Abstract and Applied Analysis

Volume 2013 (2013), Article ID 432704, 13 pages

http://dx.doi.org/10.1155/2013/432704

Research Article

Asymptotic Behavior of the Stochastic Rayleigh-van der Pol Equations with Jumps

ZaiTang Huang¹ and ChunTao Chen²

¹School of Mathematical Sciences, Guangxi Teachers Education University, Nanning 530001, China

²College of Mathematics and Information Science, Guangxi University, Nanning 530004, China

Received 20 May 2013; Revised 2 July 2013; Accepted 2 July 2013

Academic Editor: Józef Banaś

Copyright © 2013 ZaiTang Huang and ChunTao Chen. 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.

Abstract.

We study the stability, attractors, and bifurcation of stochastic Rayleigh-van der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleigh-van der Pol equations. We then examine the existence limit circle and obtain some new random attractors. We further establish stochastic bifurcation of random attractors. Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system which is significantly different from the classical Brownian motion process.

1. Introduction

Gaussian process have been used to model many physical systems for quite a long time. However, researchers have increasingly been studying models from economics and from the natural sciences, where the underlying randomness contains jumps, which exhibit jump-like behavior. Such processes are being identified in the physical environment, in engineering systems, and in financial sectors. As it will quickly become apparent, the theory of RDS with jumps is nowhere near as well developed as the theory of continuous RDS, and in fact, is still in a relatively primitive stage. In particular, the theory of stochastic bifurcation is still in its infancy. Therefore, the field is both fertile and important [1–6].

A large class of phenomena can be described using stochastic differential equations drive white noise, in such distinct domains as economics, physics, or biology. Many analytical methods have been developed to characterize the solutions of such systems. In particular, the widely applied theory of stochastic bifurcations [7–10] allows a qualitative characterization of the asymptotic regimes of random dynamical systems. This theory is very handy to study nonlinear stochastic differential equations and is used to characterize the asymptotic behavior of complicated systems. Investigating the impact of Poisson noisy perturbations on such systems is hence of great interest and is currently an active field of research. In recent years, it has been shown in many different areas that applying Poisson noisy on a random system can lead to many counter-intuitive phenomena, such as Poisson noisy-induced stabilization [11]. From a mathematical perspective, understanding the interplays between white noise and Poisson noisy perturbations is a great challenge, with many applications. Several tools have been introduced, ranging from the theory of random dynamical systems [5, 6, 12–14], the study of moment equations [3, 15]. Unfortunately, no systematic method in the flavor of bifurcation theory for the analysis of the dynamics of nonlinear SDE drive Poisson noisy exists, and this is the central problem of the present paper.

We focus here on the dynamics of stochastic Rayleigh-van der Pol equations with jumps. This type of equations may appear, for example, in diffusion approximations of physical models [16] or the classical model for a stock price [5]. The question we address is how the interplay between the stochastic jump system and diffusion system affects the behavior of the system. An important contribution of RDS theory to the field of stochastic bifurcations is characterized by a change in the shape of the invariant measure and stochastic Lyapunov exponent. We investigate the questions of stochastic stability and bifurcations, random attractors, and limits for stochastic Rayleigh-van der Pol equations with jumps, using combinations of appropriate tools such as Lyapunov direct method, Arnold asymptotic analysis approach, exponential martingale, and Lyapunov exponent.

The paper is organized of as follows. In Section 2, we give some preliminaries. We study the stochastic stability in Section 3. Random attractors are presented in Section 4. Random limits and stochastic bifurcation are further investigated in Section 5.

2. Preliminaries

In this paper, we analyze the behavior of the solutions of stochastic Rayleigh-van der Pol equations with jumps:

with initial condition in . The symbol denotes the Stratonovich calculus, is a Brownian motion and is the Poisson point process on with simultaneous jumps of probability zero with , the deterministic finite characteristic measure on a measurable space such that is a -martingale measure. We also assume that and are independent. Let be the underlying complete filtered probability space.

At the bifurcation points and , a linear change of variables yields

If and are random telegraph process with , the mean time between jumps, then stochastic equation (1) may be treated as informal derivative of the Poisson processes with intensity as

where are strength parameters and is a Poisson process with intensity over .

3. Stochastic Stability

In the section, we would like to study the stability of the linearized stochastic Rayleigh-van der Pol equations driven by Poisson noise.

Theorem 1. Suppose , if , then the diffusion system corresponding to the linearized system (2) is asymptotically unstable with the Lyapunov exponent:

However, the stochastic jump diffusion of the linearized system (2) is asymptotically stable if

where

Proof. We can rewrite the linearized system (2) as follows:

where .

Denote by the angle of . Then, , which depends on the quadrant of , and . Let . Then, is orthogonal to . Thus, in terms of the new coordinates ,

By Itô's formula, we have

Thus, the adjoint equation

admits a unique periodic solution as the invariant probability density

with

The Lyapunov exponent is as follows:

where the integrals

are the modified Bessel functions which satisfy, respectively, the zero-order and the first-order Bessel equations of a purely imaginary argument:

Note that is even, while is odd such that is between and , strictly increasing and at most linear growth. The diffusion system corresponding to (7) is asymptotically unstable with the Lyapunov exponent

if . However, the stochastic jump diffusion of (7) is asymptotically stable if

Remark. This shows the effect of Poisson noises which can stabilize the system in this respect. The alternative phenomenon also happens if the reverse inequality in (17) holds while .

Theorem 2. Suppose the condition of Theorem 1 is satisfied and the stochastic jump-diffusion linearized system (2) is asymptotically stable, that is, . Then,

Proof. Let be the integro-differential operator corresponding to the jump-diffusion process of (9), and let . Then,

Since,

where is the unique invariant probability density of the jump-diffusion process of (9), then the integro-differential equation has a unique solution . Then,

Let be a right continuous martingale defined as follows:

and define an operator on by

Note that , and are -adapted processes. Let be a -local martingale such that

are, respectively, a continuous -local martingale and a pure jump -local martingale with

Therefore, let ; then the conditional quadratic variation of is

as and are bounded on .

On the other hand,

Since

for all , then

is a martingale. Denote by the decimal part of . As

we have by the martingale inequality

In order to minimize the exponent in the inequality of (31), let

Then, we can obtain that

As , the result of Theorem 2 follows.

4. Random Attractors

In this section, we would like to study the random attractors of the stochastic Rayleigh-van der Pol equations driven by the Poisson noise.

Theorem 3. Equation (3) generates a smooth random dynamical system with the Poisson processes which is global to the forward, that is, for all . Moreover, is measurable.

Proof . We can rewrite system (3) as follows:

Choose a function

Applying Itô's formula to (35), we have

Using the Cauchy-Bunyakovsky-Schwarz inequality, then

From (36), we can obtain

Suppose that satisfies the following equation with jumps:

then is finite on any finite interval for any . Hence, (38) and (39) yield that the process is finite on for any ; that is, the random dynamical system is global to the forward.

Corollary 4. Suppose the condition of Theorem 3 is satisfied. Then the random dynamical system generated by (3) has the following additional properties:

(i)  for all ;(ii)  as .

Proof. Since all is (local) diffeomorphism of the random(open) domain into the random range where as , and , . is the set of all initial values whose orbits never explode. It may happen that .

For , Theorem 3 yields that is nonexplosive; we have the property (i) by replacing with . Theorem 3 implies that for and for , respectively. Since , the definition of yields that (ii) is true.

Lemma 5. Suppose . If , then the stochastic differential equation with the Poisson processes

has the unique invariant measure which is a Dirac measure supported by

and attracts all points with exponential speed, where and is the intensity of a Poisson process on .

Proof. By Itô's formula, we obtain

is the random dynamical system generated by (40). Since and , we have the following integrals:

Since (40) is affine with stable linear part, then, (42) and Theorem of Arnold in [7] imply that the unique invariant measure of (40) is the Dirac measure supported by

for all and attracts all points with exponential speed.

Theorem 6. Suppose the condition of Theorem 3 is satisfied and the coefficient in (3) satisfies and

Then the random dynamical system generated by (3) possesses the unique parameter-dependent tempered random attractor with domain of attractor containing the universe of sets , generated by

Moreover, the random attractor is measurable with respect to .

Proof. Choose a function

Applying Itô's formula, we have

This implies

where

Note that , where is a positive constant independent of and .

From (49), we have

Denote and ; then random dynamical systems are generated by (3) and

respectively. The random dynamical systems generated by (52) are

Thus, from (51), we have

By Lemma 5 and (45), we obtain that has the unique invariant measure which is a Dirac measure supported by

It is easy to prove that as for any initial value such that for some . Hence, we may the universe of set

that is, the random variable satisfies for any . From the results of Arnold in [7, Proposition 4.17], it yields for any .

It is clear that is closed under inclusion and for all .

Next, we will prove that the random set is forward invariant and absorbing for with respect to the universe for any .

In fact, the absorbing property follows from the definition of and Schenk-Hoppé [9, Theorem 4.1]. We only prove that for all . Since is a nonnegative function on and for , it suffices to show , it is equivalent that prove

From (53), one can obtain

It implies that (57) is true.

Since

one has

for any . It means that for any .

For any , define the subset of by

This is a nonempty compact set by the surjectivity and continuity of and the fact that preimages of bounded sets are bounded under . One can prove that the random set satisfies the assumptions of Schenk-Hoppé [9, Theorm 4.1].

The measurability of may be implied by Lemma 5.7 of Schenk-Hoppé [9]. Using the surjectivity and nonnegativity of and (54), one has

for all . It implies

that is, is forward invariant.

For any , we have that the random variable grows subexponentially fast by the definition of . yields the random variable which grows sub-exponentially fast; that is, .

As the proof of (62) and (63), we have that there is a such that, for all ,

by (59) and the set absorbing any set in . It shows the absorption of any set in .

Finally, we prove the existence of a neighborhood of in . In fact, for any , Proposition (iii) of Schenk-Hoppé [9] implies that there exists a random variable such that . It is clear that and that is a neighborhood of for all .

By Theorem 4.1 of Schenk-Hoppé [9], we finish the proof. Because of , we have that the attractor is measurable.

5. Stochastic Bifurcation

In this section, we would like to study the random limit circle and stochastic bifurcation of the stochastic Rayleigh-van der Pol equations driven by the Poisson noise.

Theorem 7. Assume that . Then the Rayleigh-van der Pol equations (3) undergoes Hopf bifurcation; thus the stochastic system (3) occurs a random limits cycle.

Proof

Step 1. Let ; then (3) becomes

The linearized RDS D is generated by the linearization of (65), namely, the white noise case by

For any invariant measure , the trace formula gives . In particular, for , we obtain .

We also need the eigenvalues of the linearization of the deterministic system at ,

which are

For , we have two real eigenvalues, while for , we have a pair of complex-conjugate eigenvalues:

where represents the “damped eigenfrequency” of the linear system:

It is a general observation that noise splits deterministic multiplicities of eigenvalues. For (which we assume) and , the deterministic linear system has two complex-conjugate eigenvalues , which amounts to just one Lyapunov exponent with multiplicity 2. For and , however, the linearized SDE

has two different simple Lyapunov exponents which satisfy . For small , we can use the asymptotic expansion given by Arnold [7, Theorem ] and others; namely, when , then the Lyapunov exponent of (71) is as follows:

where is the spectral density of , for the white noise case where . Furthermore, the rotation number expands as

where is the sine spectral density of .

Consequently, at the deterministic Hopf bifurcation point , we have for the white noise case, neglecting term,

that is, at , the top exponent has already cross and is positive. Then the Rayleigh-van der Pol equation (3) undergoesed the Hopf bifurcation.

Step 2. It follows from (72) that the top Lyapunov exponent changes sign at and the second Lyapunov exponent changes sign at .

For is stable, it is the unique invariant measure, and is the attractor of the RDS in .

We have a first bifurcation from at of a stable ergodic measure , which is a convex combination of two Dirac measures, sitting on the two “boundary points” of the one-dimensional unstable manifold of , which is a saddle point. This situation persists for . As shown before, , undergoes a P-bifurcation, and and are both Markov measures. Hence, solves the Fokker-Planck equation. The attractor in (the universe of tempered sets of) is the closure of the unstable manifold of , the two “boundary points” supporting the measure . In the punctured plane , the attractor (in the universe of simply connected tempered sets) consists of the two-point set .

At , we have a second bifurcation from of a measure , which is again a convex combination of two Dirac measures. is a saddle point that has positive and negative Lyapunov exponents, while has two positive exponents.

For , the stable measure is supported by the “boundary” of the unstable manifold of . The closure of this unstable manifold is an invariant “circle” around (random limit cycle) and supports both measures and . On this “circle,” we have hyperbolic dynamics ( is attracting and is repelling), in contrast to the deterministic case, where the dynamics on the limit cycle is just rotation and the invariant measure is unique. The interior of the “circle” is the two-dimensional unstable manifold of . Its closure is the attractor in . It carries all three invariant measures. In the punctured plane , however, the attractor is the invariant “circle,” in which we have two invariant measures, in particular, the unique Markov measure . Thus, the Rayleigh-van der Pol equation (3) occurs a random limit cycle.

Proposition 8. Suppose the condition of Theorem 3 is satisfied and the coefficient in (3) is satisfied. Denote the random attractor of the random dynamical systems generated by the stochastic Rayleigh-van der Pol equation (3). Then,

(i)the random attractor supports all invariant measures;(ii)the unstable set of any invariant measure is contained in random attractor; that is, ;(iii)there exists an invariant Markov measure supported by ; that this, is measurable.

Proof. It is the same as the proof in Proposition 7.5 of Schenk-Hoppé [9].

Proposition 9. Suppose the condition of Theorem 3 is satisfied and the coefficient in (3) is satisfied and . For , we can obtain that the Lyapunov exponents and of the linear stochastic Rayleigh-van der Pol equations (3)

are bounded by

As follows.

Proof. Define the Lyapunov function on

Applying Itô's formula to (77), we have

Since

From (78) and (79), we have

By Doléans-Dade's exponent formula, solving the corresponding linear differential equation, we have

where

Then, we can obtain

Dividing by and letting , one obtains the claimed estimate on the Lyapunov exponents.

Theorem 10. Suppose the condition of Theorem 3 is satisfied and the coefficient in (3) is satisfied and and such that

Then, the random dynamical system generated by the stochastic Rayleigh-van der Pol equation (3) possesses the unique tempered random attractor

with domain of attractor containing the universe of given by

where is the complete universe of under inclusion.

In particular, for any , the solutions and tend to zero exponentially fast as .

Proof. By Proposition 9 and assumption (91), the random dynamical system generated by (3) is stable. Let us define the nonnegative function on as follows:

Applying Itô's formula to (87), we can obtain

From (79) and (88), we can obtain

where is uniformly bounded by a positive constant .

By Lemma 5, we can obtain that the random dynamical system generated by the stochastic differential equation

has the unique invariant measure which is the Dirac measure at . From the proof of Theorem 6, is an attractor attracting any set form .

In particular, we obtain that for any and tend to zero as with an exponential speed larger than .

is immediate from Theorem 6 and the definition of .

Theorem 11. Suppose the condition of Theorem 3 is satisfied and the coefficient in (3) is satisfied and and such that

Then, the random dynamical system generated by the stochastic Rayleigh-van der Pol equations (1) possesses the unique tempered random attractor

with domain of attractor containing the universe of given by

contains any random set “living” in an annulus surrounding zero whose outer boundary does not grow exponentially fast and whose inner boundary does not tend to zero exponentially fast.

Proof. It is the same as the proof in Theorem 10.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (nos. 11201089 and 11271139) and by Guangxi Natural Science Foundation (nos. 2013GXNSFAA019014 and 2013GXNSFBA019016).

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