1 Introduction
###### Abstract

This work is concerned with the dynamics of a class of slow-fast stochastic dynamical systems with non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, eliminating the fast variables to reduce the dimension of these coupled dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps understand long time dynamics. The approximation of slow manifolds with error estimate in distribution are also considered.

Slow manifolds for stochastic systems

with non-Gaussian stable Lévy noise111This work was partly supported by the NSFC grants 11301197, 11301403, 11371367, 11271290 and 0118011074.

Shenglan Yuan, Jianyu Hu, Xianming Liu,Jinqiao Duan

Center for Mathematical Sciences,

School of Mathematics and Statistics,

Hubei Key Laboratory of Engineering Modeling and Scientific Computing,

Huazhong University of Sciences and Technology, Wuhan 430074, China

Department of Applied Mathematics

Illinois Institute of Technology, Chicago, IL 60616, USA

Keywords. Stochastic differential equations, random dynamical systems, slow manifolds, critical manifolds, dimension reduction.

## 1 Introduction

Stochastic effects are ubiquitous in complex systems in science and engineering [17, 25, 40]. Although random mechanisms may appear to be very small or very fast, their long time impacts on the system evolution may be delicate or even profound, which has been observed in, for example, stochastic bifurcation, stochastic optimal control, stochastic resonance and noise-induced pattern formation [2, 26, 37]. Mathematical modeling of complex systems under uncertainty often leads to stochastic differential equations (SDEs) [18, 24, 28]. Fluctuations appeared in the SDEs are often non-Gaussian (e.g., Lévy motion) rather than Gaussian (e.g., Brownian motion); see Schilling [5, 34].

We consider the slow-fast stochastic dynamical system where the fast dynamic is driven by -stable noise, see [15, 27]. In particular, we study

 {dxε=Sxεdt+g1(xε,yε)dt,dyε=1εFyεdt+1εg2(xε,yε)dt+σε−1αdLαt, (1.1)

where , is a small positive parameter measuring slow and fast time scale separation such that in a formal sense

 |dxεdt|≪|dyεdt|,

where we denote by the Euclidean norm. The matrix with all eigenvalues with non-negative real part, is a matrix whose eigenvalues have negative real part. Nonlinearities are Lipschitz continuous functions with . is a two-sided -valued -stable Lévy process on a probability space , where is the index of stability [1, 10]. The strength of noise in the fast equation is chosen to be to balance the stochastic force and deterministic force. is the intensity of noise.

Invariant manifolds are geometric structures in state space that are useful in investigating the dynamical behaviors of stochastic systems; see [8, 9, 12, 13]. A slow manifold is a special invariant manifold of a slow-fast system, where the fast variable is represented by the slow variable and the scale parameter is small. Moreover, it exponentially attracts other orbits. A critical manifold of a slow-fast system is the slow manifold corresponding to the zero scale parameter [16]. The theory of slow manifolds and critical manifolds provides us with a powerful tool for analyzing geometric structures of slow-fast stochastic dynamical systems, and reducing the dimension of those systems.

For a system like (1.1) based on Brownian noise (), the existence of the slow manifold and its approximation has been extensively studied [6, 11, 14, 35, 38]. The dynamics of individual sample solution paths have also been quantified; see [4, 20, 39]. Moreover, Ren and Duan [30, 31] provided a numerical simulation for the slow manifold and established its parameter estimation. The study of the dynamics generated by SDEs under non-Gaussian Lévy noise is still in its infancy, but some interesting works are emerging [10, 19, 22].

The main goal of this paper it to investigate the slow manifold of dynamical system (1.1) driven by -stable Lévy process with in finite dimensional setting, and examine its approximation and structure.

We first introduce a random transformation based on the generalised Ornstein-Uhlenbeck process, such that a solution the system of SDEs (1.1) with -stable Lévy noise can be represented as a transformed solution of random differential equations (RDEs) and vice versa. Then we prove that, for , the slow manifold with an exponential tracking property can be constructed as fixed point of the projected RDEs by using the Lyapunov-Perron method [3]. Thus as a consequence, with the inverse conversion, we can obtain the slow manifold for the original SDE system. Subsequently we convert the above RDEs to new RDEs by taking the time scaling . After that we use the Lyapunov-Perron method once again to establish the existence of the slow manifold for new RDE system, and denote as the critical manifold with zero scale parameter in particular. In addition, we show that is same as in distribution, and the distribution of converges to the distribution of , as tends to zero. Finally, we derive an asymptotic approximation for the slow manifold in distribution. Moreover, as part of ongoing studies, we try to study mean residence time on slow manifold, and generalise these results to consider system (1.1) in Hilbert spaces to study infinite dimensional dynamics.

This paper is organized as follows. In Section 2, we recall some basic concepts in random dynamical systems, and construct metric dynamical systems driven by Lévy processes with two-sided time. In Section 3, we recall random invariant manifolds and introduce hypotheses for the slow-fast system. In Section 4, we show the existence of slow manifold (Theorem 1), and measure the rate of slow manifold attract other dynamical orbits (Theorem 2). In Section 5, we prove that as the scale parameter tends to zero, the slow manifold converges to the critical manifold in distribution (Theorem 5). In Section 6, we present numerical results using examples from mathematical biology to corroborate our analytical results.

## 2 Random Dynamical Systems

We are going to introduce the main tools we need to find inertial manifolds for systems of stochastic differential equations driven by -stable Lévy noise. These tools stem from the theory of random dynamical systems; see Arnold [2].

An appropriate model for noise is a metric dynamical system , which consists of a probability space and a flow :

 θ: R×Ω→Ω, (t,ω)↦θtω; θ0=idΩ, θs+t=θs∘θt=:θsθt.

The flow is jointly measurable. All are measurably invertible with . In addition, the probability measure is invariant (ergodic) with respect to the mappings .

For example, the Lévy process with two side time represents a metric dynamical system. Let with a.s. be a Lévy process with values in defined on the canonical probability space , in which endowed with the Borel -algebra . We can construct the corresponding two-sided Lévy process defined on , see Kümmel [19, p29]. Since the paths of a Lévy process are càdlàg; see [1, Theorem 2.1.8]. We can define two-sided Lévy process on the space instead of , where is the space of càdlàg functions starting at given by

 D0={ω: for ∀t∈R, lims↑tω(s)=ω(t−), lims↓tω(s)=ω(t) exist and ω(0)=0}.

This space equipped with Skorokhod’s -topology generated by the metric is a Polish space. For functions , is given by

 dR(ω1,ω2)=inf{ε>0:|ω1(t)−ω2(λt)|≤ε, ∣∣lnarctan(λt)−arctan(λs)arctan(t)−arctan(s)∣∣≤ε for every  t,s∈R and some λ∈ΛR},

where

 ΛR={λ:R→R; λ is injective increasing, limt→−∞λ(t)=−∞, limt→∞λ(t)=∞}.

is the associated Borel -algebra , and is a separable metric space. The probability measure generated by for each . The flow is given by

 θ:R×D0→D0, θtω(⋅)↦ω(⋅+t)−ω(t),

which is a Carathéodory function. It follows that is jointly measurable. Moreover, satisfies and . Thus is a metric dynamical system generated by Lévy process with two-side time. Note that the probability measure is ergodic with respect to the flow .

In the above we define metric dynamical system first, which will step in in the complete definition of random dynamical system, that is strongly motivated by the measuability property combined with the cocycle property.

A random dynamical system taking values in the measurable space over a metric dynamical system with time space is given by a mapping

 φ:R×Ω×H→H,

that is jointly measurable and satisfies the cocycle property:

 φ(0,ω,⋅)=idH= identity on H  for each ω∈Ω;φ(t+s,ω,⋅)=φ(t,θsω,φ(s,ω,⋅))  for each s,t∈R, ω∈Ω. (2.1)

For our application, in the sequel we suppose .

Note that if satisfies the cocycle property (2.1) for almost all (where the exceptional set can depend on ), then we say forms a crude cocycle instead of a perfect cocycle. In this case, to get a random dynamical system, sometimes we can do a perfection of the crude cocycle, such that the cocycle property is valid for each and every , see Scheutzow [32].

We now recall some objects to help understand the dynamics of a random dynamical system.

A random variable with values in is called a stationary orbit (or random fixed point) for a random dynamical system if

 φ(t,ω,X(ω))=X(θtω)  for  t∈R, ω∈Ω.

Since the probability measure is invariant with respect to , the random variables have the same distribution as . Thus is a stationary process , and therefore a stationary solution to the stochastic differential equation generating the random dynamical system .

A family of nonempty closed sets is call a random set for a random dynamical system , if the mapping

 ω↦dist(z,M(ω)):=infz′∈M(ω)|z−z′|

is a random variable for every . Moreover is called an (positively) invariant set, if

 φ(t,ω,M(ω))⊆M(θtω)  for  t∈R, ω∈Ω. (2.2)

Let

 h:Ω×Rn1→Rn2, (ω,x)↦h(ω,x)

be a function such that for all , is Lipschitz continuous, and for any , is a random variable. We define

 M(ω)={(x,h(ω,x)):x∈Rn1}

such that can be represented as a graph of . It can be shown [35, Lemma 2.1] that is a random set.

If also satisfy (2.2), is called a Lipschitz continuous invariant manifold. Furthermore, is said to have an exponential tracking property if for all , there exists an such that,

 |φ(t,ω,z)−φ(t,ω,z′)|≤C(z,z′,ω)e−ct|z−z′|, t≥0, z∈Rn,

where is a positive random variable depending on and , while is a positive constant. Then is called a random slow manifold with respect to the random dynamical system .

Let and be two random dynamical systems. Then and are called conjugated, if there is a random mapping , such that for all , is a Carathéodory function, for every and , is homeomorphic, and

 ~φ(t,ω,z)=T(θtω,φ(t,ω,T−1(ω,z))), z∈Rn, (2.3)

where is the corresponding inverse mapping of . Note that provides a random transformation form to that may be simpler to treat. If is a invariant set for the random dynamical system , we define

 M(⋅):=T−1(⋅,~M(.)).

From the properties of , is also invariant set with respect to .

## 3 Slow-Fast Dynamical Systems

The theory of invariant manifolds and slow manifolds of random dynamical system are essential for the study of the solution orbits, and we can use it to simplify dynamical systems by reducing an random dynamical system on a lower-dimensional manifold.

For the slow-fast system (1.1) described by stochastic differential equations with -stable Lévy noise, the state space for slow variables is , the state space for fast variables is . To construct the slow manifolds of system (1.1), we introduce the following hypotheses.

Concerning the linear part of (1.1), we suppose

. There are constants and such that

 |eStx| ≤eγst|x|, t≤0, for~{}all x∈Rn1, |eFty| ≤eγft|y|, t≥0, for~{}all y∈Rn2.

With respect to the nonlinear parts of system (1.1), we assume

. There exists a constant such that for all ,

 |gi(x1,y1)−gi(x2,y2)|≤K(|x1−x2|+|y1−y2|),

which implies that are continuous and thus measurable with respect to all variables. If is locally Lipshitz, but the corresponding deterministic system has a bounded absorbing set. By cutting off to zero outside a ball containing the absorbing set, the modified system has globally Lipschitz drift [21].

For the proof of the existence of a random invariant manifold parametrized by , we have to assume that the following spectral gap condition.

. The decay rate of is larger than the Lipschitz constant of the nonlinear parts in system (1.1), i.e. .

###### Lemma 1.

Under hypothesis , the following linear stochastic differential equations

 dη1ε(t) =1εFη1εdt+ε−1αdLαt, η1ε(0)=η1ε0, (3.1) dξ(t) =Fξdt+dLαt, ξ(0)=ξ0, (3.2)

have càdlàg stationary solutions and defined on -invariant set of full measure, through the random variables

 η1ε(ω)=ε−1α∫0−∞e−FsεdLαs(ω), ξ(ω)=∫0−∞e−FsdLαs(ω), (3.3)

respectively. Moreover, they generate random dynamical systems.

###### Proof.

The SDE (3.2) has unique càdlàg solution

 φ(t,w,ξ0)=eFtξ0+∫t0eF(t−s)dLαs(ω), (3.4)

for details see [1, 19, 33]. It follows from (3.3) and (3.4) that

 φ(t,w,ξ(ω)) =eFtξ(ω)+∫t0eF(t−s)dLαs(ω) =eFt∫0−∞e−FsdLαs(ω)+∫t0eF(t−s)dLαs(ω) =∫t−∞eF(t−s)dLαs(ω).

By (3.3), we also see that

 ξ(θtω) =∫0−∞e−FsdLαs(θtω)=∫0−∞e−Fsd(Lαt+s(ω)−Lαt(ω)) =∫0−∞e−FsdLαt+s(ω)=∫t−∞eF(t−s)dLαs(ω).

Hence is a stationary orbit for (3.2). Then we have

 ξ(θt+sω) =∫t+s−∞eF(t+s−r)dLαr(ω)=eF(t+s)∫t−∞e−F(r+s)dLαr+s(ω) =eF(t+s)∫t−∞e−F(r+s)d(Lαr+s(ω)−Lαs(ω))=∫t−∞eF(t−r)dLαr(θsω) =ξ(θtθsω),

which implies generate a random dynamical system. Analogously we obtain the SDE (3.1) whose unique solution is the generalised Ornstein-Uhlenbeck process

 η1ε(θtω)=ε−1α∫t−∞eF(t−s)εdLαs.

###### Remark 1.

Since -stable process satisfying , and are well-defined sationary semimartingales; see [19, Remark 4.6].

###### Lemma 2.

The process has the same distribution as the process , where and are defined in Lemma 1.

###### Proof.

From -stable process are self-similar with Hurst index , i.e.,

 Lαct\lx@stackreld=c1αLαt,

where denotes equivalence (coincidence) in distribution, we have

 η1ε(θtω) =ε−1α∫t−∞eF(t−s)εdLαs(ω)=∫tε−∞eF(tε−u)ε−1αdLαεu(ω) \lx@stackreld=∫tε−∞eF(tε−u)dLαu(ω)=ξ(θtεω),

which proves that and have the same distribution. ∎

Now we will transform the slow-fast stochastic dynamical system (1.1) into a random dynamical system [29]. We introduce the random transformation

 (^xε^yε):=Tε(ω,xε,yε):=(xεyε−ση1ε(ω)). (3.5)

Then satisfies

 ⎧⎨⎩d^xε=S^xεdt+g1(^xε,^yε+ση1ε(θtω))dt,d^yε=1εF^yεdt+1εg2(^xε,^yε+ση1ε(θtω))dt. (3.6)

This can be seen by a formal differentiation of and .

For the sake of simplicity, we write Since the additional term doesn’t change the Lipschitz constant of the functions on the right hand side, the functions have the same Lipschitz constant as .

By hypotheses , system (3.6) can be solved for any contained in a -invariant set of full measure and for any initial condition such that the cocycle property is satisfied. Then the solution mapping

 (t,ω,(x0,y0))↦^φε(t,ω,(x0,y0))=(^xε(t,ω,(x0,y0)),^yε(t,ω,(x0,y0)))∈Rn, (3.7)

defines a random dynamical system. In fact, the mapping is -measurable, and for each , is a Carathéodory function.

In the following section we will show that system (3.6) generates a random dynamical system that has a random slow manifold for sufficiently small . Applying the ideas from the end of Section 2 with to the solution of (3.6), then system (1.1) also has a version satisfying the cocycle property. Clearly,

 φε(t,ω,(x0,y0)) =(Tε)−1(θtω,^φε(t,ω,Tε(ω,(x0,y0)))) =^φε(t,ω,(x0,y0))+(0,ση1ε(θtω)), t∈R, ω∈Ω (3.8)

is a random dynamical system generated by the original system (1.1). Hence, by the particular structure of if (3.6) has a slow manifold so has (1.1).

## 4 Random Slow Manifolds

To study system (3.6), for any , we introduce Banach spaces of functions with a geometrically weighted norm [36] as follows:

 Cs,−β = {φs,−:(−∞,0]→Rn1|φs,− is continuous andsupt∈(−∞,0]|e−βtφs,−t|<∞}, Cs,+β = {φs,+:[0,∞)→Rn1|φs,+ is continuous and supt∈[0,∞)|e−βtφs,+t|<∞}

with the norms

 ||φs,−||Cs,−β:=supt∈(−∞,0]|e−βtφs,−t| and ||φs,+||Cs,+β:=supt∈[0,∞)|e−βtφs,+t|

Analogously, we define Banach spaces and with the norms

 ||φf,−||Cf,−β:=supt∈(−∞,0]|e−βtφf,−t| and ||φf,+||Cf,+β:=supt∈[0,∞)|e−βtφf,+t|.

Let be the product space , . equipped with the norm

 ||(φs,±,φf,±)||C±β:=||φs,±||Cs,±β+||φf,±||Cf,±β

is a Banach space.

Letting satisfy . For the remainder of the paper, we take with sufficiently small.

###### Lemma 3.

Assume that hold. Then is in if and only if there exists a function with such that

 (^xε(t)^yε(t))=⎛⎝eStx0+∫t0eS(t−s)^g1(θεsω,^xε(s),^yε(s))ds1ε∫t−∞eF(t−s)ε^g2(θεsω,^xε(s),^yε(s))ds⎞⎠, (4.1)

where

 Mε(ω)={(x0,y0)∈Rn:^φε(⋅,ω,(x0,y0))∈C−β},  ω∈Ω. (4.2)
###### Proof.

If , by method of constant variation, system (3.6) is equivalent to the system of integral equations

 ⎧⎨⎩^xε(t)=eStx0+∫t0eS(t−s)^g1(θεsω,^xε(s),^yε(s))ds,^yε(t)=eF(t−u)ε^yε(u)+1ε∫tueF(t−s)ε^g2(θεsω,^xε(s),^yε(s))ds (4.3)

and . Moreover, by and , we have

 |eF(t−u)ε^yε(u)|≤eγf(t−u)ε|^yε(u)|≤supu∈(−∞,0]{e−βu|^yε(u)|}eγf(t−u)εeβu=||^yε||Cf,−βeγfεte(γfε−β)(−u)=||^yε||Cf,−βeγfεte−γ+γfεu→0, as u→−∞,

 ^yε(t)=1ε∫t−∞eF(t−s)ε^g2(θεsω,^xε(s),^yε(s))ds. (4.4)

Thus (4.3)-(4.4) imply that (4.1) holds.
Conversely, let satisfying (4.1), then is in by (4.2). Thus, we have finished the proof. ∎

###### Lemma 4.

Assume to be valid. Letting , if there exists an such that , the system (4.1) will have a unique solution in .

###### Proof.

For any , define two operators and satisfying

 Iεs(^φε)[t]=eStx0+∫t0eS(t−s)^g1(θεsω,^xε(s),^yε(s))ds,Iεf(^φε)[t]=1ε∫t−∞eF(t−s)ε^g2(θεsω,^xε(s),^yε(s))ds,

and the Lyapunov-Perron transform given by

 Iε(^φε)[t]=(Iεs(^φε)[t]Iεf(^φε)[t]). (4.5)

Under our assumptions above, maps into itself. Taking , then

 ||Iεs(^φε)||Cs,−β ≤Ksupt∈(−∞,0]{e−βt∫0teγs(t−s)(|^xε(s)|+|^yε(s)+ση1ε(θsω)|)ds} +supt∈(−∞,0]{e−βteγst|x0|} ≤Ksupt∈(−∞,