Slow manifolds for stochastic systems with non-Gaussian stable Lévy noise
Shenglan Yuan Jianyu Hu Xianming Liu Jinqiao Duan


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Slow manifolds for stochastic systems
with non-Gaussian stable Lévy noise[This work was partly supported
by the NSFC grants 11301197, 11301403, 11371367, 11271290 and 0118011074.]



 
Shenglan Yuan^a,b,c,[shenglanyuan@hust.edu.cn], Jianyu Hu^a,b,c,[hujianyu@hust.edu.cn],
Xianming Liu^b,c,[xmliu@hust.edu.cn],Jinqiao Duan^d,[ duan@iit.edu]
^aCenter for Mathematical Sciences,
 ^bSchool of Mathematics and Statistics,
  ^cHubei Key Laboratory of Engineering Modeling and Scientific Computing,
 Huazhong University of Sciences and Technology, Wuhan 430074, China
^dDepartment of Applied Mathematics
Illinois Institute of Technology, Chicago, IL 60616, USA






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.







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







   

 






    












§ INTRODUCTION 


Stochastic effects are ubiquitous in complex systems in science and engineering <cit.>. 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 <cit.>. Mathematical modeling of complex systems under uncertainty often leads to stochastic differential equations (SDEs) <cit.>. Fluctuations appeared in the SDEs are often non-Gaussian (e.g., Lévy motion) rather than Gaussian (e.g., Brownian motion); see Schilling <cit.>.

We consider the slow-fast stochastic dynamical system where the fast dynamic is driven by α-stable noise, see <cit.>.
In particular, we study

    {[                       dx^ε=Sx^ε dt+g_1(x^ε,y^ε)dt,; dy^ε=1/εFy^ε dt+1/ε
     g_2(x^ε,y^ε)dt+σε^-1/αdL_t^α, ].

where (x^ε,y^ε)∈ℝ^n_1×ℝ^n_2, ε is a small positive parameter measuring slow and fast time scale separation such that in a formal sense

    |dx^ε/dt|≪|dy^ε/dt|,

where we denote by |.| the Euclidean norm.
The n_1× n_1 matrix S with all eigenvalues with non-negative real part, F is a n_2× n_2
matrix whose eigenvalues have negative real part.
Nonlinearities g_i:ℝ^n_1×ℝ^n_2↦ℝ^n_i, i=1,2
are Lipschitz continuous functions with g_i(0,0)=0. {L_t^α: t∈ℝ} is a two-sided ℝ^n_2-valued α-stable
Lévy process on a probability space (Ω,ℱ,ℙ), where 1<α<2 is the index of stability <cit.>. The strength of noise in the fast equation is chosen to be ε^-1/α
to balance the stochastic force and deterministic force. σ>0 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 <cit.>. 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 ε 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 <cit.>. 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 (<ref>) based on Brownian noise (α=2), the existence of the slow manifold and its approximation has been extensively studied <cit.>. The dynamics of individual sample solution paths have also been quantified; see <cit.>. Moreover, Ren and Duan <cit.> 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 <cit.>.

The main goal of this paper it to investigate the slow manifold of dynamical system (<ref>) driven by α-stable
Lévy process with α∈(1,2) 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 (<ref>) with α-stable Lévy noise can be represented as a transformed solution of random differential equations (RDEs) and vice versa. Then we prove that, for 0<ε≪1, the slow manifold ℳ^ε with an exponential tracking property can be constructed as fixed point of the projected RDEs by using the Lyapunov-Perron method <cit.>. Thus as a consequence, with the inverse conversion, we can obtain the slow manifold M^ε for the original SDE system. Subsequently we convert the above RDEs to new RDEs by taking the time scaling t→ε t.  After that we use the Lyapunov-Perron method once again to establish the existence of the slow manifold ℳ̃^ε for new RDE system, and denote ℳ^0(ω) as the critical manifold with zero scale parameter in particular. In addition, we show that ℳ^ε is same as ℳ̃^ε in distribution, and the distribution of ℳ^ε(ω) converges to the distribution of ℳ^0(ω), as ε tends to zero.
Finally, we derive an asymptotic approximation for the slow manifold ℳ^ε 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 (<ref>) 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 <ref>), and measure the rate of slow manifold attract other dynamical orbits (Theorem <ref>). In Section 5, we prove that as the scale parameter tends to zero, the slow manifold converges to the critical manifold in distribution (Theorem <ref>). In Section 6, we present numerical results using examples from mathematical biology to corroborate our analytical results.





§ 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 α-stable Lévy noise. These tools stem from the theory of random
dynamical systems; see Arnold <cit.>.

An appropriate model for noise is a metric dynamical system Θ=(Ω,ℱ,ℙ,θ), which consists of
a probability space (Ω,ℱ,ℙ) and a flow θ={θ_t}_t∈ℝ:

    θ: ℝ×Ω→Ω, (t,ω)↦θ_tω;θ_0= id_Ω, θ_s+t=θ_s∘θ_t=:θ_sθ_t.

The flow θ is jointly ℬ(ℝ)⊗ℱ-ℱ measurable.
All θ_t are measurably invertible with θ_t^-1=θ_-t. In addition, the probability measure ℙ is
invariant (ergodic) with respect to the mappings {θ_t}_t∈ℝ.

 For example, the Lévy process with two side time represents a metric dynamical system. Let L=(L_t)_t≥0 with L_0=0 a.s. be a Lévy process with values in ℝ^n defined on the canonical probability space (E^ℝ^+,ℰ^ℝ^+,ℙ_ℝ^+), in which E=ℝ^n endowed with the Borel σ-algebra ℰ. We can construct the corresponding two-sided Lévy process L_t(ω):=ω(t), t∈ℝ defined on (E^ℝ,ℰ^ℝ,ℙ_ℝ), see Kümmel <cit.>. Since the paths of a Lévy process are càdlàg; see <cit.>. We can define two-sided Lévy process on the space
(Ω,ℱ,ℙ) instead of (E^ℝ,ℰ^ℝ,ℙ_ℝ), where Ω=𝒟_0(ℝ,ℝ^n) is the space of càdlàg functions starting at 0 given by

    𝒟_0={ω : for ∀ t∈ℝ, lim_s↑ tω(s)=ω(t-), lim_s↓ tω(s)=ω(t) exist and ω(0)=0  }.

This space equipped with Skorokhod's 𝒥_1-topology generated by the metric d_ℝ is a Polish space. For functions ω_1, ω_2∈𝒟_0, d_ℝ(ω_1, ω_2) is given by

    d_ℝ(ω_1,ω_2)=inf{ε>0:
         |ω_1(t)-ω_2(λ t)|≤ε, |lnarctan(λ t)-arctan(λ s)/arctan(t)-arctan(s)|≤ε.
         
    .for every  t, s∈ℝ and some λ∈Λ^ℝ},

where

    Λ^ℝ={λ :ℝ→ℝ; λ is injective increasing, lim_t→-∞λ(t)=-∞, lim_t→∞λ(t)=∞}.

ℱ is the associated Borel σ-algebra ℬ(𝒟_0)=ℰ^ℝ∩𝒟_0, and (𝒟_0,ℬ(𝒟_0)) is a separable metric space.
The probability measure ℙ generated by ℙ(𝒟_0∩ A):=ℙ_ℝ(A) for each A∈ℰ^ℝ. The flow θ is given by

    θ:ℝ×𝒟_0→𝒟_0, θ_tω(·)↦ω(·+t)-ω(t),

which is a Carathéodory function. It follows that θ is jointly measurable. Moreover, θ
satisfies θ_s(θ_tω(·))=ω(·+s+t)-ω(s)-(ω(s+t)-ω(s))=θ_s+tω(·) and θ_0ω(·)=ω(·). Thus (𝒟_0,ℬ(𝒟_0),ℙ,(θ_t)_t∈ℝ) is a metric dynamical system generated by Lévy process with two-side time. Note that the probability measure ℙ is ergodic with respect to the flow θ=(θ_t)_t∈ℝ.

In the above we define metric dynamical system first, which will step in  θ 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 (H,ℬ(H)) over a  metric dynamical system (Ω,ℱ,ℙ,(θ_t)_t∈ℝ) with time space ℝ is given by a mapping

    ϕ: ℝ×Ω× H → H,

 that is jointly ℬ(ℝ)⊗ℱ⊗ℬ(H)-ℬ(H) measurable and satisfies the cocycle property:

    [         ϕ(0,ω,·)= id_H= identity on H  for each ω∈Ω;; ϕ(t+s,ω,·)=ϕ(t,θ_sω,ϕ(s,ω,·))  for each s, t∈ℝ, ω∈Ω. ]

For our application, in the sequel we suppose H=ℝ^n=ℝ^n_1×ℝ^n_2.

Note that if ϕ satisfies the cocycle property (<ref>) for almost all ω∈Ω\𝒩_s,t (where the exceptional set 𝒩_s,t can depend on s,t), then we say ϕ 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 ω∈Ω, see Scheutzow <cit.>.

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

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

    ϕ(t,ω,X(ω))=X(θ_tω)  for  t∈ℝ, ω∈Ω.

Since the probability measure ℙ is
invariant with respect to {θ_t}_t∈ℝ, the random variables ω↦ X(θ_tω)
have the same distribution as ω↦ X(ω). Thus (t,ω)↦ X(θ_tω) is a stationary process , and therefore a stationary solution to the stochastic differential equation generating the random dynamical system ϕ.

A family of nonempty closed sets M={M(ω)⊂ℝ^n,ω∈Ω}
is call a random set for a random dynamical system φ, if the mapping

    ω↦ dist(z,M(ω)):=inf_z^'∈ M(ω)|z-z^'|

is a random variable for every z∈ℝ^n. Moreover M is called an (positively) invariant set, if

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


Let

    h:Ω×ℝ^n_1→ℝ^n_2, (ω,x)↦ h(ω,x)

be a function such that for all ω∈Ω, x↦ h(ω,x) is Lipschitz continuous, and for any x∈ℝ^n_1,  ω↦ h(ω,x) is a random variable. We define

    ℳ(ω)={(x,h(ω,x)):x∈ℝ^n_1}

such that ℳ={ℳ(ω)⊂ℝ^n,ω∈Ω} can be represented as a graph of h.
It can be shown <cit.> that ℳ is a random set.

If ℳ(ω), ω∈Ω also satisfy (<ref>), ℳ is called a Lipschitz continuous invariant manifold.
Furthermore, ℳ is said to have an exponential tracking property if for all ω∈Ω, there exists an z^'∈ℳ(ω) such that,

    |φ(t,ω,z)-φ(t,ω,z^')|≤ C(z,z^',ω)e^-ct|z-z^'|, t≥0, z∈ℝ^n,

where C is a positive random variable depending on z and z^', while c is a positive constant. Then ℳ is called a random slow manifold with respect to the random dynamical system φ.

Let φ and φ̃ be two random dynamical systems. Then φ and φ̃ are called conjugated, if there is a random mapping T:Ω×ℝ^n→ℝ^n, such that for all ω∈Ω, (t,z)↦ T(θ_tω,z) is a Carathéodory function, for every t∈ℝ^n and ω∈Ω, z↦ T(θ_tω,z) is homeomorphic, and

    φ̃(t,ω,z)=T(θ_tω,φ(t,ω,T^-1(ω,z))), z∈ℝ^n,

where T^-1:Ω×ℝ^n→ℝ^n is the corresponding inverse mapping of T. Note that T provides a random transformation form φ to φ̃ that may be simpler to treat. If M̃ is a invariant set for the random dynamical system φ̃, we define

    M(·):=T^-1(·,M̃(.)).

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






§ 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 (<ref>) described by stochastic differential equations
with α-stable Lévy noise, the state space for slow variables is ℝ^n_1,
the state space for fast variables is ℝ^n_2. To construct the slow manifolds of system (<ref>),
we introduce the following hypotheses.

Concerning the linear part of (<ref>), we suppose

(A_1). There are constants γ_s>0 and γ_f<0 such that

    |e^Stx|   ≤ e^γ_st|x|, t≤0,  x∈ℝ^n_1,
    
    |e^Fty|   ≤ e^γ_ft|y|, t≥0,  y∈ℝ^n_2.


With respect to the nonlinear parts of system (<ref>), we assume

(A_2). There exists a constant K>0
such that for all (x_i,y_i)∈ℝ^n, i=1,2,,

    |g_i(x_1,y_1)-g_i(x_2,y_2)|≤ K(|x_1-x_2|+|y_1-y_2|),

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

For the proof of the existence of a random invariant manifold parametrized by x∈ℝ^n_1, we have to assume that the following spectral gap condition.

(A_3).  The decay rate -γ_f of e^Ft is larger than the Lipschitz constant K of the
nonlinear parts in system (<ref>), i.e. K<-γ_f.

Under hypothesis (A_1), the following linear stochastic differential equations

    dη^1/ε(t)   =1/εFη^1/εdt+ε^-1/αdL_t^α, η^1/ε(0)=η_0^1/ε,
    
      dξ(t)   =Fξ dt+dL_t^α, ξ(0)=ξ_0,

have càdlàg stationary solutions η^1/ε(θ_tω) and ξ(θ_tω)
defined on
θ-invariant set Ω of full measure, through the random variables

    η^1/ε(ω)=ε^-1/α∫_-∞^0 e^-Fs/εdL_s^α(ω), ξ(ω)=∫_-∞^0e^-FsdL_s^α(ω),

respectively. Moreover, they generate random dynamical systems.


The SDE (<ref>) has unique càdlàg solution

    φ(t,w,ξ_0)=e^Ftξ_0+∫_0^te^F(t-s)dL_s^α(ω),

for details see <cit.>. It follows from (<ref>) and (<ref>) that

    φ(t,w,ξ(ω))   =e^Ftξ(ω)+∫_0^te^F(t-s)dL_s^α(ω)
       =e^Ft∫_-∞^0e^-FsdL_s^α(ω)+∫_0^te^F(t-s)dL_s^α(ω)
       =∫_-∞^te^F(t-s)dL_s^α(ω).

By (<ref>), we also see that

    ξ(θ_tω)   =∫_-∞^0e^-FsdL_s^α(θ_tω)=∫_-∞^0e^-Fsd(L_t+s^α(ω)-L_t^α(ω))
       =∫_-∞^0e^-FsdL_t+s^α(ω)=∫_-∞^te^F(t-s)dL_s^α(ω).

Hence φ(t,w,ξ(ω))=ξ(θ_tω) is a stationary orbit for (<ref>). Then we have

    ξ(θ_t+sω)   =∫_-∞^t+se^F(t+s-r)dL_r^α(ω)=e^F(t+s)∫_-∞^te^-F(r+s)dL_r+s^α(ω)
       =e^F(t+s)∫_-∞^te^-F(r+s)d(L_r+s^α(ω)-L_s^α(ω))=∫_-∞^te^F(t-r)dL_r^α(θ_sω)
       =ξ(θ_tθ_sω),

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

    η^1/ε(θ_tω)=ε^-1/α∫_-∞^te^F(t-s)/εdL_s^α.



Since α-stable process L_t^α satisfying Elog(1+|L_1^α|)<∞, η^1/ε(θ_tω) and ξ(θ_tω) are well-defined sationary semimartingales; see <cit.>.


The process η^1/ε(θ_tω) has the same distribution
as the process ξ(θ_t/εω), where η^1/ε and
ξ are defined in Lemma <ref>.


From α-stable process L_t^α are self-similar with Hurst index 1/α, i.e.,

    L_ct^αd=  c^1/α L_t^α,

where “d=“ denotes equivalence (coincidence) in distribution, we have

    η^1/ε(θ_tω)   =ε^-1/α∫_-∞^t e^F(t-s)/εdL_s^α(ω)=∫_-∞^t/εe^F(t/ε-u)ε^-1/αdL_ε u^α(ω)
       d=∫_-∞^t/ε e^F(t/ε-u)dL_u^α(ω)=ξ(θ_t/εω),

which proves that η^1/ε(θ_tω) and ξ(θ_t/εω) have the same distribution.

Now we will transform the slow-fast stochastic dynamical system (<ref>)
into a random dynamical system <cit.>. We introduce the random transformation

    (
      [ x̂^ε;  ŷ^ε;      ]):=T^ε(ω,x^ε,y^ε):=
      (
      [                         x^ε;               y^ε-ση^1/ε(ω);               ]).

Then (x̂^ε(t), ŷ^ε(t))=T^ε(θ_tω,x^ε(t),y^ε(t)) satisfies

    {[     dx̂^ε=Sx̂^ε dt+g_1(x̂^ε,ŷ^ε+ση^1/ε(θ_tω))dt,; dŷ^ε=1/εFŷ^ε dt+1/εg_2(x̂^ε,ŷ^ε+ση^1/ε(θ_tω))dt. ].

This can be seen by a formal differentiation of x^ε and y^ε-ση^1/ε(ω).

For the sake of simplicity, we write ĝ_i(θ_t^εω,x̂^ε,ŷ^ε) = g_i(x̂^ε,ŷ^ε+ση^1/ε(θ_tω)), i=1, 2. Since the additional term ση^1/ε doesn't change the Lipschitz constant of the functions on the right hand side, the functions ĝ_i
have the same Lipschitz constant as g_i.

By hypotheses (A_1)-(A_3), system (<ref>) can be solved for any ω contained in a θ-invariant set Ω of full measure and for any initial condition (x̂^ε(0),ŷ^ε(0))=(x_0,y_0) such that the cocycle property is
satisfied. Then the solution mapping

    (t,ω,(x_0,y_0))↦ϕ̂^ε(t,ω,(x_0,y_0))=(x̂^ε(t,ω,(x_0,y_0)),ŷ^ε(t,ω,(x_0,y_0)))∈ℝ^n,

defines a random dynamical system. In fact, the mapping ϕ̂^ε is
(ℬ(ℝ)⊗ℱ⊗ℬ(ℝ^n),ℬ(ℝ^n))-measurable, and for each ω∈Ω, ϕ̂^ε(·,ω):ℝ×ℝ^n→ℝ^n
is a Carathéodory function.

In the following section we will show that system (<ref>) generates a random dynamical system that has a random slow manifold for sufficiently
small ε>0.
Applying the ideas from the end of Section 2 with T:=T^ε to the
solution of (<ref>), then system (<ref>) also has a version satisfying the cocycle
property. Clearly,

    φ^ε(t,ω,(x_0,y_0))   =(T^ε)^-1(θ_tω,φ̂^ε(t,ω,T^ε(ω,(x_0,y_0))))  
       =ϕ̂^ε(t,ω,(x_0,y_0))+(0,ση^1/ε(θ_tω)), t∈ℝ, ω∈Ω

is a random dynamical system generated by the original system  (<ref>). Hence, by the particular structure of T_ε if (<ref>) has a slow
manifold so has (<ref>).
















































§ RANDOM SLOW MANIFOLDS 




To study system (<ref>), for any β∈ℝ, we introduce Banach spaces of functions with a geometrically weighted sup norm <cit.> as follows:

    C_β^s,-   =   {ϕ^s,-:(-∞,0]→ℝ^n_1 | ϕ^s,-  is continuous andsup_t∈ (-∞,0]
    |e^-β t ϕ_t^s,-|< ∞}, 
    
       C_β^s,+   =   {ϕ^s,+:[0,∞)→ℝ^n_1 | ϕ^s,+  is continuous and sup_t∈ [0,∞)
    |e^-β t ϕ_t^s,+|< ∞}

with the norms

    ||ϕ^s,-||_ C_β^s,-:=sup_t∈ (-∞,0]
    |e^-β t ϕ_t^s,-|       ||ϕ^s,+||_ C_β^s,+ :=sup_t∈ [0,∞)
    |e^-β t ϕ_t^s,+|

Analogously, we define Banach spaces C_β^f,- and C_β^f,+ with the norms

    ||ϕ^f,-||_ C_β^f,-:=sup_t∈ (-∞,0]
    |e^-β t ϕ_t^f,-|     ||ϕ^f,+||_ C_β^f,+ :=sup_t∈ [0,∞)
    |e^-β t ϕ_t^f,+|.

Let C_β^± be the product space C_β^±:=C_β^s,±× C_β^f,±, (ϕ^s,±,ϕ^f,±)∈ C_β^±.
C_β^± equipped with the norm

    ||(ϕ^s,±,ϕ^f,±)||_ C_β^±:=||ϕ^s,±||_ C_β^s,±+||ϕ^f,±||_ C_β^f,±
                                                                                              
is a Banach space.




Letting γ>0 satisfy K<-(γ+γ_f). For the remainder of the paper, we take β=-γ/ε with ε>0 sufficiently small.


Assume that (A_1)-(A_3) hold. Then (x_0,y_0) is in ℳ^ε(ω) if and only if there exists a function
ϕ̂^ε(t)=(x̂^ε(t),ŷ^ε(t))=(x̂^ε(t,ω,(x_0,y_0)),ŷ^ε(t,ω,(x_0,y_0)))∈ C_β^- with t≤0 such that

    (
      [ x̂^ε(t);  ŷ^ε(t);         ])
    =  (
      [                                                   e^Stx_0+∫_0^te^S(t-s)ĝ_1(θ_s^εω,x̂^ε(s),ŷ^ε(s))ds;                                                     1/ε∫_-∞^te^F(t-s)/εĝ_2(θ_s^εω,x̂^ε(s),ŷ^ε(s))ds;                                                   ]),

where

    ℳ^ε(ω)={(x_0,y_0)∈ℝ^n:ϕ̂^ε(·,ω,(x_0,y_0))∈ C_β^-},  ω∈Ω.





 If (x_0,y_0)∈ℳ^ε(ω), by method of constant variation, system (<ref>) is
equivalent to the system of integral equations
 
    {[                  x̂^ε(t)=e^Stx_0+∫_0^te^S(t-s)ĝ_1(θ_s^εω,x̂^ε(s),ŷ^ε(s))ds,; ŷ^ε(t)=e^F(t-u)/εŷ^ε(u)+
        1/ε∫_u^te^F(t-s)/εĝ_2(θ_s^εω,x̂^ε(s),ŷ^ε(s))ds ].

 and ϕ̂^ε(t)∈ C_β^-.
 Moreover, by u<0 and -(γ+γ_f)>K>0, we have
 
    [ |e^F(t-u)/εŷ^ε(u)|≤ e^γ_f(t-u)/ε|ŷ^ε(u)|
       ≤sup_u∈ (-∞,0]{e^-β u|ŷ^ε(u)|}e^γ_f(t-u)/εe^β u;  =||ŷ^ε||_C_β^f,-e^γ_f/εte^(γ_f/ε-β)(-u)
       =||ŷ^ε||_C_β^f,-e^γ_f/εte^-γ+γ_f/εu→0, as u→-∞, ]

which leads to

    ŷ^ε(t)=1/ε∫_-∞^te^F(t-s)/εĝ_2(θ_s^εω,x̂^ε(s),ŷ^ε(s))ds.

Thus (<ref>)-(<ref>) imply that (<ref>) holds.

Conversely, let ϕ̂^ε(t,ω,(x_0,y_0))∈ C_β^- satisfying (<ref>), then (x_0,y_0) is in ℳ^ε(ω) by (<ref>). Thus, we have finished the proof.



Assume (A_1)-(A_3) to be valid. Letting (x̂^ε(0),ŷ^ε(0))=(x_0,y_0), if there exists an δ such that ε∈(0,δ), the system (<ref>) will have a unique solution ϕ̂^ε(t)=(x̂^ε(t),ŷ^ε(t))=(x̂^ε(t,ω,(x_0,y_0)),ŷ^ε(t,ω,(x_0,y_0))) in C_β^-.


For any ϕ̂^ε=(x̂^ε,ŷ^ε)∈ C_β^-, define two operators ℐ_s^ε:C_β^-→ C_β^s,- and ℐ_f^ε:C_β^-→ C_β^f,- satisfying

    [ ℐ_s^ε(ϕ̂^ε)[t]=e^Stx_0+∫_0^te^S(t-s)ĝ_1(θ_s^εω,x̂^ε(s),
      ŷ^ε(s))ds,;      ℐ_f^ε(ϕ̂^ε)[t]=1/ε∫_-∞^te^F(t-s)/εĝ_2(θ_s^εω,x̂^ε(s),ŷ^ε(s))ds, ]

and the Lyapunov-Perron transform ℐ^ε given by

    ℐ^ε(ϕ̂^ε)[t]
    = (
      [ ℐ_s^ε(ϕ̂^ε)[t]; ℐ_f^ε(ϕ̂^ε)[t];                ]).

Under our assumptions above, ℐ^ε maps C_β^- into itself. Taking ϕ̂^ε=(x̂^ε,ŷ^ε)∈ C_β^-, then

    ||ℐ_s^ε(ϕ̂^ε)||_C_β^s,-   ≤ Ksup_t∈ (-∞,0]{e^-β t∫_t^0e^γ_s(t-s)(|x̂^ε(s)|+|ŷ^ε(s)+ση^1/ε(θ_sω)|)ds}
        +sup_t∈ (-∞,0]{e^-β te^γ_st|x_0|}
       ≤ Ksup_t∈ (-∞,0]{∫_t^0e^γ+εγ_s/ε(t-s)ds}||ϕ̂^ε||_C_β^-+C_1
       ≤ε K/γ+εγ_s||ϕ̂^ε||_C_β^-+C_1

and

    ||ℐ_f^ε(ϕ̂^ε)||_C_β^f,-   ≤K/εsup_t∈ (-∞,0]{e^-β t∫_-∞^te^γ_f(t-s)/ε(|x̂^ε(s)|+|ŷ^ε(s)+ση^1/ε(θ_sω)|)ds}
       ≤K/εsup_t∈ (-∞,0]{∫_-∞^te^γ+γ_f/ε(t-s)ds}||ϕ̂^ε||_C_β^-+C_2
       =-K/γ+γ_f||ϕ̂^ε||_C_β^-+C_2.

Hence, by the definition of ℐ^ε we obtain

    ||ℐ^ε(ϕ̂^ε)||_C_β^-≤ρ(ε)||ϕ̂^ε||_C_β^-+C_3,

where C_i, i=1,2,3 are constants and

    ρ(ε):=ε K/γ+εγ_s- K/γ+γ_f.


Further, we will show that ℐ^ε is a contraction.
Let ϕ̂_1^ε=(x̂_1^ε,ŷ_1^ε), ϕ̂_2^ε=(x̂_2^ε,ŷ_2^ε)∈ C_β^-. Using (A_1)-(A_2) and the definition of
C_β^-, we obtain

    ||ℐ_s^ε(ϕ̂_1^ε)-ℐ_s^ε(ϕ̂_2^ε)||_C_β^s,-   ≤ Ksup_t∈ (-∞,0]{e^-β t∫_t^0e^γ_s(t-s)(|x̂_1^ε(s)-x̂_2^ε(s)|+|ŷ_1^ε(s)-ŷ_2^ε(s)|)ds}
       ≤ Ksup_t∈ (-∞,0]{∫_t^0e^γ+εγ_s/ε(t-s)ds}||ϕ̂_1^ε-ϕ̂_2^ε||_C_β^-
       ≤ε K/γ+εγ_s||ϕ̂_1^ε-ϕ̂_2^ε||_C_β^-

and

    ||ℐ_f^ε(ϕ̂_1^ε)-ℐ_f^ε(ϕ̂_2^ε)||_C_β^f,-   ≤K/εsup_t∈ (-∞,0]{e^-β t∫_-∞^te^γ_f(t-s)/ε(|x̂_1^ε(s)-x̂_2^ε(s)|+|ŷ_1^ε(s)-ŷ_2^ε(s)|)ds}
       ≤K/εsup_t∈ (-∞,0]{∫_-∞^te^γ+γ_f/ε(t-s)ds}||
       ϕ̂_1^ε-ϕ̂_2^ε||_C_β^-
       =-K/γ+γ_f||ϕ̂_1^ε-ϕ̂_2^ε||_C_β^-.

By (<ref>) and (<ref>), we have that

    ||ℐ^ε(ϕ̂_1^ε)-ℐ^ε(ϕ̂_2^ε)||_C_β^-≤ρ(ε)||ϕ̂_1^ε-ϕ̂_2^ε||_C_β^-,

By (<ref>) and hypothesis (A_3), we have

    0<ρ(0)=-K/γ+γ_f<1,  ρ(ε)=γ K/(γ+εγ_s)^2.

Then there is a sufficiently small constant δ>0 and a constant ρ_0∈(0,1), such that

    0<ρ(ε)≤ρ_0<1    ε∈(0,δ),

which implies that ℐ^ε is strictly contractive.
Let ϕ̂^ε(t)∈ C_β^- be the unique fixed point, i.e.,
the system (<ref>) has a unique solution ϕ̂^ε(t).

In what follows we investigate the dependence of the fixed point ϕ̂^ε(t)
of the operator ℐ^ε on the intial point.

Assume the hypotheses of Lemma <ref> to be valid. Then for any (x_0,y_0),(x_0^',y_0^')∈ℝ^n, there is an δ>0 such that if ε∈(0,δ), we have

    ||ϕ̂^ε(t,ω,(x_0,y_0))-ϕ̂^ε(t,ω,(x_0^',y_0^'))||_C_β^-≤|x_0-x_0^'|/1-ρ(ε),

where ρ(ε) is defined as (<ref>).


Taking any (x_0,y_0),(x_0^',y_0^')∈ℝ^n, for simplicity we write ϕ̂^ε(t,ω,x_0), ϕ̂^ε(t,ω,x_0^') instead of ϕ̂^ε(t,ω,(x_0,y_0)), ϕ̂^ε(t,ω,(x_0^',y_0^')) in the following estimate, respectively. For every ω∈Ω, we have

    [                                                                                                       ||ϕ̂^ε(t,ω,x_0)-ϕ̂^ε(t,ω,x_0^')||_C_β^-;                                                                      ≤ ||e^St(x_0-x_0^')+∫_0^te^S(t-s)Δĝ_1(θ_s^εω,x̂^ε(s),ŷ^ε(s))ds||_C_β^s,-;                                                                                 +||1/ε∫_-∞^te^F(t-s)/εΔĝ_2(θ_s^εω,x̂^ε(s),ŷ^ε(s))ds||_C_β^f,-;                                                                      ≤          |x_0-x_0^'|+ε K/γ+εγ_s||ϕ̂^ε(t,ω,x_0)-ϕ̂^ε(t,ω,x_0^')||_C_β^-;                                                                                               -K/γ+γ_f||ϕ̂^ε(t,ω,x_0)-ϕ̂^ε(t,ω,x_0^')||_C_β^-;                                                                      =               |x_0-x_0^'|+ρ(ε)||ϕ̂^ε(t,ω,x_0)-ϕ̂^ε(t,ω,x_0^')||_C_β^-, ]

where

    [ Δĝ_i=ĝ_i(θ_s^εω,x̂^ε(s,ω,x_0),ŷ^ε(s,ω,x_0))-ĝ_i(θ_s^εω,x̂^ε(s,ω,x_0^'),ŷ^ε(s,ω,x_0^')), i=1, 2. ]

Then we obtain

    ||ϕ̂^ε(t,ω,x_0)-ϕ̂^ε(t,ω,x_0^')||_C_β^-≤1/1-ρ(ε)|x_0-x_0^'|,

which completes the proof.


By Lemma <ref>, Lemma <ref> and Lemma <ref>, we can construct the slow manifold as a random graph.


Assume that (A_1)-(A_3) hold and that ε is sufficiently small.
Then the system (<ref>) has a invariant manifold
ℳ^ε(ω)={(x_0,ĥ^ε(ω,x_0)): x_0∈ℝ^n_1}, where
ĥ^ε(·,·):Ω×ℝ^n_1↦ℝ^n_2 is a Lipschitz continuous function
with Lipschitz constant
satisfying

    Lipĥ^ε(ω,·)≤-K/γ+γ_f1/1-ρ(ε), ω∈Ω.





Taking any x_0∈ℝ^n_1, define the Lyapunov-Perron transform

    ĥ^ε(ω,x_0)= 1/ε∫_-∞^0e^-Fs/εĝ_2(θ_s^εω,x̂^ε(s),ŷ^ε(s))ds

 where (x̂^ε(s),ŷ^ε(s)) is the unique solution in C_β^- of the system (<ref>) with s≤0.
It follows from Lemma <ref>, Lemma <ref>, (<ref>) and (<ref>) that

    ℳ^ε(ω)={(x̂_0,ĥ^ε(ω,x_0)):x_0∈ℝ^n_1}.

By (<ref>) and Lemma <ref>, we have

    |ĥ^ε(ω,x_0)-ĥ^ε(ω,x_0^')|≤-K/γ+γ_f1/1-ρ(ε)|x_0-x_0^'|

for all x_0, x_0^'∈ℝ^n_1, ω∈Ω.

From Section <ref>, ℳ^ε(ω)
is a random set. Now we are going to prove that ℳ^ε(ω) is invariant in the following sense

    ϕ̂^ε(s,ω,ℳ^ε(ω))⊂ℳ^ε(θ_s^εω)  for  s≥0.

In other words, for each (x_0,y_0)∈ℳ^ε(ω), we have ϕ̂^ε(s,ω,(x _0,y_0))∈ℳ^ε(θ_s^εω).
Using the cocycle property

    ϕ̂^ε(·+s,ω,(x_0,y_0))=ϕ̂^ε(·,θ_sω,ϕ̂^ε(s,ω,(x _0,y_0)))

and the fact ϕ̂^ε(·,ω,(x_0,y_0))∈ C_β^-, it follows  that ϕ̂^ε(·,θ_sω,ϕ̂^ε(s,ω,(x_0,y_0)))∈ C_β^-. Thus, ϕ̂^ε(s,ω,(x_0,y_0))∈ℳ^ε(θ_s^εω). This completes the proof.


The invariant manifold ℳ^ε(ω) is independent of the choice of γ.


Furthermore, the invariant manifold ℳ^ε(ω) exponentially attract other dynamical orbits. Hence, ℳ^ε(ω) is a slow manifold.

Assume that (A_1)-(A_3) hold.
Then the invariant manifold ℳ^ε(ω)={(x_0,ĥ^ε(ω,x_0)):x_0∈ℝ^n_1}
for slow-fast random system (<ref>)  obtained in  Theorem <ref> has exponential tracking property in the following sense: For any z_0=(x_0,y_0)∈ℝ^n, there is a z_0^'=(x_0^',y_0^')∈ℳ^ε(ω) such that

    |ϕ̂^ε(t,ω,z_0^')-ϕ̂^ε(t,ω,z_0)|≤ Ce^-ct|z_0^'-z_0|,  t≥0,

where C>0 and c>0.


Let

    [                                                                                      ϕ̂^ε(t,ω,(x_0,y_0))=(x̂^ε(t,ω,(x_0,y_0)),ŷ^ε(t,ω,(x_0,y_0)));                                                                          ϕ̂^ε(t,ω,(x_0^',y_0^'))=(x̂^ε(t,ω,(x_0^',y_0^')),ŷ^ε(t,ω,(x_0^',y_0^'))) ]

be the two dynamical orbits of system (<ref>) with the initial condition

    ϕ̂^ε(0,ω,(x_0,_0))=(x_0,y_0),  ϕ̂^ε(0,ω,(x_0^',y_0^'))=(x_0^',y_0^').

Then

    ψ^ε(t)   =ϕ̂^ε(t,ω,(x_0^',y_0^'))-ϕ̂^ε(t,ω,(x_0,y_0))
       =(x̂^ε(t,ω,(x_0^',y_0^'))-x̂^ε(t,ω,(x_0,y_0)),
                          ŷ^ε(t,ω,(x_0^',y_0^'))-ŷ^ε(t,ω,(x_0,y_0)))
       :=(u^ε(t),v^ε(t))

satisfies the equation

    {[      du^ε=Su^εdt+Δĝ_1(θ_t^εω,u^ε,v^ε)dt,; dv^ε=1/εFv^εdt+1/εΔĝ_2(θ_t^εω,u^ε,v^ε)dt ].

with the nonlinear items

    Δĝ_i(θ_t^εω,u^ε,v^ε)   =ĝ_i(θ_t^εω,y^ε(t)+x̂^ε(t,ω,(x_0,y_0)),v^ε(t)+ŷ^ε(t,ω,(x_0,y_0)))
          -ĝ_i(θ_t^εω,x̂^ε(t,ω,(x_0,y_0)),ŷ^ε(t,ω,(x_0,y_0))), i=1, 2,

and the initial condition

    (u^ε(0),v^ε(0)) = (u_0,v_0) = (x_0^'-x_0,y_0^'-y_0).

By direct calculation, for t≥0, ψ^ε(t)=(u^ε(t),v^ε(t)) satisfying

    (
      [ u^ε(t); v^ε(t);        ])
    =  ([                                                                        ∫_+∞^te^S(t-s)Δĝ_1(θ_s^εω,u^ε(s),v^ε(s))ds;                                                          e^Ft/εv_0+1/ε∫_0^te^F(t-s)/εΔĝ_2(θ_s^εω,u^ε(s),v^ε(s))ds;                                                          ])

 is a solution of (<ref>) in C_β^+.

Now we use the Lyapunov-Perron transform again, to prove that (<ref>) has a unique solution (u^ε(t),v^ε(t)) in C_β^+ with (x_0^',y_0^')=(u_0,v_0)+(x_0,y_0)∈ℳ^ε(ω). Clearly,

    (x_0^',y_0^')∈ℳ^ε(ω)
       ⟺    y_0^'= 1/ε∫_-∞^0e^-Fs/εĝ_2(θ_s^εω,
    x̂^ε(s,ω,x_0^'),ŷ^ε(s,ω,x_0^'))ds
       ⟺    v_0=-y_0+1/ε∫_-∞^0e^-Fs/εĝ_2(θ_s^εω,
    x̂^ε(s,ω,u_0+x_0),ŷ^ε(s,ω,u_0+x_0))ds  
          =-y_0+ĥ^ε(ω,u_0+x_0).

Taking ψ^ε=(u^ε,v^ε)∈ C_β^+, define two operators 𝒥_s^ε:C_β^+→ C_β^s,+ and 𝒥_f^ε:C_β^+→ C_β^f,+ satisfying

    [                𝒥_s^ε(ψ^ε)[t]=∫_+∞^te^S(t-s)Δĝ_1(θ_s^εω,u^ε(s),v^ε(s))ds,; 𝒥_f^ε(ψ^ε)[t]= e^Ft/εv_0+1/ε∫_0^te^F(t-s)/εΔĝ_2(θ_s^εω,u^ε(s),v^ε(s))ds. ]

Moreover, the Lyapunov-Perron transform 𝒥^ε:C_β^+→ C_β^+ is given by

    𝒥^ε(ψ^ε)[t]
    = (
      [ 𝒥_s^ε(ψ^ε)[t]; 𝒥_f^ε(ψ^ε)[t];               ]).

We have the following estimates

    ||𝒥_s^ε(ψ^ε)||_C_β^s,+
       =||∫_+∞^te^S(t-s)[ĝ_1(θ_s^εω,u^ε(s)+x̂^ε(s),v^ε(s)+ŷ^ε(s))-ĝ_1(θ_s^εω,x̂^ε(s),ŷ^ε(s))]ds||_C_β^s,+
       ≤ Ksup_t∈ [0,∞){e^-β t∫_t^+∞e^γ_s(t-s)(|u^ε(s)|+|v^ε(s)|)ds}
       ≤ Ksup_t∈ [0,∞){∫_t^+∞e^γ+εγ_s/ε(t-s)ds}||ψ^ε||_C_β^+
       ≤ε K/γ+εγ_s||ψ^ε||_C_β^+,

and

    ||𝒥_f^ε(ψ^ε)||_C_β^f,+   =||e^Ft/εv_0+1/ε∫_0^te^F(t-s)/ε[ĝ_2(θ_s^εω,u^ε(s)+x̂^ε(s),v^ε(s)+ŷ^ε(s))
           -ĝ_2(θ_s^εω,x̂^ε(s),ŷ^ε(s))]ds||_C_β^f,+
       ≤K/εsup_t∈ [0,∞){e^-β t∫_0^te^γ_f(t-s)/ε(|u^ε(s)|+|v^ε(s)|)ds}+sup_t∈ [0,∞){e^-β te^γ_f/εt|v_0|}
       ≤K/εsup_t∈ [0,∞){∫_0^te^γ+γ_f/ε(t-s)ds}||ψ^ε||_C_β^++|v_0|
       =-K/γ+γ_f||ψ^ε||_C_β^++|v_0|.

Hence, by (<ref>), we obtain

    ||𝒥^ε(ψ^ε)||_C_β^+≤ρ(ε)||ψ^ε||_C_β^++|v_0|

where ρ(ε) is defined as (<ref>) in the proof of Lemma <ref>.

For any ψ^ε=(u^ε,v^ε), ψ̅^ε=(u̅^ε,v̅^ε)∈ C_β^+,

    [                                                                                                                                            ||𝒥_s^ε(ψ^ε)-𝒥_s^ε(ψ̅^ε)||_C_β^s,+;                                                                                      = ||∫_+∞^te^S(t-s)[Δĝ_1(θ_s^εω,u^ε(s),v^ε(s))
    -Δĝ_1(θ_s^εω,u̅^ε(s),v̅^ε(s))]ds||_C_β^s,+;                                                                                      =                             ||∫_+∞^te^S(t-s)[ ĝ_1(θ_s^εω,u^ε(s)+x̂^ε(s),v^ε(s)+ŷ^ε(s));                                                                                                                     - ĝ_1(θ_s^εω,u̅^ε(s)+x̂^ε(s),v̅^ε(s)+ŷ^ε(s))]ds||_C_β^s,+;                                                                                      ≤              Ksup_t∈[0,∞){e^-β t∫_t^+∞e^γ_s(t-s)(|u^ε(s)-u̅^ε(s)|+|v^ε(s)-v̅^ε(s)|)ds};                                                                                      ≤                                Ksup_t∈[0,∞){∫_t^+∞e^γ+εγ_s/ε(t-s)ds}||ψ^ε-ψ̅^ε||_C_β^+;                                                                                      =                                                          ε K/γ+εγ_s||ψ^ε-ψ̅^ε||_C_β^+. ]

On the one hand, by (<ref>), we have

    [                                                                                                                                       |e^Ft/ε(v_0-v̅_0)|≤ e^γ_f/εt Lipĥ^ε|u_0-u̅_0|;                                                                                         ≤ e^γ_f/εt Lipĥ^ε|∫_+∞^0e^S(-s)[Δĝ_1(θ_s^εω,u^ε(s),v^ε(s))-Δĝ_1(θ_s^εω,u̅^ε(s),v̅^ε(s))]ds|;                                                                                         =                   e^γ_f/εt Lipĥ^ε|∫_+∞^0e^S(-s)[ ĝ_1(θ_s^εω,u^ε(s)+x̂^ε(s),v^ε(s)+ŷ^ε(s));                                                                                                                                    - ĝ_1(θ_s^εω,u̅^ε(s)+x̂^ε(s),v̅^ε(s)+ŷ^ε(s))]ds|;                                                                                         ≤                                       Lipĥ^ε· Ke^γ_f/εt∫_0^+∞e^γ_s(-s)|ψ^ε(s)-ψ̅^ε(s)|ds, ]

which leads to

    ||e^Ft/ε(v_0-v̅_0)||_C_β^f,+   ≤ Lipĥ^ε· K||ψ^ε(s)-ψ̅^ε(s)||_C_β^+sup_t∈[0,∞){e^(γ_f/ε-β)t∫_0^+∞e^(β-γ_s)sds}
       ≤ε K Lipĥ^ε/γ+εγ_s||ψ^ε(s)-ψ̅^ε(s)||_C_β^+.

On the other hand, we observe

    [                                                                                           ||1/ε∫_0^te^F(t-s)/ε[Δĝ_2(θ_s^εω,u^ε(s),v^ε(s))-Δĝ_2(θ_s^εω,u̅^ε(s),v̅^ε(s))]ds||_C_β^f,+;                                                                                         =                             ||1/ε∫_0^te^F(t-s)/ε[ĝ_2(θ_s^εω,u^ε(s)+x̂^ε(s),v^ε(s)+ŷ^ε(s));                                                                                                                            -ĝ_2(θ_s^εω,u̅^ε(s)+x̂^ε(s),v̅^ε(s)+ŷ^ε(s))]ds||_C_β^f,+;                                                                                         ≤              K/εsup_t∈[0,∞){e^-β t∫_0^te^γ_f(t-s)/ε(|u^ε(s)-u̅^ε(s)|+|v^ε(s)-v̅^ε(s)|)ds};                                                                                         ≤                                   K/εsup_t∈[0,∞){∫_0^te^γ+γ_f/ε(t-s)ds}||ψ^ε-ψ̅^ε||_C_β^+;                                                                                         =                                                               -K/γ+γ_f||ψ^ε-ψ̅^ε||_C_β^+. ]

Using  (<ref>), (<ref>) and (<ref>), it follows that

    [                                                                                                                        ||𝒥_f^ε(ψ^ε)-𝒥_f^ε(ψ̅^ε)||_C_β^f,+;                                                                            ≤ ||e^Ft/ε(v_0-v̅_0)||_C_β^f,++||1/ε∫_0^te^F(t-s)/ε[Δĝ_2(u^ε(s),v^ε(s),θ_s^εω);                                                                                                                -Δĝ_2(u̅^ε(s),v̅^ε(s),θ_s^εω)]ds||_C_β^f,+;                                                                            ≤                                (ε K Lipĥ^ε/γ+εγ_s-K/γ+γ_f)||ψ^ε-ψ̅^ε||_C_β^+;                                                                            ≤   -(ε K^2/(γ+εγ_s)(γ+γ_f)[1-K(ε/γ+εγ_s-1/γ+γ_f)]+K/γ+γ_f)||ψ^ε-ψ̅^ε||_C_β^+. ]

By (<ref>) and (<ref>), we have

    ||𝒥^ε(ψ^ε)-𝒥^ε(ψ̅^ε)||_C_β^+≤ρ̅(ε)||ψ^ε-ψ̅^ε||_C_β^+

with

    ρ̅(ε)   =ε K/γ+εγ_s-K/γ+γ_f
       -ε K^2/(γ+εγ_s)(γ+γ_f)[1-K(ε/γ+εγ_s-1/γ+γ_f)].

Note that 0<K<-(γ+γ_f). Clearly, for small ϵ, we have 0<ρ̅(ε)<1,
which implies that 𝒥^ε is a contraction in C_β^+.
Thus, there is a unique fixed point ψ^ε:=(u^ε,v^ε) in C_β^+.
Further, ψ^ε  satisfies (x_0^',y_0^')=(u_0,v_0)+(x_0,y_0)∈ℳ^ε(ω). In fact, the solution of (<ref>) in C_β^+ if and only if it is a fixed point of the Lyapunov-Perron transform (<ref>).
Moreover, it follows from (<ref>) that

    ||ψ^ε(·)||_C_β^+≤1/1-ρ(ε)|v_0|

which leads to

    ||ϕ̂^ε(t,ω,(x_0^',y_0^'))-ϕ̂^ε(t,ω,(x_0,y_0))||_C_β^+≤1/1-ρ(ε)|y_0^'-y_0|.

And then

    |ϕ̂^ε(t,ω,z_0^')-ϕ̂^ε(t,ω,z_0)|≤e^-γ/ε t/1-ρ(ε)|z_0^'-z_0|,

with t≥0 and -γ/ε<0. Hence,   the proof has been finished.


  0<ρ(ε)<1 and 0<ρ̅(ε)<1
are the critical points in the proof of Lemma <ref> and Theorem <ref> respectively.

According to Theorem <ref> and Theorem <ref>, the system (<ref>)
has an exponential tracking slow manifold. By the relationship between of φ^ε(t,ω) and ϕ̂^ε(t,ω), so has the slow-fast system (<ref>).

Assume that (A_1)-(A_3) hold. The slow-fast system (<ref>) with jumps has a slow manifold

    M^ε(ω)=(T^ε)^-1ℳ^ε(ω)=ℳ^ε(ω)+(0,ση^1/ε(ω))
     ={(x_0,h^ε(ω,x_0)):x_0∈ℝ^n_1}

with h^ε(ω,x_0)=ĥ^ε(ω,x_0)+ση^1/ε(θ_tω), where T^ε is defined in (<ref>).


We have the relationship between φ and φ̂ given by (<ref>):

    φ^ε(t,ω,M^ε(ω))   =(T^ε)^-1(θ_tω,φ̂^ε(t,ω,T^ε(ω,M^ε(ω))))  
       =(T^ε)^-1(θ_tω,φ̂^ε(t,ω,ℳ^ε(ω)))⊂(T^ε)^-1(θ_tω,ℳ^ε(θ_tω))=M^ε(θ_tω).

Note that t→η^1/ε(θ_tω) has a sublinear growth rate for 1<α<2; see <cit.>. Thus the
transform (T^ε)^-1(θ_tω) does not change the exponential tracking property. It follows that  M^ε(ω) is a slow manifold.


It is worthy mentioning that the dynamical orbits in M^ε(ω) are cádlág and adapted.


Assume that (A_1)-(A_3) hold and that ε is sufficiently small. For any solution ϕ^ε(t,ω)=(x^ε(t,ω),y^ε(t,ω)) with initial condition
z_0=(x_0,y_0) to the fast-slow system (<ref>), there exists a solution ϕ̅^ε(t,ω)=(x̅^ε(t,ω),y̅^ε(t,ω)) with initial point z̅_0=(x̅_0,y̅_0) on the manifold M^ε(ω) which satisfies the reduction system

    dx̅^ε=Sx̅^εdt+g_1(x̅^ε,h^ε(θ_tω,x̅^ε))dt

such that

    |φ^ε(t,ω)-φ̅^ε(t,ω)|≤e^-γ/ε t/1-(ε K/γ+εγ_s-K/γ+γ_f)|z_0-z̅_0|,   ω∈Ω,

with t≥0 and -γ/ε<0.










§ SLOW MANIFOLDS 


By the scaling t→ε t, the system (<ref>) can be rewritten as

    {[ dx̂^ε=ε Sx̂^ε dt+ε g_1(x̂^ε,ŷ^ε+ση^1/ε(θ_ε tω))dt,;        dŷ^ε=Fŷ^ε dt+g_2(x̂^ε,ŷ^ε+ση^1/ε(θ_ε tω)dt. ].







If we now replace η^1/ε(θ_ε tω) by ξ(θ_tω), we get the following random dynamical system

    {[ dx̃^ε=ε Sx̃^ε dt+ε g_1(x̃^ε,ỹ^ε+σξ(θ_tω))dt,;       dỹ^ε=Fỹ^ε dt+g_2(x̃^ε,ỹ^ε+σξ(θ_tω))dt, ].

with solution of which coincides with that of the system (<ref>)  in distribution.



The slow manifold of (<ref>) can be constructed in a completely analogous procedure as Section <ref>,
so we omit the proof
and immediately state the following theorem.

Assume that (A_1)-(A_3) hold. Given (x_0,y_0)∈ℝ^n, if there exists an δ such that ε∈(0,δ) and (x̃^ε(0),ỹ^ε(0))=(x_0,y_0), then the system of integral equations

    (
      [ x̃^ε(t);  ỹ^ε(t);         ])
    =  ([                                                         e^Sε tx_0+ε∫_0^te^Sε(t-s)g_1(x̃^ε(s),ỹ^ε(s)+σξ(θ_sω))ds;                                                                    ∫_-∞^te^F(t-s)g_2(x̃^ε(s),ỹ^ε(s)+σξ(θ_sω))ds;                                                         ])

has a unique solution ϕ̃^ε(t,ω,(x_0,y_0))=(x̃^ε(t,ω,(x_0,y_0)),ỹ^ε(t,ω,(x_0,y_0)))∈ C_-γ^-. Further, system (<ref>) has a slow manifold

    ℳ̃^ε(ω)   ={(x_0,y_0)∈ℝ^n:ϕ̃^ε(·,ω,(x_0,y_0))∈ C_-γ^-}
       ={(x_0,h̃^ε(ω,x_0)): x_0∈ℝ^n_1},

where

    h̃^ε(ω,x_0)= ∫_-∞^0e^-Fsg_2(x̃^ε(s),ỹ^ε(s)+σξ(θ_sω))ds

is a Lipschitz continuous function
and Lipschitz constant

    Liph̃^ε(ω,·)≤- K/(γ+γ_f)(1 -ρ(ε)), ω∈Ω.



Now, we give the relationship of ℳ^ε(ω) and ℳ̃^ε(ω) as follows.


Assume (A_1)-(A_3) to be valid. The slow manifold
ℳ^ε(ω) (see (<ref>)) of system (<ref>) is same as the slow manifold
ℳ̃^ε(ω) (see (<ref>)) of system (<ref>) in distribution. That is, for every x_0∈ℝ^n_1,

    ĥ^ε(ω,x_0)d=h̃^ε(ω,x_0).



By the scaling s→ε s in (<ref>) and the fact that the solution of system (<ref>) coincides with that of system (<ref>) in distribution, for every x_0∈ℝ^n_1,

    ĥ^ε(ω,x_0)   =1/ε∫_-∞^0e^-Fs/εg_2(x̂^ε(s),ŷ^ε(s)+ση^1/ε(θ_ sω))ds
       =∫_-∞^0e^-Fsg_2(x̂^ε(ε s),ŷ^ε(ε s)+ση^1/ε(θ_ε sω))ds
       d=∫_-∞^0e^-Fsg_2(x̃^ε(s),ỹ^ε(s)+σξ(θ_sω))ds
       =h̃^ε(ω,x_0)

which completes the proof.




We are going to study the limiting case of the slow manifold for the system (<ref>) as ε→0
and construct an asymptotic approximation of ℳ^ε(ω) with sufficiently small ε>0 in distribution.  However, it makes also sense to study (<ref>) for ε=0. In that
case, there exists a slow manifold.

Consider the following system

    dx^0(t)=0,  
    dy^0(t)=Fy^0(t)dt+g_2(x^0(t),y^0(t)+σξ(θ_tω))dt

with the initial condition (x^0(0),y^0(0))=(x_0,y_0). As proved in Section <ref>, we
also have the following result. The system (<ref>) has  the following slow  manifold

    ℳ^0(ω)={(x_0,h^0(ω,x_0)): x_0∈ℝ^n_1}

where

    h^0(ω,x_0)=∫_-∞^0e^-Fsg_2(x_0,y^0(s)+σξ(θ_sω))ds,

whose Lipschitz constant Liph^0 satisfies

    Liph^0≤-K/γ+γ_f+K,

and y^0(t) is the unique solution in C^f,-_-γ for integral equation

    y^0(t)=∫_-∞^te^F(t-s)g_2(x_0,y^0(s)+σξ(θ_sω))ds, t≤0.


From 0<K<-(γ+γ_f), we easily have -(γ+γ_f+K)>0.


As we will show, the slow manifold of the system (<ref>) converges to the slow manifold
of the system (<ref>) in distribution. In other words, the distribution of ℳ^ε(ω)
converges to the distribution of ℳ^0(ω), as ε tends to zero.
The slow manifold ℳ^0(ω) is called the critical manifold for
the system (<ref>).

Assume that (A_1)-(A_3) hold and there exists a positive constant C
such that |g_1(x,y)|≤ C. ℳ^ε(ω) converges to ℳ^0(ω)
in distribution  as ε→0. In other words, for x_0∈ℝ^n_1, ω∈Ω,

    ĥ^ε(ω,x_0)d=h^0(ω,x_0)+𝒪(ε)  in ℝ^n_2 as ε→0.



Applying Lemma <ref>, to prove (<ref>), we can alternatively check that if

    h̃^ε(ω,x_0)d⟶h^0(ω,x_0)  in ℝ^n_2 as ε→0.



From (<ref>) and (<ref>), for sufficiently small ε, we have

    [                                                                                                                      |h̃^ε(ω,x_0)-h^0(ω,x_0)|;                                                                      = |∫_-∞^0e^-Fs[g_2(x̃^ε(s),ỹ^ε(s)+σξ(θ_sω))-g_2(x_0,y^0(s)+σξ(θ_sω))]ds|;                                                                      ≤                       K∫_-∞^0e^-γ_fs(|x̃^ε(s)-x_0|+|ỹ^ε(s)-y^0(s)|)ds. ]

According to (<ref>), for t≤0, it follows that

    [                                                    |x̃^ε(t)-x_0|=     |e^Sε tx_0-x_0+ε∫_0^te^Sε(t-s)g_1(x̃^ε(s),ỹ^ε(s)+σξ(θ_sω))ds|;                                                                 ≤ |e^Sε tx_0-x_0|+ε∫_t^0e^εγ_s(t-s)|g_1(x̃^ε(s),ỹ^ε(s)+σξ(θ_sω))|ds;                                                                 ≤                       |∫_ε t^0Sx_0e^Sudu|+ε· C·∫_t^0e^εγ_s(t-s)ds;                                                                 ≤           |Sx_0|∫_ε t^0e^γ_sudu+C/γ_s(1-e^εγ_st):=C_1(1-e^εγ_st), ]

where C_1=1/γ_s(|Sx_0|+C). Using (<ref>) and (<ref>), it is clear that

    [                                                          |ỹ^ε(t)-y^0(t)|= |∫_-∞^te^F(t-s)[g_2(x̃^ε(s),ỹ^ε(s)+σξ(θ_sω))-g_2(x_0,y^0(s)+σξ(θ_sω))]ds|;                                                                         ≤                        K∫_-∞^te^γ_f(t-s)(|x̃^ε(s)-x_0|+|ỹ^ε(s)-y^0(s)|)ds;                                                                         ≤                      K∫_-∞^te^γ_f(t-s)[C_1(1-e^εγ_s s)+|ỹ^ε(s)-y^0(s)|]ds;                                                                         =         KC_1(1/γ_f-εγ_se^εγ_st-1/γ_f)+K∫_-∞^te^γ_f(t-s)|ỹ^ε(s)-y^0(s)|ds. ]

Hence, we have

    ||ỹ^ε-y^0||_C_-γ^f,-   ≤sup_t∈ (-∞,0]e^γ t[KC_1(1/γ_f-εγ_se^εγ_st-1/γ_f)+K∫_-∞^te^γ_f(t-s)|ỹ^ε(s)-y^0(s)|ds]
       ≤ KC_1sup_t∈ (-∞,0]q(t,ε)-K/γ+γ_f||ỹ^ε-y^0||_C_-γ^f,-,

where

    q(t,ε)=1/γ_f-εγ_se^(γ+εγ_s)t-1/γ_fe^γ t,  t≤0.

Since

    dq(t,ε)/dt   =e^γ t(γ+εγ_s/γ_f-εγ_se^εγ_st-γ/γ_f)
      ≥ e^γ t(γ+εγ_s/γ_f-εγ_s-γ/γ_f)
     →0, as ε→0,

which implies  q(t,ε) is increasing with respect to the variable t for  small ε>0.
Then we immediately have

    q(t,ε)≤ q(0,ε)=1/γ_f-εγ_s-1/γ_f,  t≤0.

According to (<ref>) and (<ref>), we obtain

    ||ỹ^ε-y^0||_C_-γ^f,-≤ C_2(1/γ_f-εγ_s-1/γ_f),   with  C_2=KC_1/1+K/γ+γ_f.

Hence

    |ỹ^ε(t)-y^0(t)|≤ C_2e^-γ t(1/γ_f-εγ_s-1/γ_f),  t≤0.

It follows from (<ref>), (<ref>) and (<ref>) that

    |h̃^ε(ω,x_0)-h^0(ω,x_0)|
       ≤ K[C_1∫_-∞^0e^-γ_fs(1-e^εγ_ss)ds+ C_2(1/γ_f-εγ_s-1/γ_f)∫_-∞^0e^-(γ+γ_f)sds]
       =C_3(1/γ_f-εγ_s-1/γ_f)→0,  as  ε→0,

where C_3=K(C_1-C_2/γ+γ_f)=K(γ+γ_f)(|Sx_0|+C)/γ_s(γ+γ_f+K).
This completes the proof.


Assume the hypotheses of Theorem <ref> to be valid. Then there exists a δ such that if ε∈(0,δ), the slow manifold of system (<ref>) can be approximated in distribution as

    ℳ^ε(ω)
    
                                        d={(x_0,h^0(ω,x_0)+ε h^1(ω,x_0)+𝒪(ε^2)): X_0∈ℝ^n_1}

where h^0(ω,x_0) is defined in (<ref>),

    h^1(ω,x_0)=∫_-∞^0e^-Fs[x^1(s)g_2,x(x_0,y^0(s)+σξ(θ_sω))
                                  +y^1(s)g_2,y(x_0,y^0(s)+σξ(θ_sω))]ds,

and (x^1(t),y^1(t)) are given by (<ref>) and (<ref>).


Applying Lemma <ref>, we can alternatively prove

    h̃^ε(ω,x_0)=h^0(ω,x_0)+ε h^1(ω,x_0)+𝒪(ε^2)

For the system (<ref>), we write

    [ x̃^ε(t)=x̃^0(t)+ε x^1(t)+𝒪(ε^2),  x̃^ε(0)=x_0,;    ỹ^ε(t)=ỹ^0(t)+ε y^1(t)+𝒪(ε^2),  ỹ^ε(0)=y_0, ]

where x̃^0(t), ỹ^0(t), x^1(t) and y^1(t) will be determined in the below.
The Taylor expansions of g_i(x̃^ε(t),ỹ^ε(t)+σξ(θ_tω)), i=1, 2  at point (x̃^0(t),ỹ^0(t)+σξ(θ_tω)) are as follows.

    [                                                                                                                              g_i(x̃^ε(t),ỹ^ε(t)+σξ(θ_tω));                                                                            = g_i(x̃^0(t),ỹ^0(t)+σξ(θ_tω))+(x̃^ε(t)-x̃^0(t))g_i,x(x̃^0(t),ỹ^0(t)+σξ(θ_tω));                                                                                                     +(ỹ^ε(t)-ỹ^0(t))g_i,y(x̃^0(t),ỹ^0(t)+σξ(θ_tω))+𝒪(ε^2);                                                                            =          g_i(x̃^0(t),ỹ^0(t)+σξ(θ_tω))+ε x^1(t)g_i,x(x̃^0(t),ỹ^0(t)+σξ(θ_tω));                                                                                                           +ε y^1(t)g_i,y(x̃^0(t),ỹ^0(t)+σξ(θ_tω))+𝒪(ε^2), ]

where g_i,x(x,y) and g_i,y(x,y) denote the partial derivative of g_i(x,y) with respect to the  variables x and y respectively.


Substituting (<ref>) into (<ref>), equating the terms with the same power of ε, we deduce that

    dx̃^0(t)   =0
    
      dx^1(t)   =Sx̃^0(t)dt+g_1(x̃^0(t),ỹ^0(t)+σξ(θ_tω))dt

and

    dỹ^0(t)   =Fỹ^0(t)dt+g_2(x̃^0(t),ỹ^0(t)+σξ(θ_tω))dt
    
        dy^1(t)   =x^1(t)g_2,x(x̃^0(t),ỹ^0(t)+σξ(θ_tω))dt
         +y^1(t)[F+g_2,y(x̃^0(t),ỹ^0(t)+σξ(θ_tω))]dt.

Comparing (<ref>)    with (<ref>) and (<ref>), we immediately have

    x^0(t)=x̃^0(t),  y^0(t)=ỹ^0(t),

which implies that the system (<ref>) essentially is the system (<ref>) scaled by ε t with zero singular perturbation parameter, i.e., the system (<ref>) with ε=0.
From (<ref>) and x̃^0(0)=x_0, we get

    x^1(t)=Sx_0t+∫_0^tg_1(x_0,y^0(s)+σξ(θ_sω))dt.

According to (<ref>), (<ref>) and ỹ^0(0)=h^0(ω,x_0), we obtain

    ỹ^0(t)=e^Fth^0(ω,x_0)+∫_0^te^-F(s-t)g__2(x_0,ỹ^0(s)+σξ(θ_sω))ds.

By (<ref>)-(<ref>) and ỹ^1(0)=h^1(ω,x_0), we have

    y^1(t)   =e^Ft+∫_0^tg_2,y(x_0,ỹ^0(s)+σξ(θ_sω))dsh^1(ω,x_0)+∫_0^te^-F(s-t)+∫_s^tg_2,y(x_0,ỹ^0(u)+σξ(θ_uω))du
       · g_2,x(x_0,ỹ^0(s)+σξ(θ_sω))[Sx_0s+∫_0^sg_1(x_0,y^0(u)+σξ(θ_uω))du]ds

It follows from (<ref>), (<ref>) and (<ref>) that

    h̃^ε(ω,x_0)=   ∫_-∞^0e^-Fsg_2(x̃^ε(s),ỹ^ε(s)+σξ(θ_sω))ds 
    
    =   ∫_-∞^0e^-Fs[g_2(x̃^0(s),ỹ^0(s)+σξ(θ_sω))+ε x^1(s)g_2,x(x̃^0(s),ỹ^0(s)+σξ(θ_sω)) 
       +ε y^1(s)g_2,y(x̃^0(s),ỹ^0(t)+σξ(θ_sω))]ds+𝒪(ε^2) 
    
    =   ∫_-∞^0e^-Fs[g_2(x_0,y^0(s)+σξ(θ_sω))ds+ε∫_-∞^0e^-Fs[x^1(s)g_2,x(x_0,y^0(s)+σξ(θ_sω))
       +y^1(s)g_2,y(x_0,y^0(s)+σξ(θ_sω))]ds+𝒪(ε^2)
    
    =    h^0(ω,x_0)+ε h^1(ω,x_0)+𝒪(ε^2),

which conclude the proof.




§ EXAMPLES 

Now we present three examples from biological sciences to illustrate our analytical results.

Consider a two dimensional model of FitzHugh-Nagumo system <cit.>

    {[                    dx^ε=x^ε dt+1/3sin y^ε dt,  x^ε∈ℝ; dy^ε=-1/εy^ε dt+1/6ε(x^ε)^2 dt+σε^-1/αdL_t^α,  y^ε∈ℝ ].

where x^ε
is the “slow" component, y^ε is the “fast" component, γ_s=1, γ_f=-1, K<1, g_1(x^ε, y^ε)=1/3sin y^ε, g_2(x^ε, y^ε)=1/6(x^ε)^2.

The scaling t→ε t in (<ref>) yields

    {[        dx^ε=ε x^ε dt+1/3εsin y^ε dt,; dy^ε=-y^ε dt+1/6(x^ε)^2 dt+σ dL_t^α. ].

We can convert this two dimensional SDE system to the following system

    {[ dx̃^ε=εx̃^ε dt+ε/3sin(ỹ^ε+σξ(θ_tω))dt,;                 dỹ^ε=-ỹ^ε+1/6(x̃^ε)^2, ].

where ξ(θ_tω)=∫_-∞^t e^(t-s)dL_s^α.

Denote x̃^ε(0)=x_0, we get
h^0(ω,x_0)=x^2_0/6 and

    h^1(ω,x_0)=-x^2_0/3+x_0/9∫_-∞^0e^t[∫_0^tsin( x^2_0/6+σ∫_-∞^se^s-rdL_r^α)ds]dt.

This produces an approximated slow manifold ℳ̃^ε(ω)={(x_0,h̃^ε(ω,x_0): x_0∈[0,π]} of system (<ref>), where h̃^ε(ω,x_0)=h^0(ω,x_0)+ε h^1(ω,x_0)+𝒪(ε^2). Then we obtain

    {[ dx̅^ε=εx̅^ε dt+1/3εsiny̅^ε dt,;          y̅^ε=h̃^ε(θ_tω,x̅^ε), ].

which is the reduction system of (<ref>).








Consider a three dimensional model of FitzHugh-Nagumo system

    {[ dx_1^ε=1/2x_1^ε dt+(-(x_1^ε)^3+x_1^ε x_2^ε/20+y^ε/3)dt,  x_1^ε∈ℝ; dx_2^ε=1/3x_2^ε dt+(1/2sin x_1^εcos x_2^ε+(y^ε)^2/8)dt,  x_2^ε∈ℝ;        dy^ε=-1/εy^ε dt-1/10εx_1^ε x_2^ε dt+σε^-1/αdL_t^α,  y^ε∈ℝ ].

where (x_1^ε,x_2^ε)
is the “slow" component, y^ε is the “fast" component,γ_s=1/3, γ_f=-1, K<1, g_1(x_1^ε, x_2^ε, y^ε)=-(x_1^ε)^3+x_1^ε x_2^ε/20+y^ε/3, g_2(x_1^ε, x_2^ε, y^ε)=1/2sin x_1^εcos x_2^ε+(y^ε)^2/8, g(x_1^ε, x_2^ε, y^ε)=-1/10x_1^ε x_2^ε.

We now scale the time t→ε t, (<ref>) can be rewriten as

    {[ dx_1^ε=1/2ε x_1^ε dt+ε(-(x_1^ε)^3+x_1^ε x_2^ε/20+y^ε/3)dt,; dx_2^ε=1/3ε x_2^ε dt+ε(1/2sin x_1^εcos x_2^ε+(y^ε)^2/8)dt,;                  dy^ε=-y^ε dt-1/10x_1^ε x_2^ε dt+σ dL_t^α. ].

The corresponding system of random differential equations

    {[ dx̃_1^ε=1/2εx̃_1^ε dt+ε[-(x̃_1^ε)^3+x̃_1^εx̃_2^ε/20+1/3(ỹ^ε+σξ(θ_tω))]dt;     dx̃_2^ε=1/3εx̃_2^ε dt+ε[1/2sinx̃_1^εcosx̃_2^ε+1/8(ỹ^ε+σξ(θ_tω))^2]dt;                                       dỹ^ε=-ỹ^ε dt-1/10x̃_1^εx̃_2^ε
      dt ].

where ξ(θ_tω)=∫_-∞^t e^(t-s)dL_s^α.

Denote (x̃_1^ε(0),x̃_2^ε(0))=(x_0,x^'_0), we get
h^0(ω,(x_0,x^'_0))=-1/10x_0x_0^'
and

    h^1(ω,(x_0,x^'_0))
       =(10x_0-x_0^3/20-x_0x_0^'/12)x_0^'/10+σ(3x_0^2 -40)x_0^'/1200∫_-∞^0e^t[∫_0^t∫_-∞^se^s-rdL_r^αds]dt
       +(x^'_0/3+1/2sin x_0cos x_0^'+(x_0x_0^')^2/800)x_0/10-σ^2 x_0/80∫_-∞^0e^t[∫_0^t(∫_-∞^se^s-rdL_r^α)^2ds]dt.

We get an approximated slow manifold ℳ̃^ε(ω)={((x_0,x^'_0),h̃(ω,(x_0,x^'_0)): (x_0,x^'_0)∈ℝ^2} of system (<ref>), where h̃^ε(ω,(x_0,x^'_0))=h^0(ω,(x_0,x^'_0))+ε h^1(ω,(x_0,x^'_0))+𝒪(ε^2). The reduced system of (<ref>) given by

    {[ dx̅_1^ε=1/2εx̅_1^ε dt+ε(-(x̅_1^ε)^3+x̅_1^εx̅_2^ε/20+y̅^ε/3)dt,;   dx̅_2^ε=1/3εx̅_2^ε dt+ε(1/2sinx̅_1^εcosx̅_2^ε+(y̅^ε)^2/8)dt,;                               y̅^ε=h̃^ε(θ_tω,(x̅_1^ε,x̅_2^ε)). ].








Consider a three dimensional model of FitzHugh-Nagumo system

    {[         dx^ε=1/3x^εdt+x^ε-(x^ε)^3+sin y_1^εcos y_2^ε/50dt,  x∈ℝ; dy_1^ε=-1/εy_1^εdt+1/5εsin x^ε
      dt+σε^-1/α_1dL_t^α_1,  y_1^ε∈ℝ;  dy_2^ε=-1/εy_2^εdt-1/16ε (x^ε)^2dt+σε^-1/α_2dL_t^α_2,  y_2^ε∈ℝ;                                                                 ].

where x^ε
is the “slow" component, (y_1^ε,y_2^ε) is the “fast" component, γ_s=1/3, γ_f=-1, K<1, g(x^ε, y_1^ε, y_2^ε)=x^ε-(x^ε)^3+sin y_1^εcos y_2^ε/50, g_1(x^ε, y_1^ε, y_2^ε)=1/5sin x^ε, g_2(x^ε, y_1^ε, y_2^ε)=-(x ^ε)^2/16, and L_t^α_1, L_t^α_2 are independent two-sided α-stable Lévy motion on ℝ with 1<α<2.

By using the scaling t→ε t, we have

    {[ dx^ε=1/3ε x^εdt+εx^ε-(x^ε)^3+sin y_1^εcos y_2^ε/50dt,;           dy_1^ε=-y_1^εdt+1/5sin x^ε
      dt+σ dL_t^α_1,;            dy_2^ε=-y_2^εdt-1/16 (x^ε)^2dt+σ dL_t^α_2,;                                                       ].

which can be transformed into

    {[ dx̃^ε=1/3εx̃^ε dt+εx̃^ε-(x̃^ε)^3+sin( ỹ_1^ε+σξ_1(θ_tω))cos(ỹ_2^ε+σξ_2(θ_tω))/50dt;                                                   dỹ_1^ε=-ỹ_1^ε dt+1/5sin x̃^ε dt;                                                   dỹ_2^ε=-ỹ_2^ε dt-1/16(x̃^ε)^2dt;                                                                                   ].

where
ξ_i(θ_tω)=∫_-∞^t e^(t-s)dL_s^α_i, i=1, 2.

Denote x̃(0)=x_0, we get

    h^0(ω,x_0)
    = (
      [ 1/5sin x_0; -1/16x^2_0;            ]).

and

    h^1(ω,x_0)
       = (
      [ 1/5cos x_0[x_0^3/50-53x_0/150+∫_-∞^0e^t∫_0^tsin( 1/5sin x_0+σ∫_-∞^se^s-rdL_r^α_1ds)cos(-1/16x^2_0+σ∫_-∞^se^s-rdL_r^α_2)/50dsdt];   -1/8x_0[x_0^3/50-53x_0/150+∫_-∞^0e^t∫_0^tsin( 1/5sin x_0+σ∫_-∞^se^s-rdL_r^α_1ds)cos(-1/16 x^2_0+σ∫_-∞^se^s-rdL_r^α_2)/50dsdt];                                                                                                                                 ]).

 Then we obtain an approximated random slow manifold ℳ̃(ω)={(x_0,h̃(ω,x_0): x_0∈ℝ^2} of system (<ref>), where h̃(ω,x_0)=(h̃_1(ω,x_0),h̃_2(ω,x_0))=h^0(ω,x_0)+ε h^1(ω,x_0)+𝒪(ε^2). Moreover,
 
    {[ dx̅^ε=1/3εx̅^εdt+εx̅^ε-(x̅^ε)^3+siny̅_1^εcosy̅_2^ε/50dt,;                                  y̅_1^ε=h̃_1(θ_tω,x̅^ε),;                                  y̅_2^ε=h̃_2(θ_tω,x̅^ε),;                                                          ].

is the reduction system of (<ref>).
 






Acknowledgements. The authors are grateful to Björn Schmalfuß, René Schilling, Georg Gottwald, Jicheng
Liu and Jinlong Wei for helpful discussions on stochastic differenial equations driven by Lévy motions.





 







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