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Numerical Random Periodic Shadowing Orbits of a Class of Stochastic Differential Equations

Written By

Qingyi Zhan and Yuhong Li

Submitted: May 13th, 2016 Reviewed: November 22nd, 2016 Published: March 15th, 2017

DOI: 10.5772/67010

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Abstract

This paper is devoted to the existence of a true random periodic solution near the numerical approximate one for a kind of stochastic differential equations. A general finite-time random periodic shadowing theorem is proposed for the random dynamical systems generated by some stochastic differential equations under appropriate conditions and an estimate of shadowing distance via computable quantities is given. Application is demonstrated in the numerical simulations of random periodic orbits of the stochastic Lorenz system for certain given parameters.

Keywords

  • random chaotic system
  • stochastic differential equations
  • random periodic shadowing
  • stochastic Lorenz system

1. Introduction

The investigation for the dynamical properties of the random periodic orbits in some specific stochastic differential equations (SDEs) is a difficult problem [1]. In general, numerical computation is still one of the most feasible methods of studying random periodic orbits of SDEs describing many natural phenomena in meteorology, biology and so on [24]. As the chaotic systems is sensitive to the initial value and random noise is constantly affected the systems constantly, to estimate a particular solution of a random chaotic system by numerical solutions for a given length of time is even more difficult. Therefore, it is always difficult to infer the existence of a random periodic orbit rigorously from numerical computations. Shadowing property plays important roles in the theory and applications of random dynamical systems (RDS), especially in the numerical simulations of random chaotic systems generated by some SDEs. As we know, numerical experiments can lead to many nice discoveries, a new numerical method is presented to establish the existence of a true random periodic orbit of SDEs which lies near a numerical random periodic orbit. Furthermore, the reliability and feasibility of numerical computations is considered as well.

There are two main motivations for this work. On the one hand, it follows from the classical shadowing lemma that many studies about the periodic dynamics of deterministic chaotic systems have been performed in Ref. [3] and references therein. Many nice works on the numerical analysis of RDS had been completed in Refs. [5] and [6]. On the other hand, our results in this article have been inspired by our earlier work in Refs. [7] and [8], on shadowing orbits of SDEs where we established in a rather general setting. To the best of our knowledge, shadowing is still an interesting method for studying their random periodic dynamic behavior of SDE, and there is no investigations of the random periodic shadowing theorem of SDE exist in the literatures.

In this work, two computational issues should be considered first. One is the definition of (ω, δ)-pseudo random periodic orbit, in which a true random periodic orbit is sufficiently closed. Another issue is that in which conditions the random chaotic systems generated by some SDE possess the so-called pseudo hyperbolicity for certain given parameters. With some additional numerical computations, we can show the existence of a true random periodic orbit near the (ω, δ)-pseudo random periodic orbit under appropriate conditions. Therefore, the main difference between the existing work and the current one is that the random periodic case is concerned, and there is no hyperbolicity assumption on the original systems.

Utilizing the existence of the modified Newton equation’s solution, a random periodic shadowing theorem for some kind of SDEs is proposed. The result shows that under some appropriate conditions, there exists a true periodic orbit near the numerical approximative one and the upper bound for the shadowing distance is given.

This paper is organized as follows. In Section 2, background materials on random shadowing for random dynamical system generated by SDEs, including the definitions of (ω, δ)-pseudo random periodic orbit and the pseudo hyperbolic in mean square, are given. The main result on random periodic shadowing is then stated in Section 3. Illustrative numerical experiments for the main theorem are included in Section 4. The numerical implementations in details are presented in the following section. And, the proof for the main result is presented in Section 6. The final section is devoted to summarize the main results in the current work.

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2. Preliminaries

Let (Ω,F,P)be a canonical Wiener space, {Ft}tRbe its natural normal filtration, and W(t)(tR)is a standard one-dimensional Brownian motion defined on the space (Ω,F,P). And, we assume that Ω:={ωC(R,R):ω(0)=0}, which means that the elements of Ω can be identified with paths of a Wiener process ω(t)=Wt(ω). We consider a class of Stratonovich SDEs in the form of

dxt=f(xt)dt+μxtdWt,x(0)=x0(ω)Rd,E1

where the random variable x0(ω)is independent of F0and satisfies the inequality E|x0(ω)|2<, and μis a nonzero real number.

2.1. Basic assumptions and notations

We define the metric dynamical systems (Ω,F,P,θt)by the mapping θ:R×ΩΩ, such that for ωΩ,

θtω(s)=ω(t+s)ω(t),

where s,tR.

Let Ot(ω)be a one-dimension random stable Ornstein-Uhlenbeck process which satisfies the following linear SDE

dOt=Otdt+dWt.

And let

z(t,ω):=exp(μOt(ω))xt(ω)Rd,

then SDE (1) can be changed to a random differential equation (RDE) in the form of

dzdt=exp(μOt(ω))f(exp(μOt(ω))z)+μOtz=f1(θtω,z).E2

It follows from Doss-Sussmann Theorem in Ref. [9] that the solution of RDE (2) is the solution of SDE (1).

In this paper, we make the following assumptions:

  • We suppose that f1:Ω×RdRdbe a measurable function which is locally bounded, locally Lipschitz continuous with respect to the first variable, and be a C1 vector field on Rd.

    By Theorem 2.2.2 in Ref. [2], RDE(2) generates a unique RDS φon the metric dynamical systems (Ω,F,P,θt)as follows

    φ(s,t,ω)z=z+stf1(θτω,φ(s,τ,ω)z)dτRd,E3

    and which is C1-class with respect to zin Ref. [8].

    And there exists a diffeomorphism φ:R×R×Ω×RdRd, φ(s,t,ω,z):=φ(s,t,ω)zRd.

    We also make use of the following notations which is similar to the Ref. [8].

  • The norm of a random variable x=(x1,x2,…,xd)L2(Ω,P)is defined in the form of

    x2=[Ω[|x1(ω)|2+|x2(ω)|2+,…,+|xd(ω)|2]dP(ω)]12<,

    where L2(Ω,P)is the space of all square-integrable random variables x:ΩRd.

  • The norm of a stochastic process x(t,ω)with xt(ω)L2(Ω,P)and tRis defined as

    x(t,ω)2=suptRxt(ω)2<.

  • For a given random matrix A, and the operator norm | ⋅ |, the norm of Ais defined as follows

    AL2(Ω,P)=[E(|A|2)]12.

  • Normally, the norm 2and L2(Ω,P)are denoted as for simplicity reason, unless otherwise stated.

2.2. Some extended definitions

Definition 2.1.For a given positive number δ, if there is a sequence of positive times {tk}k=0N+1,0t0t1,…,τtN+1,τ, and a sequence of random variables

{(yk(θtkω),Ftk)}k=0N,

yk(θtkω)is Ftk-adapted, such that

f1(yk(θtkω))yk(θtkω)0,P-almostsurelyfork=0,1,2,…,N,

and the following inequalities P-almost surely hold

yk+1(θtk+1ω)φ(tk,tk+1,θtkω)yk(θtkω)]δ,k=0,1,…,N1,

and

yN(θtNω)y0(θt0ω)δ,E4

then the random variables {(yk(θtkω),Ftk)}k=0Nare said to be a (ω, δ)-pseudo random periodic orbit of RDS (3) generated by SDE (1) in mean-square sense.

Definition 2.2.For a given positive number εand a (ω, δ)-pseudo random periodic orbit {(yk(θtkω),Ftk)}k=0Nof RDS (3) generated by SDE (1) with associated times {tk}k=0N+1, if there is a sequence of times {hk}k=0N+1,h0h1,…,τhN+1,such that the following inequalities hold

yk(θtkω)xk(θhkω)ε,0tkhkε,k=0,1,…,N,

and the random variables {(xk(θhkω),Fhk)}k=0Nare on the true orbits of RDS (3) generated by SDE (1), that is

xk+1(θhk+1ω)=φ(hk,hk+1,θhkω)xk(θhkω),k=0,1,2,…,N1,

and

x0(θh0ω)=φ(hN,hN+1,θhNω)xN(θhNω),E5

then the (ω, δ)-pseudo random periodic orbit {(yk(θtkω),Ftk)}k=0Nis said to be (ω, δ)-periodic shadowed by a true orbit of RDS (3) generated by SDE (1) in mean-square sense.

Remark 2.3.As the σ-algebra Ft(t0)is nondecreasing, in order to guarantee the random variables xk(θhkω)(k=0,1,2,…,N)are Ftk-measurable, we need the shadowing condition 0tkhkεinstead of |tkhk|ε. We refer to the Ref. [2] for the deterministic counterpart. Here, we choose a sequence of times {hk}k=0N+1={tk}k=0N+1in sequels.

Definition 2.4.The RDS φ:R×R×Ω×RdRdis said to be pseudo hyperbolic in mean square if the temple variables κ1(ω),κ2(ω)1, ν1(ω),ν2(ω)0exist, such that the following inequations hold with Rd=Es(ω)Eu(ω),

Eφ(s,t1,ω)x2κ1(ω)eν1(ω)(t1t2)Eφ(s,t2,ω)x2,t1t2s,xEs(ω),Eφ(s,t2,ω)x2κ2(ω)eν2(ω)(t1t2)Eφ(s,t1,ω)x2,t1t2s,xEu(ω).

This means that there is a splitting into exponentially stable (Es(ω))and unstable (Eu(ω))components. The multiplicative ergodic theorem (MET) of Oseledets in [10] provides the stochastic analogue of the deterministic spectral theory of matrices, and a method to check the pseudo hyperbolicity.

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3. Random periodic shadowing for RDS generated by SDEs

3.1. Theoretical foundations

Let {(yk(θtkω),Ftk)}k=0Nbe a (ω, δ)-pseudo random periodic orbit of RDS (3) generated by SDE (1) and yk(θhkω)L2(Ω,P)(k=0,1,…,N). Assume that we have a sequence of d×drandom matrices {(Yk(θtkω),Ftk)}k=0Nsuch that

Yk+1(θtk+1ω)Dφ(tk,tk+1,θtkω)yk(θtkω)δ,fork=0,1,…,N1,

and

Y0(θt0ω)Dφ(tN,tN+1,θtNω)yN(θtNω)δ.E6

A sequence of d×(d− 1) random matrices (Sk(θtkω),Ftk)are chosen such that its columns form an approximate orthogonal basis for the subspace orthogonal to T(xk)and k= 0, 1, …, N, where T(xk)=f1(θtkω,xk), the approximate orthogonal means that the following inequality holds

Sk(θtkω)Sk*(θtkω)Iδ1,

for some positive number δ1(0,δ), where * denotes the transpose of matrix.

Now a sequence of (d− 1)×(d− 1) random matrices Ak(θtkω)is chosen which satisfy

Ak(θtkω)Sk+1*(θtk+1ω)Yk(θtkω)Sk(θtkω)δ,fork=0,1,…N1,

and

AN(θtNω)S0*(θt0ω)YN(θtNω)SN(θtNω)δ.

Next, a linear operator Lis defined as follows. If random variables ξ={ξk(θtkω)}k=0Nare in the space (Rd1)N+1, then we let Lξ={[Lξ]k}k=0Nto be

[Lξ]k=ξk+1(θtk+1ω)Ak(θtkω)ξk(θtkω),fork=0,1,…,N1.

and

[Lξ]N=ξ0(θt0ω)AN(θtNω)ξN(θtkω).

It follows from Section 4.2 that the operator Lhas right inverses and we choose one such right inverse L−1.

At last, we define some constants. Let Ube a convex subset of Rdcontaining the value of the (ω, δ)-pseudo orbit {(yk(θtkω),Ftk)}k=0N. Therefore, we define

Δt=inf0kNΔtk+1=inf0kN(tk+1tk).

Next, we choose a positive number 0<ε0Δtsuch that xyk(θtkω)ε0, then the solution φ(s,t,ω)x(st)is defined and remains in Ufor 0<ttk+ε0P-almost surely.

Finally, we define

M0=supxUf1(θtω,x(t)),M1=supxUDf1(θtω,x(t)),M2=supxUD2f1(θtω,x(t))

and

Θ=sup0kN1Yk(θtkω),

where

Df1=[f1(θtω,x(t))xi],

We first introduce the following lemma which has been proved in the Ref. [8] and will be applied to the main theorem [11].

Lemma 3.1Let Xand Ybe finite-dimensional random vector spaces of the same dimension, and Bbe an open subset of X. Let v0be a given element of B. Suppose that G:BYbe a C2 function and satisfy:

  1. the derivative DG(v0) of function Gat v0Bis right inverse with K;

  2. Bcontains a closed ball whose center is v0 and radius is ε¯, where ε¯=2KG(v0);

  3. the inequality 2MK2G(v0)1holds, where

    M=sup{D2G(v):vB,vv0ε¯};

Then, there is a solution v¯of the equation G(v¯)=0satisfying v¯v0ε¯.

3.2. Main results

Now, we state the main theorem and postponed its proof in the latter section.

Theorem 3.2.For a given bounded (ω, δ)-pseudo random periodic orbit of RDS (3) generated by SDE (1) {(yk(θtkω),Ftk)}k=0N, assume that

C:=max{M01(1+ΘL1),L1}.E7

If the quantities shown in Section 3.1 together with δand ε0 satisfy:

  1. C1=Cδ<12;

  2. C2=4Cδ<ε0;

  3. C3=8C2δ(M0M1+2M1exp(M1Δt)+M2Δtexp(2M1Δt))1;

Then there exists a sequence of times {hk}k=0N+1(h0h1,…,hN+1tN+1)such that the (ω, δ)-pseudo random periodic orbit {(yk(θtkω),Ftk)}k=0Nis (ω, δ)-periodic shadowed by a true random periodic orbit of SDE (1) containing points {(xk(θhkω),Fhk)}k=0Nin mean-square. Moreover, shadowing distance satisfies ε4Cδ.

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4. Numerical experiments

Here, we apply the random periodic shadowing theorem to rigorously establish the existence of random periodic orbits of the stochastic Lorenz equation. And, this section will provide numerical experiments to compute the shadowing distance.

4.1. Experimental preparation

Consider the following Stratonovich stochastic Lorenz equation (SSLE) in R3,

dXt=f(Xt)dt+μXtdWt(ω),X(0)=x0R3E8

where Xt=(x,y,z)TR3, x, yand zare the state variables, σ, ρand βare positive constant parameters, and

f(Xt)=(σx+σyρxyxzβz+xy),μXt=(μxμyμz).

Make the transformation as follows:

{x¯(t,ω)=exp(μOt(ω))xy¯(t,ω)=exp(μOt(ω))yz¯(t,ω)=exp(μOt(ω))z,

It follows from the transformation that the above SSLE (8) can be transformed to the random differential equation (RDE) in the following form

{dx¯dt=σ(x¯+y¯)+μOt(ω)x¯dy¯dt=x¯z¯+ρx¯y¯+μOt(ω)y¯dz¯dt=x¯y¯βz¯+μOt(ω)z¯.E9

The existence and uniqueness of solution of RDE (9) can be proved by the same approaches as proposed in the Refs. [2] and [12] though a normally required linear growth condition does not be satisfied. Hence, a RDS φcan be generated by the solution operator of RDE (9).

In this experiment, it appears numerically that the stochastic Lorenz equations have asymptotically stable random periodic orbit for the parameter values σ=10,ρ=100.5,β=83.

Firstly, we generate Brownian trajectories in the following way

W0=0,W(i+1)Δt=WiΔt+ψi+1

where

ψi=N(0,Δt),i=1,2,…,N

Secondly, it follows from the reference [13] that a global attractor, i.e., a forward invariant random compact set Uof RDS φgenerated by RDE (9) is the closed ball B1with center zero and radius R(ω), that is, B1={XtR3:XtR(ω)}, where

R(ω)=c2tN0exp(c1s2σWs(ω))ds

and

c1=min(1,β,σ),c2>0,2BXt,Xt<c1|Xt|2+c2,B=(σσ0ρ1000β).

It has been proved in Ref. [13] that the RDS φgenerated by Eq. (8) lies in the forward invariant random compact set Ufor P-almost surely ωΩon the finite interval.

4.2. Numerical results

We first present the results of our computations of the (ω, δ)-pseudo random periodic orbits for the stochastic Lorenz equation. To generate a good (ω, δ)-pseudo random periodic orbit, we numerically computed the orbit for some time with a rough guess of initial value. In this experiment, we take the initial value (x0, y0, z0) = (1.76, −4.48, 80.99), time step size Δt= 0.00007 and iterative step N= 100000. The (ω, δ)-pseudo random periodic orbits of Eq. (9) in Figure 1 are generated by the Euler-Maruyama scheme in Ref. [14] and the refined initial data. This also shows that there exists a forward invariant random compact set.

Figure 1.

(ω,δ)-pseudo random periodic orbits of SLS.

Secondly, we briefly describe the details of the computation of the key quantities listed in Table 2. It follows from the methods shown in Section 3, and we can determine the parameters of Theorem 3.2. Tables 1 and 2 present the important quantities and the necessary inequalities pertaining to this (ω, δ)-pseudo random periodic orbit.

ParametersValueParametersValue
Δt0.00007ε02.01
X0(1.76, −4.48, 80.99)M0≤ 9.8037
N105M1≤ 0.0185
Approx. periodτ= 0.1837M20.0014
X2623(−0.6911, −7.7293, 81.6553)Θ≤ 1.0013
X2623X04.1241δ≤ 4.1265
L−1 ≤ 4.8218e− 03

Table 1.

Value of the parameters.

InequalitiesValue
C≤ 0.1025
C1≤ 0.4229
C2≤ 1.6918
C3≤ 0.0757
Shadowing distance ε1.2688
Shadowing time t70

Table 2.

Comparison of the inequalities.

In conclusion, there is explicit dependent relationship between the shadowing distance and the pseudo orbit error, and there exists the true periodic orbit in the appropriate neighborhood of the (ω, δ)-pseudo random periodic orbit of SLS (Figure 2). Figures 3a and 3b demonstrate the relation between (ω, δ)-pseudo random periodic orbits and true periodic orbits of Eq. (8). The blue lines denote (ω, δ)-pseudo random periodic orbit for the random dynamical system, and the domain between two blue lines has at least a true orbit for the corresponding random dynamical system.

Figure 2.

The distance ∥XnX0∥.

Figure 3.

(a) The symbolic drawing of the relation between true orbit and pseudo orbit plane. (b) The approximative structure of pseudo random periodic solution projected on thezplane.

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5. Choice of the operator L−1

We are going to verify that the linear operator Lalong the obtained (ω, δ)-pseudo random periodic orbit {(yk(θtkω),Ftk)}k=0Nis invertible for P-almost surely ωΩ.

Let g={gk(θtkω)}k=0Nbe in Y. To find ξ=L1g, we have to solve the random difference equation

ξk+1(θtk+1ω)=Ak(θtkω)ξk(θtkω)+gk(θtkω),fork=0,…N1,ξ0(θt0ω)=AN(θtNω)ξN(θtNω)+gN(θtNω).

With the same choice of the parameters as Section 3, it can be shown that random matrix Ak(θtkω)is upper triangular with positive diagonal entries. Therefore, there is an integer lsuch that for most k, the first ldiagonal entries of Ak(θtkω)exceed 1 and the rest are less than 1 in mean square for P-almost surely ωΩ[15]. We can partition the random matrix Ak(θtkω)in the form

Ak(θtkω)=[Pk(θtkω)Qk(θtkω)0Rk(θtkω)],k=0,1,…,N,

where Pk(θtkω)is l× lrandom matrix,Qk(θtkω)is l× (dl−1) random matrix, and Rk(θtkω)is (dl−1) × (dl−1) random matrix.

It follows from multiplicative ergodic theorem that the Lyapunov exponents of Ak(θtkω)are nonzero. Then it suggests that the RDS φgenerated by SDE (1) along the obtained (ω, δ)-pseudo orbit {(yk(θtkω),Ftk)}k=0Nis pseudo hyperbolicity in mean square for P-almost surely ωΩ. It can be written as

{ξk+1(1)(θtk+1ω)=Pk(θtkω)ξk(1)(θtkω)+Qk(θtkω)ξk(2)(θtkω)+gk(1)(θtkω)ξk+1(2)(θtk+1ω)=Rk(θtkω)ξk(2)(θtkω)+gk(2)(θtkω)

for k= 0, 1, …, N− 1, and

{ξ0(1)(θt0ω)=PN(θtNω)ξN(1)+QN(θtNω)ξN(2)(θtNω)+gN(1)(θtNω)ξ0(2)(θt0ω)=RN(θtNω)ξN(2)(θtNω)+gN(2)(θtNω)

Let ξ0(2)(θt0ω)=0solve forwards the second equation of the first equations above. The substitute it into the first equation with ξk(2)(θtkω), and let ξN(2)(θtNω)=0, then solve it backwards. Finally, the solutions ξk(1)(θtkω)are obtained. Therefore, the right inverse L−1 is obtained as

[L1g]k=[ξk(1)(θtkω),ξk(2)(θtkω)]T,k=0,1,…,N.

Hence, invertibility of the operator Lis proved, which is an important for the application of the random shadowing lemma.

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6. Proof of the main theorem

Proof. For a given (ω, δ)-pseudo random periodic orbit {(yk(θtkω),Ftk)}k=0Nof RDS φ(3) generated by SDE (1), and an associated sequence of d×drandom matrices {Yk(θtkω)}k=0Nsatisfying Eq. (6). Our aim is to show that {(yk(θtkω),Ftk)}k=0Nis (ω, δ)-periodic shadowed by a true random periodic orbit containing {(xk(θhkω),Fhk)}k=0N, where xk(θhkω)lies in the random hyperplane Hk(θtkω)through yk(θtkω).

Suppose that the random hyperplane Hk(θtkω)is approximately normal to T(yk)=f1(θtkω,yk)at the point yk(θtkω). Therefore, we only need to find a sequence of times {hk}k=0N+1={tk}k=0N+1,h0h1,…,hN+1tN+1and a sequence of points {(xk(θhkω),FtN)}k=0Nwith xk(θhkω)Hk(θtkω)being contained in the ε-neighborhood of yk(θtkω)such that

xk+1(θhk+1ω)=φ(hk,hk+1,θhkω)xk(θhkω),fork=0,1,…,N1,

and

x0(θh0ω)=φ(hN,hN+1,θhNω)xN(θhNω).

By the assumption, we obtain that Sk(θtkω)is a d×(d− 1) random matrix whose columns form an approximative orthogonal basis for Hk(θtkω). We first define the random hyperplane Hk(θtkω)as the image of Rd1through the map zyk(θtkω)+Sk(θtkω)z, which can be viewed as a subspace of the tangent space at yk(θtkω).

Therefore, the problem of finding appropriate sequences of hkand xkbecomes that of finding a sequence of times {hk}k=0N+1:={tk}k=0N+1and a sequence of points {(zk(θhkω),FtN)}k=0Nin Rd1such that

yk+1(θtk+1ω)+Sk+1(θtk+1ω)zk+1(θhk+1ω)=φ(hk,hk+1,θhkω)(yk(θtkω)+Sk(θtkω)zk(θhkω)),k=0,1,…,N1,

and

y0(θt0ω)+S0(θt0ω)z0(θh0ω)=φ(hN,hN+1,θhNω)(yN(θtNω)+SN(θtNω)zN(θhNω)).

We next introduce the space X=RN+2×(Rd1)N+1with norm

({sk}k=0N+1,{ζk}k=0N)=max{sup0kN+1|sk|,sup0kNζk},

and the space Y=(Rd)N+1with norm

{gk}k=0N=max0kNgk,

where skR, ζkRd1and gkRd.

Now, we let Bbe a properly chosen ε-open neighborhood of v0=({tk}k=0N+1,0)in Xwhich contain the point v=({sk}k=0N+1,{ζk}k=0N). And, we introduce the function G:BYgiven by

[G(v)]k=yk+1(θtk+1ω)+Sk+1(θtk+1ω)ζk+1(θsk+1ω)φ(sk,sk+1,θskω)(yk(θtkω)+Sk(θtkω)ζk(θskω)),fork=0,1,…,N1,

and

[G(v)]N=y0(θt0ω)+S0(θt0ω)ζ0(θs0ω)φ(sN,sN+1,θsNω)(yk(θtNω)+SN(θtNω)ζN(θsNω)).E10

It is the fact that Theorem 3.2 will be proved if we can find a solution v¯=({hk}k=0N+1,{zk(θhkω)}k=0N)of the equation

G(v¯)=0,a.s.

in the closed ball of radius εabout v0=({tk}k=0N+1,0).

In order to apply Lemma 3.1, those hypotheses (i) – (iii) for the map Gas Eq. (10) should be verified.

Step I:

First and foremost, it follows from the construction of pseudo orbits that G(v0)δ.Secondly, the Gateaux derivative of the map Gat v0 with u=({τk}k=0N+1,{ξk(θtkω)}k=0N)Xis given by

[DG(v0)u]k=limε0[G(v0+εu)G(v0)]kε=τkT(yk+1)+Sk+1(θtk+1ω)ξk+1(θtk+1ω)Dφ(tk,tk+1,θtkω)yk(θtkω)Sk(θtkω)ξk(θtkω),

for k= 0, 1, …, N− 1, and

[DG(v0)u]N=τNT(yN)+S0(θt0ω)ξ0(θt0ω)Dφ(tN,tN+1,θtNω)yN(θtNω)SN(θtNω)ξN(θtNω).E11

We will approximate DG(v0) by another operator. Now, we define the operator T:XYfor uX. Let Tkube the approximation of [DG(v0)u]kin Ref. [16], we have

Tku=τkT(yk+1)+Sk+1(θtk+1ω)ξk+1(θtk+1ω)Yk(θtkω)Sk(θtkω)ξk(θtkω),k=0,1,…,N1,

and

TNu=τNT(yN)+S0(θt0ω)ξ0(θt0ω)YN(θtNω)SN(θtNω)ξN(θtNω).E12

Now, we need to prove that Tis invertible. Therefore, we must show that for all g={gk}k=0NY, there is a solution of the following equation

Tku=gk,

that is, for k= 0, 1, …, N−1,

τkT(yk+1)+Sk+1(θtk+1ω)ξk+1(θtk+1ω)Yk(θtkω)Sk(θtkω)ξk(θtkω)=gk(θtkω),

and

τNT(yN)+S0(θt0ω)ξ0(θt0ω)YN(θtNω)SN(θtNω)ξN(θtNω)=gN(θtNω).E13

As we know, the matrix

[T(yk)T(yk)|Sk(θtkω)]

is orthogonal for each k. Then this set of equations is equivalent to the following two sets of equations, one set obtained by premultiplying the kth member in Eq. (13) by T*(yk+1)and T*(y0), respectively, the other set obtained by premultiplying the kth member in Eq. (13) by Sk+1*(θtk+1ω)and S0*(θt0ω), respectively. Therefore, we obtain for k= 0, 1, …, N−1,

τkT(yk+1)2T(yk+1)*Yk(θtkω)Sk(θtkω)ξk(θtkω)=T(yk+1)*gk(θtkω),

and

τNT(y0)2T(y0)*YN(θtNω)SN(θtNω)ξN(θtNω)=T(y0)*gN(θtNω),E14
ξk+1(θtk+1ω)Ak(θtkω)ξk(θtkω)=Sk*(θtk+1ω)gk(θtkω),k=0,1,…,N1,

and

ξ0(θt0ω)AN(θtNω)ξN(θtNω)=SN*(θt0ω)gN(θtNω).E15

If we write g¯={Sk*(θtkω)gk(θtkω)}k=0N, it follows from the condition (7) that the solution of Eq. (15) is

ξk=(L1g¯)k.E16

If Eq. (16) is substituted into Eq. (14), we obtain for k= 0, 1, …, N−1,

τk=T(yk+1)*T(yk+1)2[Yk(θtkω)Sk(θtkω)L1Sk+1(θtk+1ω)+1]gk(θtkω),

and

τN=T(y0)*T(y0)2[YN(θtNω)SN(θtNω)L1S0(θt0ω)+1]gN(θtNω).E17

Taking into account Eqs. (16) and (17), we define the right inverse of Tkin the form of

Tk1g=[{τk}k=0N+1,{ξk(θtkω)}k=0N].

It follows from Eq. (17) that Tis invertible and the following inequality holds

T1C.E18

Therefore, we can construct the invertibility of DG(v0). By the operator theory, we obtain

K=[I+T1(DG(v0)T)]1T1.E19

By Eqs. (11) and (12) and the assumption (i) of Theorem 3.2, we obtain that

T1(DG(v0)T)T1DG(v0)TT1||[sup||(Dφ(tk,tk+1,θtwω)yk(θtkω)Yk(θtkω)Sk(θtkω)ξk(θtkω)||]Cδ<12.

Then the inverse [I+T1(DG(v0)T)]1exits and Kis a right inverse of DG(v0). Furthermore,

[I+T1(DG(v0)T)]12.

Therefore, we have verified hypothesis (i) of Lemma 3.1.

Step II:

It follows from Eqs. (18)–20 that we have

K2C.

and

G(v0)=supkyk+1(θtk+1ω)φ(tk,tk+1,θtkω)yk(θtkω)δ.

By the assumption (ii) of Theorem 3.2, we obtain that

ε=2KG(v0)4Cδ<ε0.

That is, the closed ball of radius εaround v0 is contained in the open set B. Therefore, we have verified hypothesis (ii) of Lemma 3.1.

Step III:

We only need to estimate D2G(v). Then we choose u¯=({rk}k=0N+1,{ηk}k=0N)and calculate the second order Gateaux differential of G(v) for k= 0, 1, …, Nas follows

[DG(v)uu¯]k:=limt0[DG(v+tu¯)uDG(v)u]k|t|=τkrkDT[yk(θtkω)+Sk(θtkω)ζk(θtkω)]T[yk(θtkω)+Sk(θtkω)ζk(θtkω)]τkDT[yk(θtkω)+Sk(θtkω)ζk(θtkω)]Dφ(tk,tk+1,θtkω)(yk(θtkω)+Sk(θtkω)ζk(θtkω))Sk(θtkω)ηk(θtkω)rkDT[yk(θtkω)+Sk(θtkω)ζk(θtkω)]Dφ(tk,tk+1,θtkω)(yk(θtkω)+Sk(θtkω)ζk(θtkω))Sk(θtkω)ξk(θtkω)D2φ(tk,tk+1,θtkω)(yk(θtkω)+Sk(θtkω)ζk(θtkω))[Sk(θtkω)ξk(θtkω)][Sk(θtkω)ηk(θtkω)].

By the norm property, i.e., subadditivity, we obtain

M=supkD2G(v)M0M1+2M1exp(M1Δt)+M2Δtexp(2M1Δt).

It follows from the assumption (iii) of Theorem 3.2 and

G(v0)δ,K24C2,

that

2MK2G(v0)1.

Then we have verified hypothesis (iii) of Lemma 3.1. Therefore, the conclusion follows from Lemma 3.1. This finishes the proof.

Remark 6.1The proof is similar to the paper [8], and we extend it to the random periodic case.

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7. Conclusion

The main result presented here is the random periodic shadowing theorem of the RDS generated by some SDEs. To conduct the study, we have extended the random shadowing theorem to the random periodic scenario by taking advantage of mean square and stochastic calculus. We show that the existence of the random periodic shadowing orbits of the SSLE so that the numerical experiments are performed and match the results of theoretical analysis. Although some progresses are made, more accurate numerical methods of estimating the shadowing distance are needed in practice, which will be presented in our further work.

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Acknowledgments

The authors like to express their gratitude to Prof. Jialin Hong for his helpful discussion. This work is partially supported by NSFC (No. 11021101, 11290142, 91130003, 91530118) and the Fundamental Research Funds for the Central Universities, HUST, No. 2016YXMS226 and Young Teacher's Education and Science Research Project in the Education Department of Fujian Province, No.JAT160182.

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Written By

Qingyi Zhan and Yuhong Li

Submitted: May 13th, 2016 Reviewed: November 22nd, 2016 Published: March 15th, 2017