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

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: 𝑑𝑥₁=𝑥₂𝑑𝑡,𝑑𝑥₂=𝛼𝑥₁+𝛽𝑥₂−𝑥³₂−𝑥²₁𝑥₂+𝜎𝑑𝑡₁𝑥₁+𝜎₂𝑥₂+𝜎₃(𝑡)+∘𝑑𝑊_𝑍𝛾₁𝑐₁(𝑧)𝑥₁(𝑡−)+𝛾₂𝑐₂(𝑧)𝑥₂(𝑡−)+𝛾₃𝑐₃(𝑧)𝑁(𝑑𝑡,𝑑𝑧), (1) with initial condition 𝑥(0)=𝑥₀ in ℜ². The symbol denotes the Stratonovich calculus, 𝑊(𝑡) is a Brownian motion and 𝑁(𝑑𝑡,𝑑𝑧) is the Poisson point process on ℜ₊×𝑍 with simultaneous jumps of probability zero with 𝜆(𝑑𝑧), the deterministic finite characteristic measure on a measurable space 𝑍 such that 𝑁(𝑑𝑡,𝑑𝑧)=𝑁(𝑑𝑡,𝑑𝑧)−𝜆(𝑑𝑧)𝑑𝑡 is a ℱ_𝑡-martingale measure. We also assume that 𝑊(𝑡) and 𝑁(𝑑𝑡,𝑑𝑧) are independent. Let (Ω,ℱ,{ℱ}_𝑡≥0ℙ) be the underlying complete filtered probability space.

At the bifurcation points 𝛼=0 and 𝜎₃=𝑐₃(𝑧)=0, a linear change of variables yields (𝑡)=𝑥(𝑡)+𝑑𝑥000𝛽𝑑𝑡𝑎𝑥³₂(𝑡)+𝑏𝑥²₂(𝑡)𝑥₁(𝑡)+𝑐𝑥₂(𝑡)𝑥²₁(𝑡)+𝑑𝑥³₁(𝑡)−𝑎𝑥³₂(𝑡)−𝑏𝑥²₂(𝑡)𝑥₁(𝑡)−𝑐𝑥₂(𝑡)𝑥²₁(𝑡)−𝑑𝑥³₁(𝑡)+𝜎𝑑𝑡₁₁𝜎₁₂𝜎₂₁𝜎₂₂𝑥(𝑡)(𝑡)+∘𝑑𝑊_𝑍𝛾₁₁(𝑧)𝛾₁₂(𝑧)𝛾₂₁(𝑧)𝛾₂₂(𝑧)𝑥(𝑡)𝑁(𝑑𝑡,𝑑𝑧). (2)

If 𝑍{−1,1} and 𝑁(𝑑𝑡,𝑑𝑧) are random telegraph process with 1/𝜆, the mean time between jumps, then stochastic equation (1) may be treated as informal derivative of the Poisson processes 𝑁(𝑡) with intensity 𝜆 as 𝑑𝑥₁=𝑥₂𝑑𝑡,𝑑𝑥₂=𝛼𝑥₁+𝛽𝑥₂−𝑥³₂−𝑥²₁𝑥₂+𝜎𝑑𝑡₁𝑥₁+𝜎₂𝑥₂+𝜎₃(𝑡)+𝛾𝑑𝑊₁𝑥₁+𝛾₁𝑥₂+𝛾₁𝑑𝑁(𝑡), (3) where 𝛾_𝑖∈ℜ(𝑖=1,2,3) are strength parameters and 𝑁(𝑡) is a Poisson process with intensity 𝜆 over (Ω,ℱ,𝑃).

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 𝜎₁₁=𝜎₂₂=𝜎₁,𝜎₁₂=𝜎₂≠0,𝜎₂₁=−𝜎₂≠0,𝛾₁₂=𝛾₂₁=0,𝛾₁₁=−𝛽𝑐(𝑧)−1,𝛾₂₂=𝛽𝑐(𝑧)−1, if 𝛽>0, then the diffusion system corresponding to the linearized system (2) is asymptotically unstable with the Lyapunov exponent: 𝛽Λ=2𝐼1+¹𝛽/2𝜎²₂𝐼₀𝛽/2𝜎²₂>0. (4) However, the stochastic jump diffusion of the linearized system (2) is asymptotically stable if 𝛽2𝐼1+¹𝛽/2𝜎²₂𝐼₀𝛽/2𝜎²₂+𝜆(𝑍)ln𝛽<−_𝑍ln|𝑐(𝑧)|𝜆(𝑑𝑧), (5) where 𝐼₁(𝑥)=12𝜋₀^2𝜋𝑒^𝑥sin𝜑𝐼𝑑𝜑,₀(𝑥)=12𝜋₀^2𝜋sin𝜑𝑒^𝑥sin𝜑𝑑𝜑. (6)

Proof. We can rewrite the linearized system (2) as follows: (𝑡)=𝑥(𝑡)𝜎𝑑𝑥000𝛽𝑑𝑡+¹𝜎₂−𝜎₂𝜎₁𝑥(𝑡)(𝑡)+∘𝑑𝑊_𝑍(𝑧)(𝑧)𝑥(𝑡)𝑁(),−𝛽𝑐−100𝛽𝑐−1𝑑𝑡,𝑑𝑧 (7) where 𝜎₁≠0,𝜎₂≠0.
Denote by 𝜑 the angle of 𝑥∈ℜ². Then, 𝜑=arctan(𝑥₂/𝑥₁)∈[0,2𝜋), which depends on the quadrant of (𝑥₁,𝑥₂)∈ℜ², and 𝜃=(cos𝜑,sin𝜑)∈𝕊. Let 𝜃^⟂=(sin𝜑,−cos𝜑). Then, 𝜃^⟂ is orthogonal to 𝜃. Thus, in terms of the new coordinates (𝜌,𝜑), 𝑥₁=𝑒^𝜌cos𝜑,𝑥₂=𝑒^𝜌sin𝜑. (8) By Itô's formula, we have 𝑑𝜌(𝑡)=𝛽sin³𝜑(𝑡)𝑑𝑡+𝜎₁(𝑡)+𝑑𝑊_𝑍𝛽|𝑐(𝑧𝑁(),(𝑡)(𝑡)(𝑡)ln)|𝑑𝑡,𝑑𝑧𝑑𝜑=𝛽sin𝜑cos𝜑𝑑𝑡−𝜎₂𝑑𝑊(𝑡)_𝑍𝜋(𝑐(𝑧𝑡1−0.5sgn))−2𝜑₋𝑁(𝑑𝑡,𝑑𝑧). (9) Thus, the adjoint equation 12𝑑²𝑑𝜑²𝜎²₂𝜓−𝑑𝛽𝑑𝜑2𝜓+sin2𝜑_𝑍𝜓𝜋1−0.5sgn(𝑐(𝑧))−𝜑𝑡₋−𝜓𝜑𝜆(𝑑𝑧)=0 (10) admits a unique periodic solution as the invariant probability density 𝜓𝜑=𝜓₀𝛽exp2𝜎₂sin2𝜑, (11) with 𝜓₀=₀^2𝜋𝛽exp2𝜎₂sin2𝜑𝑑𝜑⁻¹. (12) The Lyapunov exponent is as follows: 𝛽Λ=2+𝛽2₀^2𝜋sin2𝜑𝜓𝜑𝑑𝜑+_𝑍𝛽|𝑐(𝑧𝜆()=𝛽ln)|𝑑𝑧2+𝛽2𝐼₁𝛽/2𝜎²₂𝐼₀𝛽/2𝜎²₂+_𝑍ln𝛽|𝑐(𝑧)|𝜆(𝑑𝑧), (13) where the integrals 𝐼₁(𝑥)=12𝜋₀^2𝜋𝑒^𝑥sin𝜑𝐼𝑑𝜑,₀(𝑥)=12𝜋₀^2𝜋sin𝜑𝑒^𝑥sin𝜑𝑑𝜑, (14) are the modified Bessel functions which satisfy, respectively, the zero-order and the first-order Bessel equations of a purely imaginary argument: 𝑥²𝑦̈+𝑥𝑦̇−𝑥²𝑦=0,𝑥²𝑦̈+𝑥𝑦̇−𝑥²+1𝑦=0. (15) Note that 𝐼₀(𝑥) is even, while 𝐼₁(𝑥) is odd such that 𝐼₁(𝑥)/𝐼₀(𝑥) is between −1 and 1, strictly increasing and at most linear growth. The diffusion system corresponding to (7) is asymptotically unstable with the Lyapunov exponent Λ₀=𝛽2𝐼1+¹𝛽/2𝜎²₂𝐼₀𝛽/2𝜎²₂>0, (16) if 𝛽>0. However, the stochastic jump diffusion of (7) is asymptotically stable if 𝛽Λ=2𝐼1+¹𝛽/2𝜎²₂𝐼₀𝛽/2𝜎²₂+𝜆(𝑍)ln𝛽+_𝑍ln|𝑐(𝑧)|𝜆(𝑑𝑧)<0. (17)Remark. This shows the effect of Poisson noises which can stabilize the system in this respect. The alternative phenomenon Λ>0 also happens if the reverse inequality in (17) holds while 𝛽<0.

Theorem 2. Suppose the condition of Theorem 1 is satisfied and the stochastic jump-diffusion linearized system (2) is asymptotically stable, that is, Λ<0. Then, ℙ_𝑥0sup_𝑠≥𝑡|𝑥(𝑠(𝑧)𝑥)|≥𝐾≤𝐾Λ,𝜎,𝑐,𝜂,𝜆⁰𝐾^{−Λ/𝐾(Λ,𝜎,𝑐(𝑧),𝜂,𝜆)}−Λ×exp²𝑡(𝑧).2𝐾Λ,𝜎,𝑐,𝜂,𝜆 (18)

Proof. Let ℒ_𝜃 be the integro-differential operator corresponding to the jump-diffusion process 𝜃(𝑡) of (9), and let 𝑓∈𝐶²(𝕊). Then, ℒ_𝜃𝑓(𝑌)=ℋ₀𝑓(𝑌)+12^𝑚_𝑖=1ℋ²_𝑖𝑓(𝑌)+_𝑍𝑓(𝐶(𝑧)⋆𝑌)−𝑓(𝑌)𝜆(𝑑𝑧). (19)
Since, _𝕊(−𝑄(𝜃)+Λ)𝜓(𝜃)𝑑𝜃=0, (20) where 𝜓 is the unique invariant probability density of the jump-diffusion process of (9), then the integro-differential equation ℒ_𝜃𝜐(𝜃)=−𝑄(𝜃)+Λ has a unique solution 𝜐∈𝐶¹(𝕊^𝑑−1). Then, ℒ|𝑥(𝑡𝑥(𝑡)ln)|+𝜐|𝑥(𝑡)|=Λ<0. (21)
Let 𝑀(𝑡)=𝑀^𝑐(𝑡)+𝑀^𝑑(𝑡) be a right continuous martingale defined as follows: 𝑀^𝑐(𝑡)=^𝑡₀²_𝑖=1𝑥_𝑖(𝑡)|𝑥(𝑡)|²+𝜕_𝑖𝜐(𝜃(𝑠(𝑡(𝑠),𝑀))𝐵𝑥)𝑑𝑊_𝑑(𝑡)=^𝑡_0𝑍𝐶(𝑧)𝜃(𝐶(𝑧)()(𝜃(𝑁(),ln𝑠−)+𝜐⋆𝜃𝑠−−𝜐𝑠−))𝑑𝑠,𝑑𝑧 (22) and define an operator ⋆ on 𝕊 by 𝐶(𝑧)⋆𝜃(𝑠−)=𝐶(𝑧)𝜃(𝑠−)/|𝐶(𝑧)𝜃(𝑠−)|,𝐶(𝑧)=𝐼+𝐶(𝑧),𝜕_𝑖𝜐=𝜕𝜐/𝜕𝑥_𝑖𝜎𝐵=¹𝜎₂−𝜎₂𝜎₁𝐶(𝑧)=(𝑧)(𝑧).−𝛽𝑐−100𝛽𝑐−1 (23)
Note that 𝑏_𝑖(⋅),𝑖=1,2,…,𝑚, and 𝑐(⋅,𝑧) are ℱ_𝑡-adapted processes. Let 𝑀_𝜇,𝜈(𝑡)=𝑀^𝑐_𝜇+𝑀^𝑑_𝜈(𝑡) be a ℱ_𝑡-local martingale such that 𝑀^𝑐_𝜇(𝑡)=^𝑚_𝑖=1𝜇_𝑖^𝑡₀𝑏_𝑖(𝑠)(𝑠),𝑀𝑑𝑊_𝑑𝜈(𝑡)=𝜈^𝑡_0𝑍𝑐(𝑠,𝑧)𝑁(𝑑𝑠,𝑑𝑧)𝑑𝑠, (24) are, respectively, a continuous ℱ_𝑡-local martingale and a pure jump ℱ_𝑡-local martingale with 𝑀^𝑐_𝜇(𝑡)=^𝑚_𝑖=1𝜇²_𝑖^𝑡₀𝑏²_𝑖(𝑠)(𝑠),𝑀𝑑𝑊_𝑑𝜈(𝑡)=^𝑡_0𝑍𝑒^{𝜈𝑐(𝑠,𝑧)}−1−𝜈𝑐(𝑠,𝑧)𝑁(𝑑𝑠,𝑑𝑧)𝑑𝑠. (25)
Therefore, let 𝜅>0; then the conditional quadratic variation of 𝑀^𝑐(𝑡) is 𝜅𝑀^𝑐(𝑡)=𝜅²²_𝑖,𝑗=1^𝑡₀𝜎_𝑖𝑗(𝑥(𝑠𝑥))^𝑖(𝑠)|𝑥(𝑠)|²+𝜕_𝑖𝜐(𝜃(𝑠×𝑥))_𝑗(𝑡)|𝑥(𝑡)|²+𝜕_𝑗𝜐(𝜃(𝑠))𝑑𝑠=𝜅²²_𝑖,𝑗=1^𝑡₀𝜎_𝑖𝑗(𝑥(𝑠×𝜃))_𝑖(𝑠)+²_𝑘=1𝛿_𝑖𝑘−𝜃_𝑖(𝑠)𝜃_𝑘(𝑠(𝜃(𝑠)𝜕𝜐))𝜕𝜃_𝑘×𝜃_𝑖(𝑠)+²_𝑘=1𝛿_𝑗𝑘−𝜃_𝑗(𝑠)𝜃_𝑘(𝑠(𝜃(𝑠)𝜕𝜐))𝜕𝜃_𝑘𝑑𝑠≤𝜅²𝐾(𝐵,𝜐)𝑡≤𝜅²𝐾𝑞,𝐴,𝐵,𝐶𝑡. (26) as 𝜎_𝑖𝑗(𝜃) and 𝜕𝜐(𝜃(𝑠))/𝜕𝜃_𝑘 are bounded on 𝕊.
On the other hand, 𝜅𝑀^𝑑(𝑡)=^𝑡_0𝑍𝐶(𝑧)𝜃(𝑠𝐶(𝑧)(𝑠)(𝜃(𝑠𝐶(𝑧)𝜃(𝑠𝐶(𝑧)(𝑠)(𝜃(𝑠𝜆()≤exp𝜅ln)+𝜅𝜐⋆𝜃−𝛼𝜐))−1−𝜅ln)−𝜅𝜐⋆𝜃+𝜅𝜐))𝑑𝑧𝑑𝑠_𝑡0𝑍exp𝜅ln𝛽²𝑐²(𝑧)∨𝛽²𝑐²(𝑧)‖𝜐‖𝛽−1+2−1−𝜅ln²𝑐²(𝑧)∨𝛽²𝑐²(𝑧)()≤𝜅−1×𝜆𝑑𝑧𝑑𝑠₂𝑡2_𝑍exp𝜅ln𝛽²𝑐²(𝑧)∨𝛽²𝑐²(𝑧)‖𝜐‖𝜆()=𝜅−1+2𝑑𝑧₂𝑡2𝐾=𝛼𝑐,𝛽,𝜆,𝜐²𝑡2𝐾Λ,𝑐,𝛽,𝜆,𝜎₁,𝜎₂. (27) Since 𝔼exp𝜅𝑀^𝑐(𝑡)<∞,𝔼𝜅𝑀^𝑑(𝑡)<∞, (28) for all 𝑡>0, then ℰ()(𝑡)−1𝜅𝑀𝑡=exp𝛼𝑀2𝜅𝑀^𝑐(𝑡)−𝜅𝑀^𝑑(𝑡), (29) is a ℱ_𝑡 martingale. Denote by (𝑡) the decimal part of 𝑡. As sup_{(𝑡)𝑘≤𝑠−<𝑘+1}𝜅ln|𝑥(𝑠)|≤sup_{𝑘≤𝑠−(𝑡)<𝑘+1}𝜅|𝑥(𝑠𝑥(𝑠)ln)|+𝜐|𝑥(𝑠−)|_𝑠0ℒ|𝑥(𝑢𝑥(𝑢)ln)|+𝜐|𝑥(𝑢𝑥)|𝑑𝑢−ln₀𝑥+𝜐⁰𝑥₀+sup_{(𝑡)𝑘≤𝑠−<𝑘+1}𝜅𝑥(𝑢)−𝜐|𝑥(𝑢𝑥)|+ln₀𝑥+𝜐⁰𝑥₀+Λ𝑠≤sup_{(𝑡)𝑘≤𝑠−<𝑘+1}(𝑡)−1𝜅𝑀2𝜅𝑀^𝑐(𝑡)−𝜅𝑀^𝑑𝑡+𝜅()₂2𝐾Λ,𝛽,𝛿₁,𝜎₂((𝑡))2‖𝜐‖𝑥,𝜆𝑘++1+𝜅+ln₀+𝜅Λ(𝑘+1), (30) we have by the martingale inequality ℙ_𝑥0sup_𝑠≥𝑡|𝑥(𝑠≤)|≥𝑅_{∞[𝑡]𝑘=}ℙ_𝑥0sup_{(𝑡)𝑘≤𝑠−<𝑘+1}|𝑥(𝑠≤𝜅ln)|≥𝜅ln𝑅_{∞[𝑡]𝑘=}ℙ_𝑥0×sup_{(𝑡)𝑘≤𝑠−<𝑘+1}ℰ(𝑠)𝜅𝜅𝑀)(≥expln𝑅−ln𝑥₀‖𝜐‖((𝑡−𝜅−2−𝜅Λ𝑘+))²2𝐾Λ,𝜎₁,𝜎₂×((𝑡)≤,𝛽,𝑐,𝜆𝑘++1)_{∞[𝑡]𝑘=}𝑥₀𝑅^𝜅×𝑒^{𝜅𝐾(Λ,𝜎1,𝜎2,𝛽,𝑐,𝜆)/2+2𝜅‖𝜐‖+[𝜅Λ+𝜅2𝐾(Λ,𝜎1,𝜎2,𝛽,𝑐,𝜆)/2](𝑘+(𝑡))}. (31) In order to minimize the exponent in the inequality of (31), let Λ𝜅=−𝐾Λ,𝜎₁,𝜎₂,𝛽,𝑐,𝜆>0. (32) Then, we can obtain that ℙ_𝑥0sup_𝑠≥𝑡|𝑥(𝑠≤𝑥)|≥𝑅₀𝑅^{−Λ/𝐾(Λ,𝜎1,𝜎2,𝛽,𝑐,𝜆)}×𝑒^{(Λ2/2−2Λ‖𝜐‖)/𝐾(Λ,𝜎1,𝜎2,𝛽,𝑐,𝜆)}×_𝑘=[𝑡]𝑒^{−Λ2(𝑘+(𝑡))/𝐾(Λ,𝜎1,𝜎2,𝛽,𝑐,𝜆)}=𝑥₀𝑅^{−Λ/𝐾(Λ,𝜎1,𝜎2,𝛽,𝑐,𝜆)}×expΛ²/2−2Λ‖𝜐‖/𝐾Λ,𝜎₁,𝜎₂,𝛽,𝑐,𝜆1−exp−Λ²/2𝐾Λ,𝜎₁,𝜎₂−Λ,𝛽,𝑐,𝜆×exp²𝑡2𝐾Λ,𝜎₁,𝜎₂.,𝛽,𝑐,𝜆 (33) As ‖𝜐‖=sup_|𝜃|=1|𝜐(𝜃)|=𝐾(Λ,𝜎₁,𝜎₂,𝛽,𝑐,𝜆), 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 (𝜑(𝑡,𝜔,𝑥)) with the Poisson processes which is global to the forward, that is, 𝐷_𝑡(𝜔)=ℜ² for all 𝑡>0. Moreover, (𝜑(𝑡,𝜃_−𝑡𝜔,𝑥)) is ℱ⁰_𝑡 measurable.

Proof . We can rewrite system (3) as follows: 𝑑𝑥₁=𝑥₂𝑑𝑡,𝑑𝑥₂=𝛼𝑥₁+𝛽𝑥₂−𝑥³₂−𝑥²₁𝑥₂+𝜎𝑑𝑡₁𝑥₁+𝜎₂𝑥₂+𝜎₃(𝑡)+𝛾𝑑𝑊₁𝑥₁+𝛾₁𝑥₂+𝛾₁𝑑𝑁(𝑡). (34)
Choose a function 𝑉𝑥₁,𝑥₂=𝑥²₁+𝑥²₂. (35) Applying Itô's formula to (35), we have 𝑉𝑥₁,𝑥₂=𝑥²₁₀+𝑥²₂₀+^𝑡₀2(1+𝛼)𝑥₁𝑥₂+𝜎²₁𝑥²₁+2𝛽+𝜎²₂𝑥²₂+𝜎²₃−𝑥⁴₂−𝑥²₁𝑥²₂𝑑𝑠+2^𝑡₀𝜎₁𝑥₁𝑥₂+𝜎₂𝑥²₂+𝜎₃𝑥₂(𝑠)+𝑑𝑊_𝑡02𝛾₁𝑥₁𝑥₂+𝛾²₁𝑥²₁+2𝛾₂+𝛾²₂𝑥²₂+2𝛾₃𝑥₂+𝛾²₃𝑑𝑁(𝑠). (36)
Using the Cauchy-Bunyakovsky-Schwarz inequality, then 2(1+𝛼)𝑥₁𝑥₂+𝜎²₁𝑥²₁+2𝛽+𝜎²₂𝑥²₂≤|(1+𝛼)|+𝜎²₁𝑥²₁+|(1+𝛼)|+2𝛽+𝜎²₂𝑥²₂×|1+𝛼|+𝜎²₁∨|1+𝛼|+2𝛽+𝜎²₂𝑉𝑥₁,𝑥₂,𝜎₁𝑥₁𝑥₂+𝜎₂𝑥²₂+𝜎₃𝑥₂≤𝜎₁2𝑥²₁+𝜎₁2+𝜎₂+𝜎₃2𝑥²₂+𝜎₃2≤𝜎₁2+𝜎₂+𝜎₃2𝑉𝑥₁,𝑥₂+𝜎₃2,2𝛾₁𝑥₁𝑥₂+𝛾²₁𝑥²₁+2𝛾₂+𝛾²₂𝑥²₂+2𝛾₃𝑥₂+𝛾²₃≤𝛾₁+𝛾²₁∨2𝛾₂+𝛾²₂+𝛾₃×𝑉𝑥₁,𝑥₂+𝛾₃+𝛾²₃. (37)
From (36), we can obtain 𝑉𝑥₁,𝑥₂≤𝑉𝑡₀+^𝑡₀|(1+𝛼)|+𝜎²₁∨|(1+𝛼)|+2𝛽+𝜎²₂×𝑉𝑥₁,𝑥₂𝑑𝑠+2^𝑡₀𝜎₁2+𝜎₂+𝜎₃2𝑉𝑥₁,𝑥₂+𝜎₃2𝑑𝑊(𝑠)+^𝑡₀𝛾₁+𝛾²₁∨2𝛾₂+𝛾²₂+𝛾₃𝑉𝑥₁,𝑥₂+𝛾₃+𝛾²₃𝑑𝑁(𝑠). (38)
Suppose that 𝑌_𝑡 satisfies the following equation with jumps: 𝑌_𝑡=𝑉𝑡₀+^𝑡₀|(1+𝛼)|+𝜎²₁∨|(1+𝛼)|+2𝛽+𝜎²₂𝑌_𝑠𝑑𝑠+2^𝑡₀𝜎₁2+𝜎₂+𝜎₃2𝑌_𝑠+𝜎₃2(𝑠)+𝑑𝑊_𝑡0𝛾₁+𝛾²₁∨2𝛾₂+𝛾²₂+𝛾₃𝑌_𝑠+𝛾₃+𝛾²₃𝑑𝑁(𝑠); (39) then 𝑌_𝑡 is finite on any finite interval [0,𝑇] for any 𝑇>0. Hence, (38) and (39) yield that the process 𝑉(𝑥₁,𝑥₂) is finite on [0,𝑇] for any 0<𝑇<+∞; that is, the random dynamical system (𝜑(𝑡,𝜔,𝑥)) is global to the forward.

Corollary 4. Suppose the condition of Theorem 3 is satisfied. Then the random dynamical system (𝜑(𝑡,𝜔,𝑥)) generated by (3) has the following additional properties:(i) 𝐷_𝑡(𝜃_−𝑡)=ℜ² for all 𝑡≥0;(ii) 𝑅_𝑡(𝜃_−𝑡)↓𝐸(𝜔) as 𝑡↑∞.

Proof. Since all 𝑡∈ℜ,𝜑(𝑡,𝜔)∶𝐷_𝑡(𝜔)→𝑅_𝑡(𝜔) is (local) diffeomorphism of the random(open) domain 𝐷_𝑡(𝜔)⊂𝑅² into the random range 𝑅_𝑡(𝜔)=𝜑(𝑡,𝜔)𝐷_𝑡(𝜔)⊂ℜ² where 𝐷_𝑡(𝜔)↓ as 𝑡↑∞, and 𝜑(0,𝜔)=𝐼𝑑_ℜ², 𝐸(𝜔)=⋂_𝑡∈ℜ𝐷_𝑡(𝜔)={𝑥∶𝜑(𝑡,𝜔,𝑥)existsforalltimes}. 𝐷(𝜔) is the set of all initial values 𝑥 whose orbits never explode. It may happen that 𝐷(𝜔)=∅.
For 𝑡≥0, Theorem 3 yields that (𝜑(𝑡,𝜔,𝑥)) is nonexplosive; we have the property (i) by replacing 𝜔 with 𝜃_−𝑡𝜔. Theorem 3 implies that 𝑅_𝑡(𝜃_−𝑡𝜔)=ℜ² for 𝑡≤0 and 𝑅_𝑡(𝜃_𝑡𝜔)⊂𝑅_𝑠(𝜃_𝑠𝜔) for 0≤𝑠≤𝑡, respectively. Since 𝑅_𝑡(𝜃_−𝑡𝜔)=𝐷_−𝑡(𝜔), the definition of 𝐸(𝜔) yields that (ii) is true.

Lemma 5. Suppose 𝑘>−1. If 𝑎−𝑐²/2+𝜆log(1+𝑘)<0, then the stochastic differential equation with the Poisson processes 𝑑𝑋_𝑡=𝑎𝑋_𝑡+𝑏𝑑𝑡+𝑐𝑋_𝑡+ℎ𝑑𝑊(𝑡)+𝑘𝑋_𝑡(𝑡),𝑋+𝑓𝑑𝑁₀=𝑥, (40) has the unique invariant measure which is a Dirac measure supported by 𝜉(𝑡)=𝑐⁰_−∞𝑒^{−(𝑎−𝑐2/2)𝑡−𝑐𝑊𝑠−𝐿𝑁𝑡}𝑑𝑠+ℎ⁰_−∞𝑒^{−(𝑎−𝑐2/2)𝑡−𝑐𝑊𝑠−𝐿𝑁𝑡}𝑑𝑊(𝑠)+𝑓⁰_−∞𝑒^{−(𝑎−𝑐2/2)𝑡−𝑐𝑊𝑠−𝐿𝑁𝑡)}𝑑𝑁_𝑠, (41) and 𝜉(𝑡) attracts all points with exponential speed, where 𝐿=log(1+𝑘) and 𝜆 is the intensity of a Poisson process 𝑁 on ℜ.

Proof. By Itô's formula, we obtain 𝜑(𝑡,𝜔)𝑥=𝑒^{(𝑎−𝑐2/2)𝑡+𝑐𝑊𝑠+𝐿𝑁𝑡}𝑥+𝑐^𝑡₀𝑒^{−(𝑎−𝑐2/2)𝑡−𝑐𝑊𝑠−𝐿𝑁𝑡}𝑑𝑠+ℎ^𝑡₀𝑒^{−(𝑎−𝑐2/2)𝑡−𝑐𝑊𝑠−𝐿𝑁𝑡}𝑑𝑊(𝑠)+𝑓^𝑡₀𝑒^{−(𝑎−𝑐2/2)𝑡−𝑐𝑊𝑠−𝐿𝑁𝑡)}𝑑𝑁_𝑠, (42) is the random dynamical system generated by (40). Since lim_{𝑡→−∞}𝑁_𝑡(𝜆𝑡)=1 and 𝑎−𝑐²/2+𝜆𝐿<0, we have the following integrals: ⁰_−∞𝑒^{−(𝑎−𝑐2/2)𝑡−𝑐𝑊𝑠−𝐿𝑁𝑡}𝑑𝑠<∞,_0−∞𝑒^{−(𝑎−𝑐2/2)𝑡−𝑐𝑊𝑠−𝐿𝑁𝑡}𝑑𝑊(𝑠)<∞,_0−∞𝑒^{−(𝑎−𝑐2/2)𝑡−𝑐𝑊𝑠−𝐿𝑁𝑡}𝑑𝑁_𝑠<∞. (43) Since (40) is affine with stable linear part, then, (42) and Theorem 5.6.1 of Arnold in [7] imply that the unique invariant measure of (40) is the Dirac measure supported by 𝜉(𝑡)=lim_𝑡→∞𝜑(−𝑡,𝜔)⁻¹𝑥=𝑐⁰_−∞𝑒^{−(𝑎−𝑐2/2)𝑠−𝑐𝑊𝑠−𝐿𝑁𝑠}𝑑𝑠+ℎ⁰_−∞𝑒^{−(𝑎−𝑐2/2)𝑠−𝑐𝑊𝑠−𝐿𝑁𝑠}𝑑𝑊(𝑠)+𝑓⁰_−∞𝑒^{−(𝑎−𝑐2/2)𝑠−𝑐𝑊𝑠−𝐿𝑁𝑠)}𝑑𝑁_𝑠, (44) for all 𝑥∈ℜ and 𝜉(𝑡) attracts all points with exponential speed.

Theorem 6. Suppose the condition of Theorem 3 is satisfied and the coefficient in (3) satisfies 𝛾₁+𝛾₂>−1,𝜎₃=𝛾₃=0 and 1+𝛼+𝛽+2𝜆log1+𝛾₁+𝛾₂<𝜎₁+𝜎₂²2; (45) Then the random dynamical system generated by (3) possesses the unique parameter-dependent tempered random attractor 𝐴 with domain of attractor 𝒟(𝐴) containing the universe of sets 𝒞𝑙(𝒰), generated by 𝒰=(𝐷(𝜔))_𝜔∈Ω∶𝐷(𝜔)⊂ℜ²𝑖𝑠𝑎𝑡𝑒𝑚𝑝𝑒𝑟𝑒𝑑𝑟𝑎𝑛𝑑𝑜𝑚𝑠𝑒𝑡. (46) Moreover, the random attractor 𝐴 is measurable with respect to ℱ^0−∞=𝜎{𝑁_𝑡,𝑡≤0}.

Proof. Choose a function 𝑉𝑥₁,𝑥₂=𝑥²₁+2𝑥₁𝑥₂+𝑥²₂. (47) Applying Itô's formula, we have 𝑉𝑥₁,𝑥₂=𝑉𝑡₀+^𝑡₀21+𝛼+𝛽𝑥₁𝑥₂+2𝛼+𝜎²₁𝑥²₁+2𝛽+𝜎²₂+2𝑥²₂−𝑥⁴₂−2𝑥₁𝑥³₂−𝑥³₁𝑥₂−𝑥²₁𝑥²₂𝑑𝑠+2^𝑡₀𝜎₁𝑥²₁+𝜎₁+𝜎₂𝑥₁𝑥₂+𝜎₂𝑥²₂(𝑠)+𝑑𝑊_𝑡02𝛾₁+𝛾₂𝑥₁𝑥₂+2𝛾₁+𝛾²₁𝑥²₁+2𝛾₂+𝛾²₂𝑥²₂𝑑𝑁(𝑠). (48) This implies 𝑉𝑥₁,𝑥₂=𝑉𝑡₀+^𝑡₀1+𝛼+𝛽𝑉𝑥₁,𝑥₂+𝐴𝑥₁,𝑥₂+𝑑𝑠_𝑡0𝜎₁+𝜎₂𝑉𝑥₁,𝑥₂+𝐵𝑥₁,𝑥₂𝑑𝑊(𝑠)+^𝑡₀𝛾₁+𝛾₂𝑉𝑥₁,𝑥₂+𝐶𝑥₁,𝑥₂𝑑𝑁(𝑠), (49) where 𝐴𝑥₁,𝑥₂=𝛼+𝜎²₁−1−𝛽𝑥²₁+𝛽+𝜎²₂+1−𝛼𝑥²₂−𝑥⁴₂−2𝑥₁𝑥³₂−𝑥³₁𝑥₂−𝑥²₁𝑥²₂,𝐵𝑥₁,𝑥₂=𝜎₁−𝜎₂𝑥₁−𝑥₂,𝐶𝑥₁,𝑥₂=𝛾₁+𝛾²₁−𝛾₂𝑥²₁+𝛾₂+𝛾²₂−𝛾₁𝑥²₂. (50) Note that 𝐴(𝑥₁,𝑥₂)≤𝐾(𝛼,𝛽,𝜎₁,𝜎₂,𝛾₁,𝛾₂)=∶𝐾,𝐵(𝑥₁,𝑥₂)≤𝐾(𝛼,𝛽,𝜎₁,𝜎₂,𝛾₁,𝛾₂),𝐶(𝑥₁,𝑥₂)≤𝐾(𝛼,𝛽,𝜎₁,𝜎₂,𝛾₁,𝛾₂), where 𝐾 is a positive constant independent of 𝑥₁ and 𝑥₂.
From (49), we have 𝑉𝑥₁,𝑥₂≤𝑉𝑡₀+^𝑡₀1+𝛼+𝛽𝑉𝑥₁,𝑥₂++𝐾𝑑𝑠_𝑡0𝜎₁+𝜎₂𝑉𝑥₁,𝑥₂(𝑠)++𝐾𝑑𝑊_𝑡0𝛾₁+𝛾₂𝑉𝑥₁,𝑥₂+𝐾𝑑𝑁(𝑠). (51) Denote 𝜑(𝑡,𝜔,(𝑥₁,𝑥₂)) and 𝜓(𝑡,𝜔,(𝑥₁,𝑥₂)); then random dynamical systems are generated by (3) and 𝑍_𝑡≤𝑉𝑡₀+^𝑡₀1+𝛼+𝛽𝑍_𝑠++𝐾𝑑𝑠_𝑡0𝜎₁+𝜎₂𝑍_𝑠(𝑠)++𝐾𝑑𝑊_𝑡0𝛾₁+𝛾₂𝑍_𝑠+𝐾𝑑𝑁(𝑠), (52) respectively. The random dynamical systems generated by (52) are 𝜓(𝑡,𝜔)𝑥=𝑒^{[(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑡+(𝜎1+𝜎2)𝑊𝑡+log(1+𝛾1+𝛾2)𝑁𝑡}×𝑥+𝐾^𝑡₀𝑒^{−[(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑠−(𝜎1+𝜎2)𝑊𝑠−log(1+𝛾1+𝛾2)𝑁𝑠}×𝑑𝑠+𝐾_𝑡0𝑒^{−[(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑠−(𝜎1+𝜎2)𝑊𝑠−log(1+𝛾1+𝛾2)𝑁𝑠}𝑑𝑊(𝑠)+𝐾×∫^𝑡₀𝑒^{[−(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑠−(𝜎1+𝜎2)𝑊𝑠−log(1+𝛾1+𝛾2)𝑁𝑠}𝑑𝑁_𝑠. (53) Thus, from (51), we have 𝑉𝜑(𝑡,𝜔),𝑥₁,𝑥₂≤𝑉𝜓(𝑡,𝜔),𝑥₁,𝑥₂. (54)
By Lemma 5 and (45), we obtain that 𝜓(𝑡,𝜔) has the unique invariant measure which is a Dirac measure supported by 𝑟(𝜔)=𝐾𝑒^{[(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑡+(𝜎1+𝜎2)𝑊𝑡+log(1+𝛾1+𝛾2)𝑁𝑡}×^𝑡₀𝑒^{−[(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑠−(𝜎1+𝜎2)𝑊𝑠−log(1+𝛾1+𝛾2)𝑁𝑠}(𝑠)+𝑑𝑊_𝑡0𝑒^{[−(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑠−(𝜎1+𝜎2)𝑊𝑠−log(1+𝛾1+𝛾2)𝑁𝑠}𝑑𝑁_𝑠>0. (55)
It is easy to prove that 𝜓(𝑡,𝜃_−𝑡𝜔)𝑥(𝜃_−𝑡𝜔)→𝑟(𝜔) as 𝑡→∞ for any initial value 𝑥(𝜔)∈ℜ₊ such that 𝑒^{[−(1+𝛼+𝛽)+(𝜎1+𝜎2)2/2]𝑡}𝑥(𝜃_−𝑡𝜔)→0 for some 𝑎>0. Hence, we may the universe of set 𝒰=𝐼(𝜔)⊂ℜ₊isatemperedrandomset; (56) that is, the random variable 𝜂(𝜔)=sup_{𝑥∈𝐼(𝜔)}𝑥 satisfies lim_𝑡→∞log⁺[𝜂(𝜃_𝑡𝜔)/𝑡]=0 for any 𝐼=𝒰. From the results of Arnold in [7, Proposition 4.17], it yields lim_𝑡→∞𝑒^−𝜀𝑡𝜂(𝜃_−𝑡𝜔)=0 for any 𝜀>0.
It is clear that 𝒰 is closed under inclusion and Ω×{𝑥}∈𝒰 for all 𝑥∈ℜ⁺.
Next, we will prove that the random set [0,(1+𝜀)𝑟] is forward invariant and absorbing for 𝜓(𝑡,𝜔) with respect to the universe 𝒰 for any 𝜀>0.
In fact, the absorbing property follows from the definition of 𝒰 and Schenk-Hoppé [9, Theorem 4.1]. We only prove that 𝜓(𝑡,𝜔)[0,(1+𝜀)𝑟(𝜔)]⊂[0,(1+𝜀)𝑟(𝜃_𝑡𝜔)] for all 𝑡≥0. Since 𝜓(𝑡,𝜔) is a nonnegative function on ℜ and 𝜓(𝑡,𝜔)𝑥<𝜓(𝑡,𝜔)𝑦 for 𝑥<𝑦, it suffices to show 𝜓(𝑡,𝜔)(1+𝜀)𝑟(𝜔)≤(1+𝜀)𝑟(𝜃_𝑡𝜔), it is equivalent that prove 𝜓𝑡,𝜃_−𝑡𝜔(1+𝜀)𝑟𝜃_−𝑡𝜔≤(1+𝜀)𝑟(𝜔). (57) From (53), one can obtain 𝜓𝑡,𝜃_−𝑡𝜔(1+𝜀)𝑟𝜃_−𝑡𝜔(𝜔)×=𝑟+𝜀𝐾_{−𝑡−∞}𝑒^{−[(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑠−(𝜎1+𝜎2)𝑊𝑠−log(1+𝛾1+𝛾2)𝑁𝑠}(𝑠)+𝑑𝑊_{−𝑡−∞}𝑒^{[−(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑠−(𝜎1+𝜎2)𝑊𝑠−log(1+𝛾1+𝛾2)𝑁𝑠}𝑑𝑁_𝑠. (58) It implies that (57) is true.
Since 𝑟𝜃_−𝑡𝜔=𝐾𝑒^{−[(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑡+(𝜎1+𝜎2)𝑊−𝑡+log(1+𝛾1+𝛾2)𝑁−𝑡}×^−𝑡_−∞𝑒^{−[(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑠−(𝜎1+𝜎2)𝑊𝑠−log(1+𝛾1+𝛾2)𝑁𝑠}(𝑠)+𝑑𝑊_{−𝑡−∞}𝑒^{[−(1+𝛼+𝛽)−(𝜎1+𝜎2)2/2]𝑠−(𝜎1+𝜎2)𝑊𝑠−log(1+𝛾1+𝛾2)𝑁𝑠}𝑑𝑁_𝑠≤𝑟(𝜔), (59) one has 𝑒^−𝜖𝑡𝑟𝜃_−𝑡𝜔⟶0,𝑡⟶∞, (60) for any 𝜖>0. It means that [0,(1+𝜀)𝑟(𝜔)]∈𝒰 for any 𝜀>0.
For any 𝜀>0, define the subset 𝐵(𝜔) of ℜ² by 𝐵(𝜔)=𝑉⁻¹([0,(1+𝜀)]𝑟(𝜔)). (61) This is a nonempty compact set by the surjectivity and continuity of 𝑉(𝑥,𝑦) and the fact that preimages of bounded sets are bounded under 𝑉(𝑥,𝑦). One can prove that the random set 𝐵(𝜔) satisfies the assumptions of Schenk-Hoppé [9, Theorm 4.1].
The measurability of 𝐵(𝜔) may be implied by Lemma 5.7 of Schenk-Hoppé [9]. Using the surjectivity and nonnegativity of 𝑉 and (54), one has 𝜓𝑡,𝜃_−𝑡𝜔0,(1+𝜀)𝑟𝜃_−𝑡𝜔⊂[0,(1+𝜀)𝑟(𝜔)]⟺𝜓𝑡,𝜃_−𝑡𝜔𝑉𝐵𝜃_−𝑡𝜔(𝐵(𝜔𝜓⊂𝑉))⟹𝑉𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔⊂𝑉(𝐵(𝜔)), (62) for all 𝑡≥0. It implies 𝜓𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔⊂𝑉⁻¹𝑉𝜓𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔⊂𝑉⁻¹(𝑉(𝐵(𝜔))=𝐵(𝜔); (63) that is, 𝐵 is forward invariant.
For any 𝐷∈𝒰, we have that the random variable 𝜂(𝜔)=sup{‖(𝑥,𝑦)‖²∶(𝑥,𝑦)∈𝐷(𝜔)} grows subexponentially fast by the definition of 𝒰.𝑉(𝑥,𝑦)≤3(𝑥²+𝑦²)+5≤3‖(𝑥,𝑦)‖²+5 yields the random variable sup{𝑥∈𝑉(𝐷(𝜔))}≤𝜂(𝜔)+5 which grows sub-exponentially fast; that is, 𝑉(𝐷)∈𝒰.
As the proof of (62) and (63), we have that there is a 𝑡(𝜔,𝐷)>0 such that, for all 𝑡>𝑡(𝜔,𝐷), 𝑉𝜑𝑡,𝜃_−𝑡𝜔𝐷𝜃_−𝑡𝜔⊂𝜓𝑡,𝜃_−𝑡𝜔𝑉𝐷𝜃_−𝑡𝜔⊂[0,(1+𝜀)𝑟(𝜔)]=𝑉(𝐵(𝜔)), (64) by (59) and the set [0,(1+𝜀)𝑟(𝜔)] absorbing any set in 𝒰. It shows the absorption of any set in 𝒰.
Finally, we prove the existence of a neighborhood of 𝐵 in 𝒰. In fact, for any 𝐵∈𝒰, Proposition 3.2(iii) of Schenk-Hoppé [9] implies that there exists a random variable 𝜌(𝜔)>0 such that 𝐵(𝜔)⊂𝑆(𝜌(𝜔))={𝑥∈ℜ²,‖𝑥‖𝜌(𝜔)}. It is clear that 𝑆(𝑏𝜌(𝜔))∈𝒰 and that 𝑆(𝜌(𝜔)) is a neighborhood of 𝐵 for all 𝑏>1.
By Theorem 4.1 of Schenk-Hoppé [9], we finish the proof. Because of 𝑟(𝜔)∈ℱ⁰_−∞, we have that the attractor 𝐴 is ℱ⁰_−∞ 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 𝜎₃=𝛾₁=𝛾₂=𝛾₃=0. 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 𝑐₁(𝑧)=𝑐₂(𝑧)=𝑐₃(𝑧)=𝜎₃=𝛾₁=𝛾₂=𝛾₃=0; then (3) becomes 𝜎𝑑𝑥=01𝛼−¹2𝜎𝛽−²2+0𝑥𝑑𝑡−𝑥²₁𝑥₂−𝑥³₂𝜎𝑑𝑡+00₁𝜎₂𝑥∘𝑑𝑊(𝑡). (65) The linearized RDS D𝜑 is generated by the linearization of (65), namely, the white noise case by ̇𝜎𝜐=01𝛼−¹2𝜎𝛽−²2𝜐+00−2𝑥₁𝑥₂−3𝑥²₂𝜎𝜐+00₁𝜎₂𝜐∘𝑊̇(𝑡). (66) For any invariant measure 𝜈, the trace formula gives 𝜆₁(𝜈)+𝜆₂(𝜈)=𝛽−𝜎₂/2−3𝛾𝔼_𝜈𝑥²₁. In particular, for 𝜈=𝛿₀, we obtain 𝜆₁(𝜈)+𝜆₂(𝜈)=𝛽−𝜎₂/2.
We also need the eigenvalues of the linearization of the deterministic system 𝜎₁=𝜎₂=0 at 𝑥=0, ̇𝜎𝜐=01𝛼−¹2𝜎𝛽−²2𝜐, (67) which are 𝜆_1,2=𝛽−𝜎₂/22±𝛽−𝜎₂/2²4𝜎+𝛼−¹2. (68) For (𝛽−𝜎₂/2)²/4+𝛼−𝜎₁/2>0, we have two real eigenvalues, while for (𝛽−𝜎₂/2)²/4+𝛼−𝜎₁/2<0, we have a pair of complex-conjugate eigenvalues: 𝜆_1,2=𝛽−𝜎₂/22±𝑖𝜔_𝑑,𝜔_𝑑∶=−𝛽−𝜎₂/2²4𝜎−𝛼+¹2, (69) where 𝜔_𝑑 represents the “damped eigenfrequency” of the linear system: 𝑋̈−𝛽𝑋̇−𝛼𝑋=0. (70)
It is a general observation that noise splits deterministic multiplicities of eigenvalues. For 𝛽²<−4𝛼 (which we assume) and 𝜎₁=𝜎₂=0, the deterministic linear system has two complex-conjugate eigenvalues 𝛽/2±𝜔_𝑑, which amounts to just one Lyapunov exponent 𝜆₁(0,𝛽)=𝛽/2 with multiplicity 2. For 𝜎₁≠0 and 𝜎₂≠0, however, the linearized SDE ̇𝜎𝜐=01𝛼−¹2𝜎𝛽−²2𝜎𝜐+00₁𝜎₂𝜐∘𝑊̇(𝑡), (71) has two different simple Lyapunov exponents 𝜆_𝑖(𝜇,𝛼,𝜎₁,𝜎₂)(𝑖=1,2) which satisfy 𝜆₁(𝛼,𝛽,𝜎₁,𝜎₂)+𝜆₂(𝛼,𝛽,𝜎₁,𝜎₂)=𝛽−𝜎₂/2. For small 𝜎₁,𝜎₂, we can use the asymptotic expansion given by Arnold [7, Theorem 9.4.11] and others; namely, when 𝜎₁→0,𝜎₂→0, then the Lyapunov exponent of (71) is as follows: 𝜆_1,2=𝛽−𝜎₂/22±𝜎²₁+𝜎²₂8𝜔²_𝑑𝑆2𝜔_𝑑+𝑂max𝜎⁴₁+𝜎⁴₂, (72) where 𝑆(⋅) is the spectral density of 𝑊̇(𝑡), for the white noise case where 𝑆(𝑢)≡1. Furthermore, the rotation number expands as 𝜌𝜇,𝛼=𝜔_𝑑−𝑐𝜎²₁+𝜎²₂8𝜔²_𝑑𝑇2𝜔_𝑑+𝑂max𝜎⁴₁+𝜎⁴₂, (73) where 𝑇(⋅) is the sine spectral density of 𝑊̇(𝑡),𝑐>0.
Consequently, at the deterministic Hopf bifurcation point 𝛽=0, we have for the white noise case, neglecting 𝑂(max{𝜎⁴₁+𝜎⁴₂}) term, 𝜆₂𝜎₁,𝜎₂𝜎,0=−²¹+𝜎²₂8𝜔²_𝑑𝑆2𝜔_𝑑<0<𝜆₁𝜎₁,𝜎₂=𝜎,0²¹+𝜎²₂8𝜔²_𝑑𝑆2𝜔_𝑑; (74) that is, at 𝛽=𝜎₂/2, the top exponent has already cross 0 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 𝛽_𝐷1=((𝜎²₁+𝜎²₂)/4𝛼)𝑆(2𝜔_𝑑)+𝑂(max{𝜎⁴₁+𝜎⁴₂})<0 and the second Lyapunov exponent changes sign at 𝛽_𝐷2=−((𝜎²₁+𝜎²₂)/4𝛼)𝑆(2𝜔_𝑑)+𝑂(max{𝜎⁴₁+𝜎⁴₂})>0.
For 𝛽−𝜎₂/2<𝛽_𝐷1,𝛿₀ is stable, it is the unique invariant measure, and 𝐴_𝛽={0} is the attractor of the RDS 𝜑_𝛽 in ℝ².
We have a first bifurcation from 𝛿₀ at 𝛽_𝐷1=𝛽−𝜎₂/2<0 of a stable ergodic measure 𝜈¹(𝜎₁,𝜎₂,𝛽)_𝜔=(1/2)(𝛿_𝑥1(𝜔)+𝛿_{−𝑥1(𝜔)}), which is a convex combination of two Dirac measures, sitting on the two “boundary points” of the one-dimensional unstable manifold of 𝑥=0, which is a saddle point. This situation persists for 𝛽−𝜎₂/2∈(𝛽_𝐷1,𝛽_𝐷2). As shown before, 𝛽_𝑃=0∈(𝛽_𝐷1,𝛽_𝐷2), 𝜈¹ undergoes a P-bifurcation, and 𝛿₀ and 𝜈¹ are both Markov measures. Hence, 𝜌=𝔼𝜈¹ solves the Fokker-Planck equation. The attractor 𝐴_𝛼 in (the universe of tempered sets of) ℝ² is the closure of the unstable manifold of 𝑥=0, the two “boundary points” supporting the measure 𝜈¹. In the punctured plane ℝ²⧵{0}, the attractor 𝐴⁰_𝛽 (in the universe of simply connected tempered sets) consists of the two-point set supp𝜈¹.
At 𝛽−𝜎₂/2=𝛽_𝐷2, we have a second bifurcation from 𝛿₀ of a measure 𝜈²(𝜎₁,𝜎₂,𝛽)_𝜔=(1/2)(𝛿_𝑥2(𝜔)+𝛿_{−𝑥2(𝜔)}), which is again a convex combination of two Dirac measures. 𝜈² is a saddle point that has positive and negative Lyapunov exponents, while 𝛿₀ has two positive exponents.
For 𝛽−𝜎₂/2>𝛽_𝐷2, the stable measure 𝜈¹ is supported by the “boundary” of the unstable manifold of 𝜈². The closure of this unstable manifold is an invariant “circle” around 𝑥=0 (random limit cycle) and supports both measures 𝜈¹ and 𝜈². On this “circle,” we have hyperbolic dynamics (𝜈¹ is attracting and 𝜈² 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 0. Its closure is the attractor 𝐴_𝛽 in ℝ². It carries all three invariant measures. In the punctured plane ℝ²⧵{0}, however, the attractor 𝐴⁰_𝛽 is the invariant “circle,” in which we have two invariant measures, in particular, the unique Markov measure 𝜈¹. 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 𝐴(𝜔) the random attractor of the random dynamical systems 𝜑 generated by the stochastic Rayleigh-van der Pol equation (3). Then,(i)the random attractor 𝐴 supports all invariant measures;(ii)the unstable set of any invariant measure 𝜇 is contained in random attractor; that is, 𝑀⁺_𝜇(𝜔)⊂𝐴(𝜔);(iii)there exists an invariant Markov measure 𝜇 supported by 𝜕𝐴(𝜔); that this, 𝜇_𝜔 is ℱ⁰_−∞ 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 𝛼=−1,𝜎₁≥0,𝜎₂≥0,𝛾₁>𝛾₂≥0. For −2<𝛽<2, we can obtain that the Lyapunov exponents Λ₁ and Λ₂ of the linear stochastic Rayleigh-van der Pol equations (3)𝑑𝑥₁=𝑥₂𝑑𝑡,𝑑𝑥₂=−𝑥₁+𝛽𝑥₂𝑑𝑡+𝜎₁𝑥₁+𝜎₂𝑥₂(𝑡)+𝛾𝑑𝑊₁𝑥₁+𝛾₂𝑥₂𝑑𝑁(𝑡), (75) are bounded by 𝜎𝛽+max²₁,𝜎²₂𝛽1+2−2𝜎₂+𝜎₁+𝛽𝜎₁+𝜎₂/2/1+𝛽/2²2+𝜆log1+2+𝛾²₁+𝛾₁𝛽+𝛾²₂+2𝛾₂+𝛽𝛾₂/21+𝛽/2≤Λ₂≤Λ₁≤𝜎𝛽+max²₁,𝜎²₂𝛽−1−/22𝜎₂+𝜎₁+𝛽𝜎₁+𝜎₂/2/1−𝛽/2²2+𝜆log1+2+𝛾²₁+𝛾₁𝛽+𝛾²₂+2𝛾₂+𝛽𝛾₂/2𝛽.1−/2 (76)
As follows.

Proof. Define the Lyapunov function on ℜ²𝑉𝑥₁,𝑥₂=𝑥²₁−𝛽𝑥₁𝑥₂+𝑥²₂. (77) Applying Itô's formula to (77), we have 𝑑𝑉𝑥₁,𝑥₂=𝛽𝑥²₂+𝛽𝑥²₁−𝛽²𝑥₁𝑥₂+𝜎²₁𝑥²₁+𝜎²₂𝑥²₂+𝑑𝑡2𝜎₂𝑥²₂+2𝜎₁−𝛽𝜎₂𝑥₁𝑥₂−𝛽𝜎₁𝑥²₁(𝑡)+𝛾𝑑𝑊₂₂+2𝛾₂𝑥²₂+2𝛾₁−𝛾₂𝛽𝑥₁𝑥₂+𝛾²₁−𝛾₁𝛽𝑥²₁𝑑𝑁(𝑡). (78) Since 2𝜎₂𝑥²₂+2𝜎₁𝑥₁𝑥₂≤2𝜎₂+𝜎₁𝑥²₁+𝑥²₂,−𝛽𝜎₂𝑥₁𝑥₂−𝛽𝜎₁𝑥²₁≤𝛽𝜎₁+𝜎₂2𝑥²₁+𝑥²₂,2𝑥₁𝑥₂+𝛾²₁−𝛾₁𝛽𝑥²₁≤1+𝛾²₁+𝛾₁𝛽𝑥²₁+𝑥²₂,𝛾²₂+2𝛾₂𝑥²₂−𝛽𝛾₂𝑥₁𝑥₂≤1+𝛾²₂+2𝛾₂+𝛽𝛾₂2𝑥²₁+𝑥²₂,𝛽1−2𝑥²₁+𝑥²₂≤𝑉𝑥₁,𝑥₂≤𝛽1+2𝑥²₁+𝑥²₂. (79) From (78) and (79), we have 𝑑𝑉𝑥₁,𝑥₂≤𝜎𝛽+max²₁,𝜎²₂𝛽𝑉𝑥1−/2¹,𝑥₂+𝑑𝑡2𝜎₂+𝜎₁+𝛽𝜎₁+𝜎₂/2𝛽𝑉𝑥1−/2¹,𝑥₂(𝑡)+𝑑𝑊2+𝛾²₁+𝛾₁𝛽+𝛾²₂+2𝛾₂+𝛽𝛾₂/2𝛽𝑥1−/2×𝑉₁,𝑥₂(𝑡),𝑥𝑑𝑁𝑑𝑉₁,𝑥₂≥𝜎𝛽+max²₁,𝜎²₂𝛽𝑉𝑥1+/2¹,𝑥₂+𝑑𝑡2𝜎₂+𝜎₁+𝛽𝜎₁+𝜎₂/2𝛽𝑥1+/2×𝑉₁,𝑥₂(𝑡)+𝑑𝑊2+𝛾²₁+𝛾₁𝛽+𝛾²₂+2𝛾₂+𝛽𝛾₂/2𝛽𝑥1+/2×𝑉₁,𝑥₂𝑑𝑁(𝑡). (80)
By Doléans-Dade's exponent formula, solving the corresponding linear differential equation, we have 𝑉𝑥₁,𝑥₂=≤𝑉𝑡₀(𝑡)()𝑁(𝑡,𝑡exp𝑎𝑡+𝑏𝑊+log1+𝑐)≥𝑉₀exp𝑘𝑡+ℎ𝑊(𝑡)+log(1+𝑟)𝑁(𝑡), (81) where 𝜎𝑎=𝛽+max²₁,𝜎²₂𝛽−1−/22𝜎₂+𝜎₁−𝛽𝜎₁+𝜎₂/2/1−𝛽/2²2,𝑏=2𝜎₂+𝜎₁−𝛽𝜎₁+𝜎₂/2𝛽,1−/2𝑐=2+𝛾²₁−𝛾₁𝛽+𝛾²₂+2𝛾₂−𝛽𝛾₂/2𝛽,𝜎1−/2𝑘=𝛽+max²₁,𝜎²₂𝛽−1+/22𝜎₂+𝜎₁−𝛽𝜎₁+𝜎₂/2/1+𝛽/2²2,𝑘=2𝜎₂+𝜎₁−𝛽𝜎₁+𝜎₂/2𝛽,1+/2𝑟=2+𝛾²₁−𝛾₁𝛽+𝛾²₂+2𝛾₂−𝛽𝛾₂/2𝛽.1+/2 (82) Then, we can obtain lim_𝑡→∞log𝑉𝑥₁,𝑥₂𝑡=≤lim_𝑡→∞log𝑎𝑡+𝑏𝑊(𝑡)+log(1+𝑐)𝑁(𝑡)𝑡=𝑎+𝜆log(1+𝑐),≥lim_𝑡→∞log𝑘𝑡+ℎ𝑊(𝑡)+log(1+𝑟)𝑁(𝑡)𝑡=𝑘+𝜆log(1+𝑟). (83) Dividing by 𝑡 and letting 𝑡→∞, 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 𝛼=−1 and 𝛽∈(−2,0) such that 𝜎Λ=𝛽+max²₁,𝜎²₂𝛽−1−/22𝜎₂+𝜎₁+𝛽𝜎₁+𝜎₂/2/1−𝛽/2²2+𝜆log1+2+𝛾²₁+𝛾₁𝛽+𝛾²₂+2𝛾₂+𝛽𝛾₂/21−𝛽/2<0. (84) Then, the random dynamical system generated by the stochastic Rayleigh-van der Pol equation (3) possesses the unique tempered random attractor 𝐴(𝜔)={0}, (85) with domain of attractor 𝒟(𝐴) containing the universe of 𝒞𝑙(𝒰) given by 𝒰=(𝐷(𝜔))_𝜔∈Ω∶𝐷(𝜔)⊂ℜ²𝑖𝑠𝑎𝑡𝑒𝑚𝑝𝑒𝑟𝑒𝑑𝑟𝑎𝑛𝑑𝑜𝑚𝑠𝑒𝑡, (86) where 𝒞𝑙(𝒰) is the complete universe of 𝒰 under inclusion.
In particular, for any 𝑥∈ℜ², the solutions 𝜑(𝑡,𝜃_−𝑡𝜔)𝑥 and 𝜑(𝑡,𝜔)𝑥 tend to zero exponentially fast as 𝑡→∞.

Proof. By Proposition 9 and assumption (91), the random dynamical system generated by (3) is stable. Let us define the nonnegative function on ℜ² as follows: 𝑉𝑥₁,𝑥₂=−𝛽𝑥⁴₁+𝑥²₁−𝛽𝑥₁𝑥₂+𝑥²₂. (87) Applying Itô's formula to (87), we can obtain 𝑑𝑉𝑥₁,𝑥₂=𝛽𝑥²₂+𝛽𝑥²₁−𝛽²𝑥₁𝑥₂+𝜎²₁𝑥²₁+𝜎²₂𝑥²₂+𝛽𝑥₁𝑥³₂−2𝑥⁴₂−2𝑥²₁𝑥²₂+𝑑𝑡2𝜎₂𝑥²₂+2𝜎₁−𝛽𝜎₂𝑥₁𝑥₂−𝛽𝜎₁𝑥²₁(𝑡)+𝛾𝑑𝑊₂₂+2𝛾₂𝑥²₂+2𝛾₁−𝛾₂𝛽𝑥₁𝑥₂+𝛾²₁−𝛾₁𝛽𝑥²₁𝑑𝑁(𝑡). (88) From (79) and (88), we can obtain 𝑑𝑉𝑥₁,𝑥₂≤𝜎𝛽+max²₁,𝜎²₂𝛽𝑉𝑥1−/2¹,𝑥₂𝛽++𝑅𝑑𝑡2𝜎₂+𝜎₁+𝛽𝜎₁+𝜎₂/2𝛽𝑉𝑥1−/2¹,𝑥₂(𝑡)+𝑑𝑊2+𝛾²₁+𝛾₁𝛽+𝛾²₂+2𝛾₂+𝛽𝛾₂/2𝛽𝑥1−/2×𝑉₁,𝑥₂𝑑𝑁(𝑡), (89) where 𝑅(𝑥₁,𝑥₂)=𝛽𝑥₁𝑥³₂−2𝑥⁴₂−2𝑥²₁𝑥²₂ is uniformly bounded by a positive constant 𝑅(𝛽).
By Lemma 5, we can obtain that the random dynamical system generated by the stochastic differential equation 𝑑𝑋_𝑡≤𝜎𝛽+max²₁,𝜎²₂𝛽𝑋1−/2^𝑡𝛽++𝑅𝑑𝑡2𝜎₂+𝜎₁+𝛽𝜎₁+𝜎₂/2𝛽𝑋1−/2^𝑡(𝑡)+𝑑𝑊2+𝛾²₁+𝛾₁𝛽+𝛾²₂+2𝛾₂+𝛽𝛾₂/2𝛽𝑋1−/2^𝑡𝑑𝑁(𝑡), (90) has the unique invariant measure 𝛿₀ which is the Dirac measure at 0. From the proof of Theorem 6, 𝐴(𝜔)={0}=𝑉⁻({0}) is an attractor attracting any set form 𝒰.
In particular, we obtain that for any 𝑥∈ℜ²,𝜑(𝑡,𝜗_−𝑡,𝑥) and 𝜑(𝑡,𝜔,𝑥) tend to zero as 𝑡→∞ with an exponential speed larger than Λ.
𝐴(𝜔)=𝐴^ℜ2(𝜔) is immediate from Theorem 6 and the definition of 𝒰.

Theorem 11. Suppose the condition of Theorem 3 is satisfied and the coefficient in (3) is satisfied and 𝛼=−1 and 𝛽∈(0,2) such that 𝜎Λ=𝛽+max²₁,𝜎²₂𝛽−1+/22𝜎₂+𝜎₁+𝛽𝜎₁+𝜎₂/2/1+𝛽/2²2+𝜆log1+2+𝛾²₁+𝛾₁𝛽+𝛾²₂+2𝛾₂+𝛽𝛾₂/21+𝛽/2>0. (91) Then, the random dynamical system generated by the stochastic Rayleigh-van der Pol equations (1) possesses the unique tempered random attractor 𝐴(𝜔)=ℜ²⧵{0}, (92) with domain of attractor 𝒟(𝐴) containing the universe of 𝒞𝑙(𝒰) given by 𝒰=𝑎𝑙𝑙𝑟𝑎𝑛𝑑𝑜𝑚𝑠𝑒𝑡(𝐷(𝜔))_𝜔∈Ω⊂ℜ²⧵{0}∶sup_{(𝜔)𝑥∈𝐷}‖𝑥‖+1‖𝑥‖𝑖𝑠𝑡𝑒𝑚𝑝𝑒𝑟𝑒𝑑. (93)
𝒰 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).