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

Authors: ZaiTang Huang; ChunTao Chen
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 Section2, 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:(1)dx1=x2dt,dx2=[αx1+βx2-x23-x12x2]dt+[σ1x1+σ2x2+σ3]∘dW(t)+∫Z‍[γ1c1(z)x1(t-)+γ2c2(z)x2(t-)+γ3c3(z)]N(dt,dz),
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 ℱt-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 pointsα=0 and σ3=c3(z)=0, a linear change of variables yields
(2)dx(t)=(000β)x(t)dt+(ax23(t)+bx22(t)x1(t)+cx2(t)x12(t)+dx13(t)-ax23(t)-bx22(t)x1(t)-cx2(t)x12(t)-dx13(t))dt+(σ11σ12σ21σ22)x(t)∘dW(t)+∫Z(γ11(z)γ12(z)γ21(z)γ22(z))‍x(t)N(dt,dz).IfZ{-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
(3)dx1=x2dt,dx2=[αx1+βx2-x23-x12x2]dt+[σ1x1+σ2x2+σ3]dW(t)+[γ1x1+γ1x2+γ1]dN(t),
where γi∈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.
Supposeσ11=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:
(4)Λ=β2(1+I1(β/2σ22)I0(β/2σ22))>0.
However, the stochastic jump diffusion of the linearized system (2) is asymptotically stable if
(5)β2(1+I1(β/2σ22)I0(β/2σ22))+λ(Z)lnβ<-∫Z‍ln|c(z)|λ(dz),
where
(6)I1(x)=12π∫02π‍exsinφdφ,I0(x)=12π∫02π‍sinφexsinφdφ.Proof.
We can rewrite the linearized system (2) as follows:
(7)dx(t)=(000β)x(t)dt+(σ1σ2-σ2σ1)x(t)∘dW(t)+∫Z‍(-βc(z)-100βc(z)-1)x(t)N(dt,dz),
where σ1≠0,σ2≠0.
Denote byφ 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),
(8)x1=eρcosφ,x2=eρsinφ.
By Itô's formula, we have
(9)dρ(t)=βsin3φ(t)dt+σ1dW(t)+∫Z‍ln(β|c(z)|)N(dt,dz),dφ(t)=βsinφ(t)cosφ(t)dt-σ2dW(t)∫Z‍[π(1-0.5sgn(c(z)))-2φ(t-)(1-0.5sgn(c(z)))]N(dt,dz).
Thus, the adjoint equation
(10)12d2dφ2(σ22ψ)-ddφ[(β2sin2φ)ψ]+∫Z‍[ψ(π(1-0.5sgn(c(z)))-φ(t-))-ψ(φ)(1-0.5sgn(c(z)))]λ(dz)=0
admits a unique periodic solution as the invariant probability density
(11)ψ(φ)=ψ0exp{β2σ2sin2φ},
with
(12)ψ0=(∫02π‍exp{β2σ2sin2φ}dφ)-1.
The Lyapunov exponent is as follows:
(13)Λ=β2+β2∫02π‍sin2φψ(φ)dφ+∫Z‍ln(β|c(z)|)λ(dz)=β2+β2I1(β/2σ22)I0(β/2σ22)+∫Z‍ln(β|c(z)|)λ(dz),
where the integrals
(14)I1(x)=12π∫02π‍exsinφdφ,I0(x)=12π∫02π‍sinφexsinφdφ,
are the modified Bessel functions which satisfy, respectively, the zero-order and the first-order Bessel equations of a purely imaginary argument:
(15)x2y¨+xy˙-x2y=0,x2y¨+xy˙-(x2+1)y=0.
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
(16)Λ0=β2(1+I1(β/2σ22)I0(β/2σ22))>0,
if β>0. However, the stochastic jump diffusion of (7) is asymptotically stable if
(17)Λ=β2(1+I1(β/2σ22)I0(β/2σ22))+λ(Z)lnβ+∫Z‍ln|c(z)|λ(dz)<0.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 Theorem1 is satisfied and the stochastic jump-diffusion linearized system (2) is asymptotically stable, that is, Λ<0. Then,
(18)ℙx0{sups≥t|x(s)|≥K}≤K(Λ,σ,c(z),η,λ)(|x0|K)-Λ/K(Λ,σ,c(z),η,λ)×exp(-Λ2t2K(Λ,σ,c(z),η,λ)).Proof.
Letℒθ be the integro-differential operator corresponding to the jump-diffusion process θ(t) of (9), and let f∈C2(𝕊). Then,
(19)ℒθf(Y)=ℋ0f(Y)+12∑i=1m‍ℋi2f(Y)+∫Z‍{f(C(z)⋆Y)-f(Y)}λ(dz).
Since,(20)∫𝕊‍(-Q(θ)+Λ)ψ(θ)dθ=0,
where ψ is the unique invariant probability density of the jump-diffusion process of (9), then the integro-differential equation ℒθυ(θ)=-Q(θ)+Λ has a unique solution υ∈C1(𝕊d-1). Then,
(21)ℒ{ln|x(t)|+υ(x(t)|x(t)|)}=Λ<0.
LetM(t)=Mc(t)+Md(t) be a right continuous martingale defined as follows:
(22)Mc(t)=∫0t‍∑i=12‍(xi(t)|x(t)|2+∂iυ(θ(s))Bx(t))dW(s),Md(t)=∫0t‍∫Z‍{ln|C~(z)θ(s-)|+υ(C~(z)⋆θ(s-))-υ(θ(s-))(C~(z)⋆θ(s-))}N~(ds,dz),
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/∂xi(23)B=(σ1σ2-σ2σ1)C(z)=(-βc(z)-100βc(z)-1).
Note thatbi(·),i=1,2,…,m, and c(·,z) are ℱt-adapted processes. Let Mμ,ν(t)=Mμc+Mνd(t) be a ℱt-local martingale such that
(24)Mμc(t)=∑i=1m‍μi∫0t‍bi(s)dW(s),Mνd(t)=ν∫0t‍∫Z‍c(s,z)N~(ds,dz)ds,
are, respectively, a continuous ℱt-local martingale and a pure jump ℱt-local martingale with
(25)〈Mμc〉(t)=∑i=1m‍μi2∫0t‍bi2(s)dW(s),⌊Mνd⌋(t)=∫0t‍∫Z‍{eνc(s,z)-1-νc(s,z)}N~(ds,dz)ds.
Therefore, letκ>0; then the conditional quadratic variation of Mc(t) is
(26)〈κMc〉(t)=κ2∑i,j=12‍∫0t‍σij(x(s))(xi(s)|x(s)|2+∂iυ(θ(s)))×(xj(t)|x(t)|2+∂jυ(θ(s)))ds=κ2∑i,j=12‍∫0t‍σij(x(s))×{θi(s)+∑k=12‍[δik-θi(s)θk(s)]∂υ(θ(s))∂θk}×{θi(s)+∑k=12‍[δjk-θj(s)θk(s)]∂υ(θ(s))∂θk}ds≤κ2K(B,υ)t≤κ2K(q,A,B,C)t.
as σij(θ) and ∂υ(θ(s))/∂θk are bounded on 𝕊.
On the other hand,(27)⌊κMd⌋(t)=∫0t‍∫Z‍[exp{(C~(z)⋆θ(s))κln|C~(z)θ(s)|+κυ(C~(z)⋆θ(s))-αυ(θ(s))}-1-κln|C~(z)θ(s)|-κυ(C~(z)⋆θ(s))+κυ(θ(s)){(C~(z)⋆θ(s))κln|C~(z)θ(s)|]λ(dz)ds≤∫0t‍∫Z‍[exp{κ(ln|β2c2(z)|∨|β2c2(z)-1|+2∥υ∥β2c2(z))β2c2(z)}-1-κ(ln|β2c2(z)|∨|β2c2(z)-1|)]×λ(dz)ds≤κ2t2∫Z‍[exp{κ(ln|β2c2(z)|∨|β2c2(z)-1|+2∥υ∥β2c2)}]λ(dz)=κ2t2K(c,β,λ,υ)=α2t2K(Λ,c,β,λ,σ1,σ2).
Since
(28)𝔼exp{〈κMc〉(t)}<∞,𝔼⌊κMd⌋(t)<∞,
for all t>0, then
(29)ℰ(κM)t=exp{αM(t)-12〈κMc〉(t)-⌊κMd⌋(t)},
is a ℱt martingale. Denote by (t) the decimal part of t. As
(30)supk≤s-(t)<k+1κln|x(s)|≤supk≤s-(t)<k+1κ{ln|x(s)|+υ(x(s)|x(s)|)-∫0s‍ℒ[ln|x(u)|+υ(x(u)|x(u)|)]du-ln|x0|+υ(x0|x0|)}+supk≤s-(t)<k+1κ{-υ(x(u)|x(u)|)+ln|x0|+υ(x0|x0|)+Λs}≤supk≤s-(t)<k+1{κM(t)-12〈κMc〉(t)-⌊κMd⌋(t)}+κ22K(Λ,β,δ1,σ2,λ)(k+(t)+1)+κ(2∥υ∥+ln|x0|)+κΛ(k+1),
we have by the martingale inequality
(31)ℙx0{sups≥t|x(s)|≥R}≤∑k=[t]∞‍ℙx0{supk≤s-(t)<k+1κln|x(s)|≥κlnR}≤∑k=[t]∞‍ℙx0×{supk≤s-(t)<k+1ℰ(κM)(s)≥exp[κ22κ(lnR-lnx0-2∥υ∥)-κΛ(k+(t))-κ22K(Λ,σ1,σ2,β,c,λ)×(k+(t)+1)κ22]}≤∑k=[t]∞‍(|x0|R)κ×eκK(Λ,σ1,σ2,β,c,λ)/2+2κ∥υ∥+[κΛ+κ2K(Λ,σ1,σ2,β,c,λ)/2](k+(t)).
In order to minimize the exponent in the inequality of (31), let
(32)κ=-ΛK(Λ,σ1,σ2,β,c,λ)>0.
Then, we can obtain that
(33)ℙx0{sups≥t|x(s)|≥R}≤(|x0|R)-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)=(|x0|R)-Λ/K(Λ,σ1,σ2,β,c,λ)×exp{(Λ2/2-2Λ∥υ∥)/K(Λ,σ1,σ2,β,c,λ)}1-exp{-Λ2/2K(Λ,σ1,σ2,β,c,λ)}×exp{-Λ2t2K(Λ,σ1,σ2,β,c,λ)}.
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, Dt(ω)=ℜ2 for all t>0. Moreover, (φ(t,θ-tω,x)) is ℱt0 measurable.Proof .
We can rewrite system (3) as follows:
(34)dx1=x2dt,dx2=[αx1+βx2-x23-x12x2]dt+[σ1x1+σ2x2+σ3]dW(t)+[γ1x1+γ1x2+γ1]dN(t).
Choose a function(35)V(x1,x2)=x12+x22.
Applying Itô's formula to (35), we have
(36)V(x1,x2)=x102+x202+∫0t‍[2(1+α)x1x2+σ12x12+(2β+σ22)x22+σ32-x24-x12x22]ds+2∫0t‍[σ1x1x2+σ2x22+σ3x2]dW(s)+∫0t‍[2γ1x1x2+γ12x12+(2γ2+γ22)x22+2γ3x2+γ32]dN(s).
Using the Cauchy-Bunyakovsky-Schwarz inequality, then(37)2(1+α)x1x2+σ12x12+(2β+σ22)x22≤[|(1+α)|+σ12]x12+[|(1+α)|+|(2β+σ22)|]x22×([|1+α|+σ12]∨[|1+α|+|2β+σ22|])V(x1,x2),σ1x1x2+σ2x22+σ3x2≤|σ1|2x12+(|σ1|2+|σ2|+|σ3|2)x22+|σ3|2≤(|σ1|2+|σ2|+|σ3|2)V(x1,x2)+|σ3|2,2γ1x1x2+γ12x12+(2γ2+γ22)x22+2γ3x2+γ32≤([|γ1|+γ12]∨[2|γ2|+γ22+|γ3|])×V(x1,x2)+|γ3|+γ32.
From (36), we can obtain
(38)V(x1,x2)≤V(t0)+∫0t‍([|(1+α)|+σ12]∨[|(1+α)|+|(2β+σ22)|])×V(x1,x2)ds+2∫0t‍[(|σ1|2+|σ2|+|σ3|2)V(x1,x2)+|σ3|2]dW(s)+∫0t‍{([|γ1|+γ12]∨[2|γ2|+γ22+|γ3|])V(x1,x2)+|γ3|+γ32}dN(s).
Suppose thatYt satisfies the following equation with jumps:
(39)Yt=V(t0)+∫0t‍([|(1+α)|+σ12]∨[|(1+α)|+|(2β+σ22)|])Ysds+2∫0t‍[(|σ1|2+|σ2|+|σ3|2)Ys+|σ3|2]dW(s)+∫0t‍{([|γ1|+γ12]∨[2|γ2|+γ22+|γ3|])Ys+|γ3|+γ32}dN(s);
then Yt 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 Theorem3 is satisfied. Then the random dynamical system (φ(t,ω,x)) generated by (3) has the following additional properties:(i)
Dt(θ-t)=ℜ2 for all t≥0;(ii)
Rt(θ-t)↓E(ω) as t↑∞.Proof.
Since allt∈ℜ,φ(t,ω):Dt(ω)→Rt(ω) is (local) diffeomorphism of the random(open) domain Dt(ω)⊂R2 into the random range Rt(ω)=φ(t,ω)Dt(ω)⊂ℜ2 where Dt(ω)↓ as t↑∞, and φ(0,ω)=Idℜ2, E(ω)=⋂t∈ℜ‍Dt(ω)={x:φ(t,ω,x)exists  for  all  times}. D(ω) is the set of all initial values x whose orbits never explode. It may happen that D(ω)=∅.
Fort≥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 Rt(θ-tω)=ℜ2 for t≤0 and Rt(θtω)⊂Rs(θsω) for 0≤s≤t, respectively. Since Rt(θ-tω)=D-t(ω), the definition of E(ω) yields that (ii) is true.Lemma 5.
Supposek>-1. If a-c2/2+λlog(1+k)<0, then the stochastic differential equation with the Poisson processes
(40)dXt=(aXt+b)dt+(cXt+h)dW(t)+(kXt+f)dN(t),X0=x,
has the unique invariant measure which is a Dirac measure supported by
(41)ξ(t)=[c∫-∞0‍e-(a-c2/2)t-cWs-LNtds+h∫-∞0‍e-(a-c2/2)t-cWs-LNtdW(s)+f∫-∞0‍e-(a-c2/2)t-cWs-LNt)dNs],
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(42)φ(t,ω)x=e(a-c2/2)t+cWs+LNt[x+c∫0t‍e-(a-c2/2)t-cWs-LNtds+h∫0t‍e-(a-c2/2)t-cWs-LNtdW(s)+f∫0t‍e-(a-c2/2)t-cWs-LNt)dNs],
is the random dynamical system generated by (40). Since limt→-∞Nt(λt)=1 and a-c2/2+λL<0, we have the following integrals:
(43)∫-∞0‍e-(a-c2/2)t-cWs-LNtds<∞,∫-∞0‍e-(a-c2/2)t-cWs-LNtdW(s)<∞,∫-∞0‍e-(a-c2/2)t-cWs-LNtdNs<∞.
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
(44)ξ(t)=limt→∞φ(-t,ω)-1x=[c∫-∞0e-(a-c2/2)s-cWs-LNsds+h∫-∞0‍e-(a-c2/2)s-cWs-LNsdW(s)+f∫-∞0‍e-(a-c2/2)s-cWs-LNs)dNs],
for all x∈ℜ and ξ(t) attracts all points with exponential speed.Theorem 6.
Suppose the condition of Theorem3 is satisfied and the coefficient in (3) satisfies γ1+γ2>-1,σ3=γ3=0 and
(45)(1+α+β)+2λlog[1+(γ1+γ2)]<(σ1+σ2)22;
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
(46)𝒰={(D(ω))ω∈Ω:D(ω)⊂ℜ2isatemperedrandomset}.
Moreover, the random attractor A is measurable with respect to ℱ0-∞=σ{Nt,t≤0}.Proof.
Choose a function(47)V(x1,x2)=x12+2x1x2+x22.
Applying Itô's formula, we have
(48)V(x1,x2)=V(t0)+∫0t‍[x12x222(1+α+β)x1x2+(2α+σ12)x12+(2β+σ22+2)x22-x24-2x1x23-x13x2-x12x22]ds+2∫0t‍[σ1x12+(σ1+σ2)x1x2+σ2x22]dW(s)+∫0t‍[2(γ1+γ2)x1x2+(2γ1+γ12)x12+(2γ2+γ22)x22]dN(s).
This implies
(49)V(x1,x2)=V(t0)+∫0t‍[(1+α+β)V(x1,x2)+A(x1,x2)]ds+∫0t‍[(σ1+σ2)V(x1,x2)+B(x1,x2)]dW(s)+∫0t‍[(γ1+γ2)V(x1,x2)+C(x1,x2)]dN(s),
where
(50)A(x1,x2)=(α+σ12-1-β)x12+(β+σ22+1-α)x22-x24-2x1x23-x13x2-x12x22,B(x1,x2)=(σ1-σ2)(x1-x2),C(x1,x2)=(γ1+γ12-γ2)x12+(γ2+γ22-γ1)x22.
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
(51)V(x1,x2)≤V(t0)+∫0t‍[(1+α+β)V(x1,x2)+K]ds+∫0t‍[(σ1+σ2)V(x1,x2)+K]dW(s)+∫0t‍[(γ1+γ2)V(x1,x2)+K]dN(s).
Denote φ(t,ω,(x1,x2)) and ψ(t,ω,(x1,x2)); then random dynamical systems are generated by (3) and
(52)Zt≤V(t0)+∫0t‍[(1+α+β)Zs+K]ds+∫0t‍[(σ1+σ2)Zs+K]dW(s)+∫0t‍[(γ1+γ2)Zs+K]dN(s),
respectively. The random dynamical systems generated by (52) are
(53)ψ(t,ω)x=e[(1+α+β)-(σ1+σ2)2/2]t+(σ1+σ2)Wt+log(1+γ1+γ2)Nt×[x+K∫0t‍e-[(1+α+β)-(σ1+σ2)2/2]s-(σ1+σ2)Ws-log(1+γ1+γ2)Nsds+K×∫0t‍e-[(1+α+β)-(σ1+σ2)2/2]s-(σ1+σ2)Ws-log(1+γ1+γ2)NsdW(s)+K×∫0te[-(1+α+β)-(σ1+σ2)2/2]s-(σ1+σ2)Ws-log(1+γ1+γ2)NsdNs].
Thus, from (51), we have
(54)V(φ(t,ω),(x1,x2))≤V(ψ(t,ω),(x1,x2)).
By Lemma5 and (45), we obtain that ψ(t,ω) has the unique invariant measure which is a Dirac measure supported by
(55)r(ω)=Ke[(1+α+β)-(σ1+σ2)2/2]t+(σ1+σ2)Wt+log(1+γ1+γ2)Nt×{∫0t‍e-[(1+α+β)-(σ1+σ2)2/2]s-(σ1+σ2)Ws-log(1+γ1+γ2)NsdW(s)+∫0t‍e[-(1+α+β)-(σ1+σ2)2/2]s-(σ1+σ2)Ws-log(1+γ1+γ2)NsdNs}>0.
It is easy to prove thatψ(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]tx(θ-tω)→0 for some a>0. Hence, we may the universe of set
(56)𝒰¯={I(ω)⊂ℜ+is  a  tempered  random  set};
that is, the random variable η(ω)=supx∈I(ω)x satisfies limt→∞log+[η(θtω)/t]=0 for any I=𝒰¯. From the results of Arnold in [7, Proposition 4.17], it yields limt→∞e-ɛtη(θ-tω)=0 for any ɛ>0.
It is clear that𝒰¯ is closed under inclusion and Ω×{x}∈𝒰¯ for all x∈ℜ+.
Next, we will prove that the random set[0,(1+ɛ)r] is forward invariant and absorbing for ψ(t,ω) with respect to the universe 𝒰¯ for any ɛ>0.
In fact, the absorbing property follows from the definition of𝒰¯ and Schenk-Hoppé [9, Theorem 4.1]. We only prove that ψ(t,ω)[0,(1+ɛ)r(ω)]⊂[0,(1+ɛ)r(θtω)] 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+ɛ)r(ω)≤(1+ɛ)r(θtω), it is equivalent that prove
(57)ψ(t,θ-tω)(1+ɛ)r(θ-tω)≤(1+ɛ)r(ω).
From (53), one can obtain
(58)ψ(t,θ-tω)(1+ɛ)r(θ-tω)=r(ω)+ɛK×{∫-∞-t‍e-[(1+α+β)-(σ1+σ2)2/2]s-(σ1+σ2)Ws-log(1+γ1+γ2)NsdW(s)+∫-∞-t‍e[-(1+α+β)-(σ1+σ2)2/2]s-(σ1+σ2)Ws-log(1+γ1+γ2)NsdNs∫-∞-t‍}.
It implies that (57) is true.
Since(59)r(θ-tω)=Ke-[(1+α+β)-(σ1+σ2)2/2]t+(σ1+σ2)W-t+log(1+γ1+γ2)N-t×[∫-∞-te-[(1+α+β)-(σ1+σ2)2/2]s-(σ1+σ2)Ws-log(1+γ1+γ2)NsdW(s)+∫-∞-t‍e[-(1+α+β)-(σ1+σ2)2/2]s-(σ1+σ2)Ws-log(1+γ1+γ2)NsdNs∫-∞-t‍]≤r(ω),
one has
(60)e-ϵtr(θ-tω)⟶0,t⟶∞,
for any ϵ>0. It means that [0,(1+ɛ)r(ω)]∈𝒰¯ for any ɛ>0.
For anyɛ>0, define the subset B(ω) of ℜ2 by
(61)B(ω)=V-1([0,(1+ɛ)]r(ω)).
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 ofB(ω) may be implied by Lemma 5.7 of Schenk-Hoppé [9]. Using the surjectivity and nonnegativity of V and (54), one has
(62)ψ(t,θ-tω)[0,(1+ɛ)r(θ-tω)]⊂[0,(1+ɛ)r(ω)]⟺ψ(t,θ-tω)V(B(θ-tω))⊂V(B(ω))⟹V(ψ(t,θ-tω)B(θ-tω))⊂V(B(ω)),
for all t≥0. It implies
(63)ψ(t,θ-tω)B(θ-tω)⊂V-1(V(ψ(t,θ-tω)B(θ-tω)))⊂V-1(V(B(ω))=B(ω);
that is, B is forward invariant.
For anyD∈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),
(64)V(φ(t,θ-tω)D(θ-tω))⊂ψ(t,θ-tω)V(D(θ-tω))⊂[0,(1+ɛ)r(ω)]=V(B(ω)),
by (59) and the set [0,(1+ɛ)r(ω)] absorbing any set in 𝒰¯. It shows the absorption of any set in 𝒰.
Finally, we prove the existence of a neighborhood ofB 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)∈ℱ-∞0, we have that the attractor A is ℱ-∞0 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 thatσ3=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
(65)dx=(01α-σ12β-σ22)xdt+(0-x12x2-x23)dt+(00σ1σ2)x∘dW(t).
The linearized RDS Dφ is generated by the linearization of (65), namely, the white noise case by
(66)υ˙=(01α-σ12β-σ22)υ+(00-2x1x2-3x22)υ+(00σ1σ2)υ∘W˙(t).
For any invariant measure ν, the trace formula gives λ1(ν)+λ2(ν)=β-σ2/2-3γ𝔼νx12. 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 systemσ1=σ2=0 at x=0,
(67)υ˙=(01α-σ12β-σ22)υ,
which are
(68)λ1,2=β-σ2/22±(β-σ2/2)24+α-σ12.
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:
(69)λ1,2=β-σ2/22±iωd,ωd:=-(β-σ2/2)24-α+σ12,
where ωd represents the  “damped eigenfrequency” of the linear system:
(70)X¨-βX˙-αX=0.
It is a general observation that noise splits deterministic multiplicities of eigenvalues. Forβ2<-4α (which we assume) and σ1=σ2=0, the deterministic linear system has two complex-conjugate eigenvalues β/2±ωd, 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
(71)υ˙=(01α-σ12β-σ22)υ+(00σ1σ2)υ∘W˙(t),
has two different simple Lyapunov exponents λi(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:
(72)λ1,2=β-σ2/22±σ12+σ228ωd2S(2ωd)+O(max{σ14+σ24}),
where S(·) is the spectral density of W˙(t),  for the white noise case where S(u)≡1. Furthermore, the rotation number expands as
(73)ρ(μ,α)=ωd-c(σ12+σ22)8ωd2T(2ωd)+O(max{σ14+σ24}),
where T(·) is the sine spectral density of W˙(t),c>0.
Consequently, at the deterministic Hopf bifurcation pointβ=0, we have for the white noise case, neglecting O(max{σ14+σ24}) term,
(74)λ2(σ1,σ2,0)=-σ12+σ228ωd2S(2ωd)<0<λ1(σ1,σ2,0)=σ12+σ228ωd2S(2ωd);
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 βD1=((σ12+σ22)/4α)S(2ωd)+O(max{σ14+σ24})<0 and the second Lyapunov exponent changes sign at βD2=-((σ12+σ22)/4α)S(2ωd)+O(max{σ14+σ24})>0.
Forβ-σ2/2<βD1,δ0 is stable, it is the unique invariant measure, and Aβ={0} is the attractor of the RDS φβ in ℝ2.
We have a first bifurcation fromδ0 at βD1=β-σ2/2<0 of a stable ergodic measure ν1(σ1,σ2,β)ω=(1/2)(δx1(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∈(βD1,βD2). As shown before, βP=0∈(βD1,βD2), ν1 undergoes a P-bifurcation, and δ0 and ν1 are both Markov measures. Hence, ρ=𝔼ν1 solves the Fokker-Planck equation. The attractor Aα 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 Aβ0 (in the universe of simply connected tempered sets) consists of the two-point set suppν1.
Atβ-σ2/2=βD2, we have a second bifurcation from δ0 of a measure ν2(σ1,σ2,β)ω=(1/2)(δx2(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.
Forβ-σ2/2>βD2, 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 Aβ in ℝ2. It carries all three invariant measures. In the punctured plane ℝ2∖{0}, however, the attractor Aβ0 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 Theorem3 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 attractorAsupports all invariant measures;(ii)
the unstable set of any invariant measureμis contained in random attractor; that is,Mμ+(ω)⊂A(ω);(iii)
there exists an invariant Markov measureμsupported by∂A(ω); that this,μωisℱ-∞0measurable.Proof.
It is the same as the proof in Proposition 7.5 of Schenk-Hoppé [9].Proposition 9.
Suppose the condition of Theorem3 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)(75)dx1=x2dt,dx2=[-x1+βx2]dt+[σ1x1+σ2x2]dW(t)+[γ1x1+γ2x2]dN(t),
are bounded by
(76)[β+max{σ12,σ22}1+|β|2]-[((2σ2+σ1)+|β|(σ1+σ2/2))/(1+|β|/2)]22+λlog(1+2+γ12+γ1|β|+γ22+2γ2+|β|γ2/21+|β|/2)≤Λ2≤Λ1≤[β+max{σ12,σ22}1-|β|/2]-[((2σ2+σ1)+|β|(σ1+σ2/2))/(1-|β|/2)]22+λlog(1+2+γ12+γ1|β|+γ22+2γ2+|β|γ2/21-|β|/2).
As follows.Proof.
Define the Lyapunov function onℜ2(77)V(x1,x2)=x12-βx1x2+x22.
Applying Itô's formula to (77), we have
(78)dV(x1,x2)=[βx22+βx12-β2x1x2+σ12x12+σ22x22]dt+[2σ2x22+(2σ1-βσ2)x1x2-βσ1x12]dW(t)+[(γ22+2γ2)x22+(2γ1-γ2β)x1x2+(γ12-γ1β)x12]dN(t).
Since
(79)2σ2x22+2σ1x1x2≤(2σ2+σ1)(x12+x22),-βσ2x1x2-βσ1x12≤|β|(σ1+σ22)(x12+x22),2x1x2+(γ12-γ1β)x12≤(1+γ12+γ1|β|)(x12+x22),(γ22+2γ2)x22-βγ2x1x2≤(1+γ22+2γ2+|β|γ22)(x12+x22),(1-|β|2)(x12+x22)≤V(x1,x2)≤(1+|β|2)(x12+x22).
From (78) and (79), we have
(80)dV(x1,x2)≤[β+max{σ12,σ22}1-|β|/2]V(x1,x2)dt+[(2σ2+σ1)+|β|(σ1+σ2/2)1-|β|/2]V(x1,x2)dW(t)+[2+γ12+γ1|β|+γ22+2γ2+|β|γ2/21-|β|/2]×V(x1,x2)dN(t),dV(x1,x2)≥[β+max{σ12,σ22}1+(|β|/2)]V(x1,x2)dt+[(2σ2+σ1)+|β|(σ1+σ2/2)1+|β|/2]×V(x1,x2)dW(t)+[2+γ12+γ1|β|+γ22+2γ2+|β|γ2/21+|β|/2]×V(x1,x2)dN(t).
By Doléans-Dade's exponent formula, solving the corresponding linear differential equation, we have(81)V(x1,x2)={≤V(t0)exp{at+bW(t)+log(1+c)N(t)},≥V(t0)exp{kt+hW(t)+log(1+r)N(t)},
where
(82)a=[β+max{σ12,σ22}1-|β|/2]-[((2σ2+σ1)-β(σ1+σ2/2))/(1-|β|/2)]22,b=(2σ2+σ1)-β(σ1+σ2/2)1-|β|/2,c=2+γ12-γ1β+γ22+2γ2-βγ2/21-|β|/2,k=[β+max{σ12,σ22}1+|β|/2]-[((2σ2+σ1)-β(σ1+σ2/2))/(1+|β|/2)]22,k=(2σ2+σ1)-β(σ1+σ2/2)1+|β|/2,r=2+γ12-γ1β+γ22+2γ2-βγ2/21+|β|/2.
Then, we can obtain
(83)limt→∞log(V(x1,x2))t={≤limt→∞log(at+bW(t)+log(1+c)N(t))t=a+λlog(1+c),≥limt→∞log(kt+hW(t)+log(1+r)N(t))t=k+λlog(1+r).
Dividing by t and letting t→∞, one obtains the claimed estimate on the Lyapunov exponents.Theorem 10.
Suppose the condition of Theorem3 is satisfied and the coefficient in (3) is satisfied and α=-1 and β∈(-2,0) such that
(84)Λ=[β+max{σ12,σ22}1-|β|/2]-[((2σ2+σ1)+|β|(σ1+σ2/2))/(1-|β|/2)]22+λlog(1+2+γ12+γ1|β|+γ22+2γ2+|β|γ2/21-|β|/2)<0.
Then, the random dynamical system generated by the stochastic Rayleigh-van der Pol equation (3) possesses the unique tempered random attractor
(85)A(ω)={0},
with domain of attractor 𝒟(A) containing the universe of 𝒞l(𝒰) given by
(86)𝒰={(D(ω))ω∈Ω:D(ω)⊂ℜ2isatemperedrandomset},
where 𝒞l(𝒰) is the complete universe of 𝒰 under inclusion.
In particular, for anyx∈ℜ2, the solutions φ(t,θ-tω)x and φ(t,ω)x tend to zero exponentially fast as t→∞.Proof.
By Proposition9 and assumption (91), the random dynamical system generated by (3) is stable. Let us define the nonnegative function on ℜ2 as follows:
(87)V(x1,x2)=-βx14+x12-βx1x2+x22.
Applying Itô's formula to (87), we can obtain
(88)dV(x1,x2)=[βx22+βx12-β2x1x2+σ12x12+σ22x22+βx1x23-2x24-2x12x22]dt+[2σ2x22+(2σ1-βσ2)x1x2-βσ1x12]dW(t)+[(γ22+2γ2)x22+(2γ1-γ2β)x1x2+(γ12-γ1β)x12]dN(t).
From (79) and (88), we can obtain
(89)dV(x1,x2)≤{[β+max{σ12,σ22}1-|β|/2]V(x1,x2)+R(β)}dt+[(2σ2+σ1)+|β|(σ1+σ2/2)1-|β|/2]V(x1,x2)dW(t)+[2+γ12+γ1|β|+γ22+2γ2+|β|γ2/21-|β|/2]×V(x1,x2)dN(t),
where R(x1,x2)=βx1x23-2x24-2x12x22 is uniformly bounded by a positive constant R(β).
By Lemma5, we can obtain that the random dynamical system generated by the stochastic differential equation
(90)dXt≤{[β+max{σ12,σ22}1-|β|/2]Xt+R(β)}dt+[(2σ2+σ1)+|β|(σ1+σ2/2)1-|β|/2]XtdW(t)+[2+γ12+γ1|β|+γ22+2γ2+|β|γ2/21-|β|/2]XtdN(t),
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 anyx∈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
ℜ
2
(
ω
) is immediate from Theorem 6 and the definition of 𝒰.Theorem 11.
Suppose the condition of Theorem3 is satisfied and the coefficient in (3) is satisfied and α=-1 and β∈(0,2) such that
(91)Λ=[β+max{σ12,σ22}1+|β|/2]-[((2σ2+σ1)+|β|(σ1+σ2/2))/(1+|β|/2)]22+λlog(1+2+γ12+γ1|β|+γ22+2γ2+|β|γ2/21+|β|/2)>0.
Then, the random dynamical system generated by the stochastic Rayleigh-van der Pol equations (1) possesses the unique tempered random attractor
(92)A(ω)=ℜ2∖{0},
with domain of attractor 𝒟(A) containing the universe of 𝒞l(𝒰) given by
(93)𝒰={(∥x∥+1∥x∥)allrandomset(D(ω))ω∈Ω⊂ℜ2∖{0}:supx∈D(ω)(∥x∥+1∥x∥)istempered}.
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 Theorem10.



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*Source: 432704-2013-09-29.xml*