜆≥1.
However, the difference (Δ𝑋_𝑡,Δ𝑌_𝑡)=(𝑋¹_𝑡−𝑋²_𝑡,𝑌¹_𝑡−𝑌²_𝑡) of any pair of solutions satisfies the system of random ordinary differential equations 𝑑𝑑𝑡₊Δ𝑋_𝑡=𝑓(𝑋¹_𝑡)−𝑓(𝑋²_𝑡)+𝜆(Δ𝑌_𝑡−Δ𝑋_𝑡𝑑),𝑑𝑡₊Δ𝑌_𝑡=𝑔(𝑌¹_𝑡)−𝑔(𝑌²_𝑡)−𝜆(Δ𝑌_𝑡−Δ𝑋_𝑡), (3.9) so 𝑑𝑑𝑡₊(|Δ𝑋_𝑡|²+|Δ𝑌_𝑡|²)=2(Δ𝑋_𝑡,𝑓(𝑋¹_𝑡)−𝑓(𝑋²_𝑡))+2(Δ𝑌_𝑡,𝑔(𝑌¹_𝑡)−𝑔(𝑌²_𝑡))−2𝜆|Δ𝑋_𝑡−Δ𝑌_𝑡|²≤−2𝑙(|Δ𝑋_𝑡|²+|Δ𝑌_𝑡|²) (3.10) from which we obtain |Δ𝑋_𝑡|²+|Δ𝑌_𝑡|²≤(|Δ𝑋₀|²+|Δ𝑌₀|²)𝑒^−2𝑙𝑡 (3.11) which means all solutions converge pathwise to each other as 𝑡→∞. Thus the random attractor consists of a singleton set formed by an ordered pair of stationary processes (𝑋^𝜆_𝑡(𝜔),𝑌^𝜆_𝑡(𝜔)) or equivalently (𝑋^𝜆(𝜃_𝑡𝜔),𝑌^𝜆(𝜃_𝑡𝜔)).

Lemma 3.2 (a property of stationary solutions). The stationary solutions of the coupled stochastic system (2.19) have the following asymptotic behavior: 𝑋^𝜆_𝑡(𝜔)−𝑌^𝜆_𝑡(𝜔)→0as𝜆→∞ (3.12) pathwise on any bounded time interval [𝑇₁,𝑇₂] of ℝ.

Proof. Since 𝑑(𝑋^𝜆_𝑡−𝑌^𝜆_𝑡)=(−2𝜆(𝑋^𝜆_𝑡−𝑌^𝜆_𝑡)+𝑓(𝑋^𝜆_𝑡)−𝑔(𝑌^𝜆_𝑡))𝑑𝑡+𝑎𝑑𝐿¹_𝑡−𝑏𝑑𝐿²_𝑡, (3.13) we have 𝑑(𝐷^𝜆_𝑡𝑒^2𝜆𝑡)=𝑒^2𝜆𝑡(𝑓(𝑋^𝜆_𝑡)−𝑔(𝑌^𝜆_𝑡))+𝑎𝑒^2𝜆𝑡𝑑𝐿¹_𝑡−𝑏𝑒^2𝜆𝑡𝑑𝐿²_𝑡, (3.14) where 𝐷^𝜆_𝑡=𝑋^𝜆_𝑡−𝑌^𝜆_𝑡, so pathwise |𝐷^𝜆_𝑡|≤𝑒^{−2𝜆(𝑡−𝑇1)}|𝐷^𝜆_𝑇1|+^𝑡_𝑇1𝑒^{−2𝜆(𝑡−𝑠)}(|𝑓(𝑋^𝜆_𝑠)|+|𝑔(𝑌^𝜆_𝑠)|)𝑑𝑠+|𝑎||^𝑡_𝑇1𝑒^{−2𝜆(𝑡−𝑠)}𝑑𝐿¹_𝑡|+|𝑏||^𝑡_𝑇1𝑒^{−2𝜆(𝑡−𝑠)}𝑑𝐿²_𝑡|. (3.15) By Lemma 2.1 we see that the radius 𝑅_𝜆(𝜃_𝑡𝜔) is pathwise uniformly bounded on each bounded time interval [𝑇₁,𝑇₂], so we see that the right hand of above inequality converge to 0 as 𝜆→∞ pathwise on the bounded time interval [𝑇₁,𝑇₂].

We now present the main result of this paper.

Theorem 3.3 (synchronization under Lévy noise). Suppose that the coupled stochastic system in ℝ^2𝑑𝑑𝑋_𝑡=(𝑓(𝑋_𝑡)+𝜆(𝑌_𝑡−𝑋_𝑡))𝑑𝑡+𝑎𝑑𝐿¹_𝑡,𝑑𝑌_𝑡=(𝑔(𝑌_𝑡)+𝜆(𝑋_𝑡−𝑌_𝑡))𝑑𝑡+𝑏𝑑𝐿²_𝑡 (3.16) defines a random dynamical system (𝜃,𝜑). In addition, assume that the continuous functions 𝑓,𝑔 satisfy the integrability condition (2.20) as well as the one-sided Lipschitz dissipative condition (2.16), then the coupled stochastic system (3.16) is synchronized to a single averaged SDE in ℝ^𝑑𝑑𝑍_𝑡=12[𝑓(𝑍_𝑡)+𝑔(𝑍_𝑡𝑎)]𝑑𝑡+2𝑑𝐿¹_𝑡+𝑏2𝑑𝐿²_𝑡, (3.17) in the sense that the stationary solutions of (3.16) pathwise converge to that of (3.17), that is, (𝑋^𝜆_𝑡,𝑌^𝜆_𝑡)→(𝑍^∞_𝑡,𝑍^∞_𝑡) pathwise on any bounded time interval [𝑇₁,𝑇₂] as parameter 𝜆→∞.

Proof. It is enough to demonstrate the result for any sequence 𝜆_𝑛→∞. Define 𝑍^𝜆_𝑡1∶=2[𝑋^𝜆_𝑡+𝑌^𝜆_𝑡],𝑡∈ℝ. (3.18) Note that 𝑍^𝜆_𝑡(𝜔)=𝑍^𝜆(𝜃_𝑡𝜔) satisfies the equation 𝑑𝑍^𝜆_𝑡=12[𝑓(𝑋^𝜆_𝑡)+𝑔(𝑌^𝜆_𝑡𝑎)]𝑑𝑡+2𝑑𝐿¹_𝑡+𝑏2𝑑𝐿²_𝑡. (3.19) Also we define 𝑍_𝑡(𝜔)=𝑍(𝜃_𝑡1𝜔)∶=2[𝑋_𝑡(𝜔)+𝑌_𝑡(𝜔)],𝑡∈ℝ, (3.20) where 𝑋_𝑡 and 𝑌_𝑡 are the (stationary) solutions of the Langevin equations 𝑑𝑋_𝑡=−𝑋_𝑡𝑑𝑡+𝑎𝑑𝐿¹_𝑡,𝑑𝑌_𝑡=−𝑌_𝑡𝑑𝑡+𝑏𝑑𝐿²_𝑡, (3.21) that is, 𝑋_𝑡=𝑎𝑒^−𝑡𝑡_−∞𝑒^𝑠𝑑𝐿¹_𝑡,𝑌_𝑡=𝑏𝑒^−𝑡𝑡_−∞𝑒^𝑠𝑑𝐿²_𝑡. (3.22) The difference 𝑍^𝜆_𝑡−𝑍_𝑡 satisfies 2(𝑍^𝜆_𝑡−𝑍_𝑡)=2(𝑍^𝜆−𝑍)+^𝑡₀(𝑓(𝑋^𝜆_𝑠)+𝑔(𝑌^𝜆_𝑠)+𝑋_𝑠+𝑌_𝑠)𝑑𝑠. (3.23) By Lemma 2.1, and the fact that these solutions belong to the common compact ball and every càdlàg function is bounded on finite closed intervals, we obtain |𝑓(𝑋^𝜆_𝑡(𝜔))+𝑔(𝑌^𝜆_𝑡(𝜔))|+|𝑋_𝑡(𝜔)+𝑌_𝑡(𝜔)|≤𝑀_𝑇1,𝑇2(𝜔)<∞, (3.24) which implies uniform boundedness as well as equicontinuity. Thus by the Ascoli-Arzela theorem [13], we conclude that for any sequence 𝜆_𝑛→∞, there is a random subsequence 𝜆_𝑛𝑗(𝜔)→∞, such that 𝑍^𝜆𝑛𝑗_𝑡(𝜔)−𝑍_𝑡(𝜔)→𝑍^∞_𝑡(𝜔)−𝑍_𝑡(𝜔) as 𝑗→∞. Thus 𝑍^𝜆𝑛𝑗_𝑡(𝜔)→𝑍^∞_𝑡(𝜔) as 𝑗→∞. Now, by Lemma 3.2𝑍^𝜆𝑛𝑗_𝑡(𝜔)−𝑌^𝜆𝑛𝑗_𝑡𝑋(𝜔)=^{𝜆𝑛𝑗𝑡}(𝜔)−𝑌^𝜆𝑛𝑗_𝑡(𝜔)2𝑍→0,_{𝜆𝑛𝑗𝑡}(𝜔)−𝑋^𝜆𝑛𝑗_𝑡𝑌(𝜔)=^{𝜆𝑛𝑗𝑡}(𝜔)−𝑋^𝜆𝑛𝑗_𝑡(𝜔)2→0, (3.25) as 𝜆_𝑛𝑗→∞, so 𝑋^𝜆𝑛𝑗_𝑡(𝜔)=2𝑍^𝜆𝑛𝑗_𝑡(𝜔)−𝑌^𝜆𝑛𝑗_𝑡(𝜔)→𝑍^∞_𝑡𝑌(𝜔),_{𝜆𝑛𝑗𝑡}(𝜔)=2𝑍^𝜆𝑛𝑗_𝑡(𝜔)−𝑋^𝜆𝑛𝑗_𝑡(𝜔)→𝑍^∞_𝑡(𝜔), (3.26) as 𝜆_𝑛𝑗→∞.
Using the integral representation of the equation, it can be verified that 𝑍^∞_𝑡 is a solution of the averaged random differential equation (3.17) for all 𝑡∈ℝ. The drift of this SDE satisfies the dissipative one-sided condition (2.16). It has a random attractor consisting of a singleton set formed by a stationary orbit, which must be equal to 𝑍^∞_𝑡.
Finally, we note that all possible subsequences of 𝑍^𝜆𝑛_𝑡 have the same pathwise limit. Thus the full sequence 𝑍^𝜆𝑛_𝑡 converges to 𝑍^∞_𝑡, as 𝜆_𝑛→∞. This completes the proof.

3.1. An Example

Consider two scalar SDEs:

𝑑𝑋_𝑡=−(𝑋_𝑡+1)𝑑𝑡+𝑑𝐿¹_𝑡,𝑑𝑌_𝑡=−(𝑌_𝑡+3)𝑑𝑡+2𝑑𝐿²_𝑡, (3.27) which we rewrite as

𝑑𝑋_𝑡=−𝑋_𝑡𝑑𝑡+𝑑𝐿³_𝑡,𝑑𝑌_𝑡=−𝑌_𝑡𝑑𝑡+2𝑑𝐿⁴_𝑡, (3.28) where 𝐿³_𝑡=−𝑡+𝐿¹_𝑡 and 𝐿⁴_𝑡=−3𝑡/2+𝐿²_𝑡.

The corresponding coupled system (3.16) is

𝑑𝑋_𝑡=−𝑋_𝑡𝑑𝑡+𝜆(𝑌_𝑡−𝑋_𝑡)𝑑𝑡+𝑑𝐿³_𝑡,𝑑𝑌_𝑡=−𝑌_𝑡𝑑𝑡+𝜆(𝑋_𝑡−𝑌_𝑡)𝑑𝑡+2𝑑𝐿⁴_𝑡 (3.29) with the stationary solutions

𝑋^𝜆_𝑡=^𝑡_−∞𝑒^{−(𝜆+1)(𝑡−𝑠)}cosh𝜆(𝑡−𝑠)𝑑𝐿³_𝑠+2^𝑡_−∞𝑒^{−(𝜆+1)(𝑡−𝑠)}sinh𝜆(𝑡−𝑠)𝑑𝐿⁴_𝑠,𝑌^𝜆_𝑡=^𝑡_−∞𝑒^{−(𝜆+1)(𝑡−𝑠)}sinh𝜆(𝑡−𝑠)𝑑𝐿³_𝑠+2^𝑡_−∞𝑒^{−(𝜆+1)(𝑡−𝑠)}cosh𝜆(𝑡−𝑠)𝑑𝐿⁴_𝑠. (3.30) Let 𝜆→∞, then

(𝑋^𝜆_𝑡,𝑌^𝜆_𝑡)→(𝑍^∞_𝑡,𝑍^∞_𝑡), (3.31) where 𝑍^∞_𝑡, given by

𝑍^∞_𝑡=^𝑡_−∞12𝑒^{−(𝑡−𝑠)}𝑑𝐿³_𝑠+^𝑡_−∞𝑒^{−(𝑡−𝑠)}𝑑𝐿⁴_𝑠, (3.32) is the stationary solution of the following averaged SDE:

𝑑𝑍_𝑡=−𝑍_𝑡1𝑑𝑡+2𝑑𝐿³_𝑡+𝑑𝐿⁴_𝑡, (3.33) which is equivalent to the following SDE, in terms of the original Lévy motions 𝐿¹ and 𝐿²,

𝑑𝑍_𝑡=−(𝑍_𝑡1+2)𝑑𝑡+2𝑑𝐿¹_𝑡+𝑑𝐿²_𝑡. (3.34)

Acknowledgments

The authors would like to thank Peter Imkeller and Bjorn Schmalfuss for helpful discussions and comments. This research was partly supported by the NSF Grants 0620539 and 0731201, NSFC Grant 10971225, and the Cheung Kong Scholars Program.