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

Volume 2010, Article ID 502803,13pages doi:10.1155/2010/502803

Research Article

Synchronization of Dissipative Dynamical Systems Driven by Non-Gaussian L ´evy Noises

Xianming Liu,

Jinqiao Duan,

Jicheng Liu,

and Peter E. Kloeden

1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

2Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

3Institut f ¨ur Mathematik, Johann Wolfgang Goethe-Universit¨at, D-60054, Frankfurt am Main, Germany

Correspondence should be addressed to Xianming Liu,mathliuxm@yahoo.cn Received 17 September 2009; Accepted 15 January 2010

Academic Editor: Salah-Eldin Mohammed

Copyrightq2010 Xianming Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 L´evy 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 book1. Recently Caraballo and Kloeden2,3 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 L´evy motions.

This paper is organized as follows. We first recall some basic facts about random dynamical systemsRDSsas well as formulate the problem of synchronization of stochastic dynamical systems driven by L´evy noises inSection 2. The main resultTheorem 3.3and an example are presented inSection 3.

Throughout this paper, the norm of a vectorxin Euclidean spaceR^dis always denote by|x|.

2. Dynamical Systems Driven by L ´evy Noises

Dynamical systems driven by nonGaussian L´evy motions have attracted much attention recently4,5. Under certain conditions, the SDEs driven by L´evy motion generate stochastic flows4,6, and also generate random dynamical systemsor cocyclesin 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 L´evy motion. This shows some qualitatively different dynamical behavior between SDEs driven by Gaussian and nonGaussian noises.

2.1. L ´evy Processes

A L´evy process or motion onR^d is characterized by a drift parameterγ ∈R^d, a covariance d×dmatrix A, and a nonnegative Borel measureν, defined onR^d,BR^dand concentrated onR^d\ {0}, which satisfies

R^d\{0}

y²∧1 ν

dy

<∞, 2.1

or equivalently

R^d\{0}

y² 1y²ν

dy

<∞. 2.2

This measureνis the so-called the L´evy jump measure of the L´evy processLt. Moreover L´evy processL_thas the following L´evy-It ˆo decomposition:

LtγtBt

|x|<1xNt, dx

|x|≥1xNt, dx, 2.3

whereNdt, dxis Poisson random measure,

Ndt, dx Ndt, dx−νdxdt, 2.4

is the compensated Poisson random measure of Lt, and Bt is an independent Brownian motion onR^dwith covariance matrixAsee4,10–12. We callA, ν, γthe generating triplet.

General semimartingales, especially L´evy motions, are thought to be appropriate models for nonGaussian processes with jumps11. Let us recall that a L´evy motion Lt is a nonGaussian process with independent and stationary increments. Moreover, its sample paths are only continuous in probability, namely, P|Lt −L_t0| ≥ δ → 0 as t → t₀ for any positiveδ. With a suitable modification 4, these paths may be taken as c`adl`ag, 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 c`adl`ag function has finite or at most countable discontinuities on any time intervalsee, e.g.,4, page 118. This generalizes the Brownian motionBt, sinceBtsatisfies all these three conditions, but additionally,ialmost every sample path of the Brownian motion is continuous in time in the usual sense, andiithe increments of Brownian motion are Gaussian distributed.

The next useful lemma provides some important pathwise properties ofLtwith two- sided timet∈R.

Lemma 2.1pathwise boundedness and convergence. LetLtbe a two-sided L´evy motion onR^d for whichE|L1|<∞andEL1 0. Then we have the following.

ilim_{t→ ±∞}1/tLt0, a.s.

iiThe integrals ^t_−∞e^−λt−s dL_sωare pathwisely uniformly bounded inλ >1 on finite time intervalsT1, T2inR.

iiiThe integrals ^t_T

1e^−λt−s dLsω → 0 asλ → ∞, pathwise on finite time intervalsT1, T2 inR.

Proof. i This convergence result comes from the law of large numbers, in11, Theorem 36.5.

iiSince the functionht e^−λtis continuous in t, integrating by parts we obtain _t

−∞e^−λt−s dLsω Ltω−λ _t

−∞e^−λt−sLsωds. 2.5

Then byiand the fact that every c`adl`ag function is bounded on finite closed intervals, we concludeii.

iiiIntegrating again by parts, it follows that _t

T1

e^−λt−s dL_sω Lt−L_T1e^−λt−T1λ _t

T1

e^−λt−sLtω−L_sωds, 2.6

from which the result follows.

Remark 2.2. The assumptions onL_tin the above lemma are satisfied by a wide class of L´evy processes, for instance, the symmetricα-stable L´evy motion onR^dwith 1< α <2. Indeed, in this case, we have _|x|>1|x|νdx<∞, and thenE|L1|<∞, see11, Theorem 25.3.

For the canonical sample space of L´evy processes, that is,Ω DR,R^dof c`adl`ag 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 DR,R^d a Skorohod space.

2.2. Random Dynamical Systems

Following Arnold 7, a random dynamical systemRDS on a probability spaceΩ,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 mappingϕ : R×Ω×R^d → R^d, namely,ϕ satisfies the conditions

ϕ0, ω id_Rd, ϕts, ω ϕt, θsω◦ϕs, ω 2.7

for allω∈Ωand alls, t∈R. This cocycle is required to be at least measurable from theσ-field BR× F × BR^dto theσ-fieldBR^d.

For random dynamical systems driven by L´evy noise we takeΩ DR,R^d with the Skorohod metric as the canonical sample space and denote by F : BDR,R^d the associated Borelσ-field. LetμL be the L´evyprobability measure onF which is given by the distribution of a two-sided L´evy process with paths inDR,R^d.

The driving systemθ θt, t∈RonΩis defined by the shift

θtωs:ωts−ωt. 2.8

The map t, ω → θ_tω is continuous, thus measurable 7, page 545, and the L´evy probability measure isθ-invariant, that is,

μ_L

θ⁻¹_t A

μ_LA 2.9

for allA∈ F, see4, page 325.

We say that a familyA {Aω, ω ∈ Ω}of nonempty measurable compact subsets AωofR^disinvariantfor a RDSθ, ϕ, ifϕt, ω, Aω Aθtωfor allt >0 and that it is a random attractor if in addition it is pathwise pullback attracting in the sense that

H_d^∗

ϕt, θ−tω, Dθ−tω, Aω

−→0 ast−→ ∞ 2.10

for all suitable familiescalled the attracting universeofD {Dω, ω ∈Ω}of nonempty measurable bounded subsets Dω of R^d, where H_d^∗A, B sup_x∈Ainf_y∈B|x − y| is the Hausdorffsemi-distance onR^d.

The following result about the existence of a random attractor may be proved similarly as in2,15–18.

Lemma 2.3random attractor for c`adl`ag RDS. Let (θ, ϕ) be an RDS onΩ×R^d and letϕ be continuous in space, but c`adl`ag in time. If there exits a familyB {Bω, ω ∈ Ω} of nonempty measurable compact subsetsBωofR^dand aT_D,ω ≥0 such that

ϕt, θ_−tω, Dθ_−tω⊂Bω, ∀t≥T_D,ω , 2.11

for all familiesD {Dω, ω∈Ω}in a given attracting universe, then the RDS (θ, ϕ) has a random attractorA{Aω, ω∈Ω}with the component subsets defined for eachω∈Ωby

Aω

s>0

t≥s

ϕt, θ_−tω, Bθ_−tω. 2.12

Forevermore if the random attractor consists of singleton sets, that is, Aω {X^∗ω} for some random variableX^∗, thenX^∗_tω X^∗θtωis a stationary stochastic process.

We also need the following Gronwall’s lemma from19.

Lemma 2.4. Letxtsatisfy the differential inequality d

dtx≤gtxht, 2.13

whered/dtx:lim_h↓0xth−xt/his right-hand derivative ofx. Then

xt≤x0exp _t

0

grdr

_t

0

exp _t

s

grdr

hsds. 2.14

2.3. Dissipative Synchronization

Suppose that we have two autonomous ordinary differential equations inR^d, dx

dt fx, dy dt g

y

, 2.15

where the vector fields f and g are sufficiently regular e.g., differentiableto ensure the existence and uniqueness of local solutions, and additionally satisfy one-sided dissipative Lipschitz conditions

max

x1−x2, fx1−fx2 ,

x1−x2, gx1−gx2

≤ −l|x1−x2|² 2.16

onR^dfor somel >0. These dissipative Lipschitz conditions ensure existence and uniqueness of global solutions. Each of the systems has a unique globally asymptotically stable equilibria, xandy, respectively18. Then, the coupled deterministic dynamical system inR^2d

dx

dt fx λ y−x

, dy

dt gx λ

x−y 2.17

with parameterλ > 0 also satisfies a one-sided dissipative Lipschitz condition and, hence, also has a unique equilibriumx^λ, y^λ, which is globally asymptotically stable18. Moreover, x^λ, y^λ → z, zasλ → ∞, wherezis the unique globally asymptotically stable equilibrium of the “averaged” system inR^d

dz dt 1

2

fz gz

. 2.18

This phenomenon is known as synchronization for the coupled deterministic system2.17.

The parameterλoften 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; see20.

Caraballo and Kloeden2, and Caraballo et al.3showed 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 solutionX^∗ of a SDE system may be characterized as a stationary orbit of the corresponding random dynamical systemθ, ϕ defined by the SDE system, namely,ϕt, ω, X^∗ω X^∗θtω.

The aim of this paper is to investigate synchronization under nonGaussian L´evy noise.

In particular, we consider a coupled SDE system inR^d, driven by L´evy motion dX_t

fXt λYt−X_t

dtadL¹_t, dY_t

gYt λXt−Y_t

dtbdL²_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 L´evy motion as inLemma 2.1, andf, gsatisfy the one-sided dissipative Lipschitz conditions2.16.

In addition to the one-sided Lipschitz dissipative condition2.16on the functionsf andg, as in2we further assume the following integrability condition. There existsm0>0 such that for anym ∈ 0, m0, and any c`adl`ag function u : R → R^d with subexponential growth it follows

_t

−∞e^msfus²ds <∞, _t

−∞e^msgus²ds <∞. 2.20 Without loss of generality, we assume that the one-sided dissipative Lipschitz constantl≤m0. In the next section we will show that the coupled system2.19has a unique stationary solutionX_t^λ,Y_t^λwhich is pathwise globally asymptotically stable withX_t^λ,Y_t^λ → 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 inR^d

dZ_t 1 2

fZt gZt dt1

2adL¹_t1

2bdL²_t. 2.21

3. Systems Driven by L ´evy 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 conditions2.16with Lipschitz constantl, then the coupled stochastic system2.19has a unique stationary solution.

Proof. First, the stationary solutions of the Langevin equations4,21

dX_t−λXtdtadL¹_t, dY_t−λYtdtbdL²_t 3.1

are given by

X^λ_t ae^−λt _t

−∞e^λsdL¹_t, Y^λ_t be^−λt _t

−∞e^λsdL²_t. 3.2

The differences of the solutions of 2.19and these stationary solutions satisfy a system of random ordinary differential equations, with right-hand derivative in time

d dt

X_t−X^λ_t

fXt λYt−X_t λX^λ_t, d

dt

Y_t−Y^λ_t

gYt λXt−Y_t λY^λ_t.

3.3

The equations3.3are equivalent to

d

dtU^λ_t fXt λ

V_t^λ−U_t^λ

λY^λ_t, d

dtV_t^λgYt λ

U_t^λ−V_t^λ

λX^λ_t, 3.4

whereU_t^λX_t−X^λ_t andV_t^λY_t−Y^λ_t. Thus,

1 2

d dt

U_t^λ2V_t^λ2

U^λ_t, f

U_t^λX^λ_t

−f

X^λ_t

V_t^λ, g

V_t^λY^λ_t

−g

Y^λ_t

U_t^λ, f

X^λ_t

λY^λ_t

V_t^λ, g

Y^λ_t

λX^λ_t

−λU_t^λ−V_t^λ2

≤ −l 2

U^λ_t²V_t^λ2

2 l

f

X^λ_t

λY^λ_t ²2

l g

Y^λ_t

λX^λ_t

². 3.5

Hence, byLemma 2.4,

U^λ_t²V_t^λ2≤

U^λ_t0²V_t^λ₀²

e^lt−t0

4e^−lt l

_t

t0

e^ls f

X^λ_t

λY^λ_t

² g

Y^λ_t

λX^λ_t

²

ds.

3.6

Define

|Rλω|²14 l

₀

−∞e^ls f

X^λθsω

λY^λθsω ²

g

Y^λθsω

λX^λθsω ²

ds

3.7 and letB^λ_2dωbe a random closed ball inR^2dcentered on the origin and of radiusRλω.

Now we can use pathwise pullback convergencei.e., witht₀ → −∞to show that

|U_t^λ|²|V_t^λ|²is pathwise absorbed by the familyB^λ_2d{B^λ_2dω, ω∈Ω}, that is, for appropriate familiesD, there exists T_D,ω ≥0 such that

ϕt, θ−tω, Dθ−tω⊂B_2d^λ ω, ∀t≥T_D,ω . 3.8

Hence, byLemma 2.3, the coupled system has a random attractorA^λ {A^λω, ω∈ Ω}withA^λω⊂B_2d^λ ω.

Note that, byLemma 2.1, it can be shown that the random compact absorbing balls B^λ_2dωare contained in the common compact ball forλ≥1.

However, the differenceΔXt,ΔYt X_t¹−X²_t, Y_t¹−Y_t²of any pair of solutions satisfies the system of random ordinary differential equations

d

dtΔXtf X_t¹

−f X_t²

λΔYt−ΔXt, d

dtΔYtg Y_t¹

−g Y_t²

−λΔYt−ΔXt,

3.9

so d dt

|ΔXt|²|ΔYt|² 2

ΔXt, f X_t¹

−f X_t²

2 ΔYt, g

Y_t¹

−g Y_t²

−2λ|ΔXt−ΔYt|²

≤ −2l

|ΔXt|²|ΔYt|²

3.10

from which we obtain

|ΔXt|²|ΔYt|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_t^λω,Y_t^λωor equivalentlyX^λθtω,Y^λθtω.

Lemma 3.2a property of stationary solutions. The stationary solutions of the coupled stochastic system2.19have the following asymptotic behavior:

X^λ_tω−Y_t^λω−→0 asλ−→ ∞ 3.12

pathwise on any bounded time intervalT1, T₂ofR.

Proof. Since

d

X_t^λ−Y_t^λ

−2λ

X_t^λ−Y_t^λ f

X^λ_t

−g Y_t^λ

dtadL¹_t−bdL²_t, 3.13

we have

d

D^λ_te^2λt e^2λt

f X_t^λ

−g Y_t^λ

ae^2λtdL¹_t −be^2λtdL²_t, 3.14

whereD_t^λX_t^λ−Y_t^λ, so pathwise D_t^λ≤e^−2λt−T1D^λ_T

1

_t

T1

e^−2λt−sf

X_s^λg Y_s^λ

ds

|a|

_t

T1

e^−2λt−sdL¹_t |b|

_t

T1

e^−2λt−sdL²_t .

3.15

By Lemma 2.1 we see that the radius R_λθtω is pathwise uniformly bounded on each bounded time intervalT1, T2, so we see that the right hand of above inequality converge to 0 asλ → ∞pathwise on the bounded time intervalT1, T₂.

We now present the main result of this paper.

Theorem 3.3synchronization under L´evy noise. Suppose that the coupled stochastic system in R^2d

dXt

fXt λYt−Xt

dtadL¹_t, dY_t

gYt λXt−Y_t

dtbdL²_t 3.16

defines a random dynamical system θ, ϕ. In addition, assume that the continuous functionsf, g satisfy the integrability condition2.20as well as the one-sided Lipschitz dissipative condition2.16, then the coupled stochastic system3.16is synchronized to a single averaged SDE inR^d

dZ_t 1 2

fZt gZt dta

2dL¹_tb

2dL²_t, 3.17

in the sense that the stationary solutions of 3.16 pathwise converge to that of 3.17, that is, X_t^λ,Y_t^λ → Z^∞_t , Z^∞_t pathwise on any bounded time intervalT1, T₂as parameterλ → ∞.

Proof. It is enough to demonstrate the result for any sequenceλ_n → ∞. Define

Z^λ_t : 1 2

X^λ_t Y_t^λ

, t∈R. 3.18

Note thatZ^λ_tω Z^λθtωsatisfies the equation

dZ^λ_t 1 2

f X_t^λ

g Y_t^λ

dta

2dL¹_tb

2dL²_t. 3.19

Also we define

Z_tω Zθtω: 1 2

X_tω Y_tω

, t∈R, 3.20

whereXtandYtare thestationarysolutions of the Langevin equations

dX_t−XtdtadL¹_t, dY_t−YtdtbdL²_t, 3.21

that is,

X_tae^−t _t

−∞e^sdL¹_t, Y_tbe^−t _t

−∞e^sdL²_t. 3.22

The differenceZ^λ_t −Z_tsatisfies

2 Z^λ_t −Zt

2

Z^λ−Z

_t

0

f

X^λ_s g

Y_s^λ

XsYs

ds. 3.23

ByLemma 2.1, and the fact that these solutions belong to the common compact ball and every c`adl`ag function is bounded on finite closed intervals, we obtain

f

X_t^λω g

Y_t^λωXtω Y_tω≤M_T1,T2ω<∞, 3.24 which implies uniform boundedness as well as equicontinuity. Thus by the Ascoli-Arzela theorem13, we conclude that for any sequenceλ_n → ∞, there is a random subsequence

λ_njω → ∞, such thatZ_t^λnjω−Ztω → Z^∞_t ω−Ztωasj → ∞. ThusZ_t^λnjω → Z_t^∞ω asj → ∞. Now, byLemma 3.2

Z_t^λnjω−Y_t^λnjω X_t^λnjω−Y_t^λnjω

2 −→0,

Z^λ_t^njω−X^λ_t^njω Y_t^λnjω−X_t^λnjω

2 −→0,

3.25

asλ_nj → ∞, so

X_t^λnjω 2Z^λ_t^njω−Y_t^λnjω−→Z_t^∞ω, Y_t^λnjω 2Z^λ_t^njω−X^λ_t^njω−→Z_t^∞ω,

3.26

asλ_nj → ∞.

Using the integral representation of the equation, it can be verified thatZ^∞_t is a solution of the averaged random differential equation3.17for allt∈R. The drift of this SDE satisfies the dissipative one-sided condition2.16. It has a random attractor consisting of a singleton set formed by a stationary orbit, which must be equal toZ^∞_t .

Finally, we note that all possible subsequences ofZ_t^λn have the same pathwise limit.

Thus the full sequenceZ^λ_tⁿ converges toZ^∞_t , asλ_n → ∞. This completes the proof.

3.1. An Example Consider two scalar SDEs:

dX_t−Xt1dtdL¹_t, dY_t−Yt3dt2dL²_t, 3.27

which we rewrite as

dX_t−XtdtdL³_t, dY_t−Ytdt2dL⁴_t, 3.28

whereL³_t −tL¹_tandL⁴_t −3t/2L²_t.

The corresponding coupled system3.16is

dXt −XtdtλYt−XtdtdL³_t,

dYt −YtdtλXt−Ytdt2dL⁴_t 3.29

with the stationary solutions

X_t^λ _t

−∞e^−λ1t−scoshλt−sdL³_s2 _t

−∞e^−λ1t−ssinhλt−sdL⁴_s, Y_t^λ

_t

−∞e^−λ1t−ssinhλt−sdL³_s2 _t

−∞e^−λ1t−scoshλt−sdL⁴_s.

3.30

Letλ → ∞, then

X^λ_t,Y_t^λ

−→Z_t^∞, Z^∞_t , 3.31

whereZ^∞_t , given by

Z_t^∞ _t

−∞

1

2e^−t−sdL³_{s t}

−∞e^−t−sdL⁴_s, 3.32

is the stationary solution of the following averaged SDE:

dZt−Ztdt1

2dL³_tdL⁴_t, 3.33

which is equivalent to the following SDE, in terms of the original L´evy motionsL¹andL²,

dZt−Zt2dt1

2dL¹_tdL²_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.

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