Synchronization of dissipative dynamical systems driven by non-Gaussian Levy noises

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 non- Gaussian Levy noises is considered. After discussing cocycle prop- erty, stationary orbits and random attractors, a synchronization phe- nomenon is shown to occur, when the drift terms of the coupled system satisfy certain dissipativity and integrability conditions. The synchro- nization result implies that coupled dynamical systems share a dy- namical feature in some asymptotic sense.


Introduction
Synchronization of coupled dynamical systems is an unbiquitous 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 [33]. Recently Caraballo and Kloeden [6,7] have 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 non-Gaussian noises (in terms of Lévy motion). We show that couple dissipative systems exhibits synchronization for a class of Lévy motions.
Gaussian processes like Brownian motion have been widely used to model fluctuations in engineering and science. The sample paths of a particle driven by Brownian motion are continuous in time almost surely (i.e., no jumps), the mean square displacement increases linearly in time (i.e., normal diffusion), and the probability density function decays exponentially in space (i.e., light tail or exponential relaxation) [20]. But some complex phenomena involve non-Gaussian fluctuations, with peculiar properties such as anomalous diffusion (mean square displacement is a nonlinear power law of time) [4] and heavy tail (non-exponential relaxation) [35]. For instance, it has been argued that diffusion in a case of geophysical turbulence [28] is anomalous. Loosely speaking, the diffusion process consists of a series of "pauses", when the particle is trapped by a coherent structure, and "flights" or "jumps" or other extreme events, when the particle moves in a jet flow. Moreover, anomalous electrical transport properties have been observed in some amorphous materials such as insulators, semiconductors and polymers, where transient current is asymptotically a power law function of time [26,12]. Finally, some paleoclimatic data [9] indicates heavy tail distributions and some DNA data [28] shows long range power law decay for spatial correlation.
Lévy motions are thought to be appropriate models for non-Gaussian processes with jumps [25]. Let us recall that a Lévy motion L(t), or L t , is a non-Gaussian process with independent and stationary increments, i.e., increments ∆L(t, ∆t) = L(t + ∆t) − L(t) are stationary (therefore ∆L has no statistical dependence on t) and independent for any non overlapping time lags ∆t. Moreover, its sample paths are only continuous in probability, namely, P(|L(t) − L(t 0 )| ≥ δ) → 0 as t → t 0 for any positive δ. With a suitable modification [1], these paths may be taken as càdlàg, i.e., 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., p.118, [1]). This generalizes the Brownian motion B(t) or 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. This paper is organized as follows. We first recall some basic facts about stochastic differential equations (SDEs) driven by Lévy noise in section 2, including a fact that the solution mappings of such SDEs generate random dynamical systems (RDS). In section 3, we formulate the problem of synchronization of stochastic dynamical systems driven by Lévy noises. The main result (Theorem 1) and an example are presented in section 4.

Dynamical systems driven by Lévy noises
Dynamical systems driven by non-Gaussian Lévy motions have attracted much attention recently [1,27]. Under certain conditions, the SDEs driven by Lévy motion generate stochastic flows [1,18], and also generate random dynamical systems (or cocycles) in the sense of Arnold [2]. Recently, exit time estimates have been investigated by Imkeller & Pavlyukevich [13,14] , and Yang & Duan [34] for SDEs driven by Lévy motion. This shows some qualitatively different dynamical behaviors between SDEs driven by Gaussian and non-Gaussian noises.

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 non-negative Borel measure ν, defined on (R d , B(R d )) and concentrated on R d \ {0}, which satisfies 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 where N(dt, dx) is Poisson random measure, 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 [1,25,24,34]). We call (A, ν, γ) the generating triplet.
The next useful lemma provides states some important pathwise properties of L t with two-sided time t ∈ R. Here | · | denotes the usual Euclidean norm in R d .

Lemma 1. (Pathwise boundedness and convergence)
Let L t be a two-sided Lévy motion on R d for which E|L 1 | < ∞ and EL 1 = γ.
Proof. (i) This convergence result comes from the law of large numbers, in [25], Theorem 36.5.
(ii) Due to the continuous of function h(t) = e −λt , on integrating by parts we obtain Then we use (i) to conclude (ii).
(iii) Integrating again by parts, it follows that from which the result follows.
Remark 1. The assumptions on L t in the above lemma are satisfied by a wide class of Lévy processes, for instance, the α-stable symmetric Lévy motion on R d with 1 < α < 2. Indeed, in this case, we have |x|>1 |x|ν(dx) < ∞, and then E|L 1 | < ∞; see [25] Theorem 25.3.
Let us introduce the canonical sample space for Lévy processes, the space Ω = D(R, R d ) of càdlàg functions, i.e., continuous on the right and have limits on the left, defined on R and taking values in R d .
If we use the usual compact-open metric, D(R, R d ) is not separable. However, it is complete and separable when endowed with the Skorohod metric [3,29], in which case we call D(R, R d ) a Skorohod space. The Skorohod metric on D(R, R d ) is defined as where Λ denotes the set of strictly increasing, continuous functions from R to itself.
Similarly, we can define a Skorohod space on a bounded time interval We recall the following compactness result (see [3], where the infimum is taken over all the finite sets {t i } of points satisfying Then, a set B has compact closure in the Skorohod space

SDE driven by Lévy processes
We consider the following stochastic differential equation (SDE) driven by Lévy motion, which has continuous drift and Brownian motion components, namely whereÑ(dt, dx) and N(dt, dx) are defined above, and the coefficients b, σ, F, G are all assumed to be measurable. Here F and G may be different, while the positive parameter c may be different from 1, which allows greater generality. We introduce the d × d matrix and define We make the following general assumptions for the SDE (3): A.3 There exits δ > 2 and K 3 > 0 such that, for all y 1 , From [1], Theorem 6.23, page 304, we have the following existence and uniqueness result for solutions of such SDE driven by Lévy motion Note that a càdlàg solution process has finite or at most countable discontinuities on any time interval (see, e.g., p.118, [1]). For more details about SDEs driven by Lévy motions, see [11,19,18]. Due to the Lévy-Itô decomposition (1), the following SDE, which we consider in the sequel, is a special case of (3).
Remark 2. The reason to take the left limit in Y (t−) in the equation (3) is to make sure that the càdlàg solution process Y is predictable and unique [21]. For typographical convenience, however, we will write Y (t) instead of Y (t−) for the rest of this paper. Moreover, in the case of additive noise, i.e., if the noise intensity g(·) does not depend on the state Y , the distinction for left limit or not is not necessary, when we consider the integral form of the equation (3), as dt for continuous f . In this case f (Y (t−)) has only countable discontinuous points and is thus Riemann and Lebesgue integrable.
Remark 3. The above global assumptions do not hold for SDE, which we consider in the sequel, with a nonlinear dissipative drift f term such as x T f (x) ≤ K − l|x| 2 for some constants K ≥ 0 and l > 0. However analogous global existence and uniqueness results also hold in this case since the dissipativity condition prevents explosions and hence ensures otherwise local existence is global. See [29] for more details.

Random dynamical systems
Following Arnold [2], 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, i.e., θ t is a deterministic dynamical system; and a cocycle mapping ϕ : R × Ω × R d → R d , namely, ϕ satisfies the conditions: 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 ).

Lemma 4. (RDS generated by SDEs driven by Lévy motion) Suppose in addition to the assumptions of Lemma 3 that condition (A3) is satisfied. Then there exists a unique càdlàg adapted solution to (3), and the solution mapping defines a RDS, which is continuous in x but càdlàg in time.
Proof. Let Φ s,t satisfy (3) with initial condition Φ s,s (y) = y, i.e.

Remark 4. In view of Remark 3 and analogous result holds for SDE with a nonlinear dissipative drift term
We say that a familyÂ = {A(ω), ω ∈ Ω} of non-empty measurable compact subsets A(ω) of R d is invariant for a RDS (θ, ϕ), if ϕ(t, ω, A(ω)) = A(θ t ω) for all t > 0 and that it is a random attractor if in addition it is pathwise pullback attracting in the sense that ), A(ω)) → 0 as t → ∞ for all suitable families (called the attracting universe) ofD = {D(ω), ω ∈ Ω} of non-empty measurable bounded subsets The following result about the existence of a random attractor may be proved similarly as in [30,6,8,31,17].
We also need the following Gronwall's lemma from [22].

Dissipative synchronization
Suppose we have two autonomous ordinary differential equations in R d , 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-side dissipative Lipschitz conditions on R d for some l > 0. These dissipative Lipschitz conditions ensure existence and uniqueness of global solutions; see Remark 3 above. Each of the systems has a unique globally asymptotically stable equilibria, x and y, respectively [17]. Then, the coupled deterministic dynamical system in R 2d with parameter λ > 0 also sastisfies a one-sided dissipative Lipschitz condition and, hence, also has a unique equilibrium (x λ , y λ ), which is globally asymptotically stable [17]. 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 dt = 1 2 (f (z) + g(z)) .
This phenomena is known as synchronization for the coupled deterministic system (10). The parameter λ often appears naturally in the context of the problem under consideraiton. For example in control theory it is a control parameter which can be chosen by the engineer, whereas in chemical reactions in thin layers separted by a membrane it is the reciprocal of the thickness of the layers, see [5].
Caraballo et al. [6,7] 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 t ω).
They considered a coupled system of stochastic differential equations (SDEs) in R 2d .
where α, β ∈ R d are constant vectors with no components equal to zero, B 1 t , B 2 t are independent two-sided scalar Brownian motions, and f, g satisfy the one-side dissipative Lipschitz conditions (9). This coupled system has a unique stationary solution (X λ t , Y λ t ), which is pathwise globally asymptotically stable. Moreover, the coupled system (12) is synchronized to the "averaged" SDE in R d t is the unique pathwise globally asymptotically stable stationary solution of (13).
The aim of this paper is to investigate synchronization under non-Gaussian Lévy noise. In particular, we consider a coupled SDE system in R d , driven by non-Gaussian Lévy noise in R 2d where α, β, f, g are as above, and L 1 t , L 2 t are independent two-sided scalar Lévy processes satisfying conditions in Lemma 1. We assume that this coupled system defines a random dynamical system ϕ (i.e.., it satisfies the assumptions in Lemma 4 or some generalization of it).
In addition to the one-side Lipschitz dissipative condition (9) on the functions f and g, as in [6] we further assume the following integrability condition: There exists m 0 > 0 such that for any m ∈ (0, m 0 ], and any càdlàg function u : R → R d with sub-exponential growth it follows t −∞ e ms |f (u(s))| 2 ds < ∞, t −∞ e ms |g(u(s))| 2 ds < ∞.
Without loss of generality, we assume that the one-sided dissipative Lipschitz constant l ≤ m 0 .
In the next section we will show that the coupled system (14) has a unique stationary solution (X λ t , Y λ t ) which is pathwise globally asymptotically stable with ( t is the unique pathwise globally asymptotically stable stationary solution of the "averaged" SDE in R d 4 Systems driven by Lévy noise For the coupled system (14), we have the follow two lemmas about its stationary solutions.

Lemma 7. (Existence of stationary solutions)
If the Assumption (15) holds, and f and g satisfy the one-side Lipschitz dissipative conditions (9), then the coupled stochastic system (14) has a unique stationary solution.
Proof. First, the stationary solutions of the Langevin equations are given byX The differences of the solutions of (14) and these stationary solutions satisfy a system of random ordinary differential equations, with right-hand derivative in time: 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)).
Remark 5. Using Lemma 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.

Lemma 8. (A property of stationary solutions)
The stationary solutions of the coupled stochastic system (14) have the following asymptotic behavior:

Theorem 1. (Synchronization under non-Gaussian Lévy noise)
Suppose that the coupled stochastic system in R 2d defines a random dynamical system (θ, ϕ). In addition, assume that f and g satisfy the integrability condition (15) as well as the one-side Lipschitz dissipative condition (9). Then the coupled stochastic system (21) is synchronized to a single averaged SDE in R d in the sense that the stationary solutions of (21) pathwise converge to that of (22), i.e. (X By Lemma 1, we obtain on the interval [T 1 , T 2 ]. Therefore, Z ∞ t is a solution of the averaged SDE (22) for all t ∈ R. The drift of this SDE satisfies the dissipative one-side condition (9). 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 is the stationary solution of the following averaged SDE dZ t = −Z t dt + 1 2 which is equivalent to the following SDE, in terms of the original Lévy motions L 1 and L 2 :