This paper is devoted to a new numerical approach for the possibility of
The investigation of random periodic orbits at large and in specific stochastic differential equations (SDEs) is a difficult dynamical problem [
Due to the sensitivity of the initial value and random noise pumped into the systems constantly, it is difficult to expect that a particular solution of chaotic systems of SDEs can be well approximated by a numerical solution for any given length of time. Therefore, it is always difficult to infer rigorously the existence of a random periodic orbit from the numerical computations. Shadowing property has an important position in theory and application of random dynamical systems (RDS), especially in the numerical simulations of chaotic systems of SDEs. We present here a new method for establishing the existence of a true random periodic orbit of SDEs which lies near a computed random periodic orbit.
In this work the main motivations are twofold. On the one hand, it follows from the classical results about random periodic solutions of SDEs [
For example, the conditions which can assure the stochastic shadowing in a class of SDEs have been constructed in [
It is well known that Lipschitz shadowing is the extension of classical shadowing, which is widely used in numerical analysis and computation. Therefore it is very meaningful to extend this definition to the stochastic periodic case, which is defined as
Utilizing forward infinite horizon stochastic integral equations, we propose the finitetime random periodic Lipschitz shadowing theorem of SDEs. By random Romberg algorithm and random numerical computation, the shadowing distance is obtained. These results show that under some appropriate conditions the numerical approximative random periodic orbits of SDEs are close to the true ones and shadowing distance can be well estimated.
A more detailed outline of this paper is as follows. Section
Let
In this paper, we make the following assumptions which are made for the theoretical analysis.
(i) The initial value
(ii) Assume that the function
(iii) The function
(iv) The function
We define
Here the expression
It follows from the conclusions in [
We also utilize the notations as follows.
For any random vector
For a stochastic process
The norm of random matrix is defined in the form of
where
In the continuation the norm
We will extend the definitions of
For a given positive number
For a given positive number
As the
Suppose that
Then there exists a sequence of points
Firstly, for a given constant
This claim is proved by a truncation procedure. As we have done in [
Now we define
Secondly, we only need to prove that a
It follows from the above that the random periodic solution
The difference from [
We show the numerical method in detail, which we use for the approximation of shadowing distance, and this method consists of three steps as follows.
Utilizing the onestep numerical scheme (EulerMaruyama (EM) scheme, Milstein scheme [
It follows from Theorem
In order to make this article selfcontained we outline the numerical method which is used for approximating the random periodic solution. The reader is referred to the paper [
Firstly, we need to obtain the initial value for the sake of the approximation. It follows from (
Secondly, we need to obtain the approximation of the improper integral (
By means of reselecting the corresponding starting time and
Finally, in order to improve the accuracy of the integral, the random Romberg algorithm is applied to (
Let
For any given presupposed error tolerance
It follows from Steps
We can divide the time interval
We will show that the shadowing distance (
Suppose that
First and foremost, it follows from (
By the conclusions (
Secondly, from the expression of
Then the fact that
Assume that we are working in a onedimensional space of real numbers and consider the following stochastic logistic equation:
It follows from Theorem
As shown in [
First and foremost, utilizing Theorem
Random periodic solutions with the starting points
Secondly, we need to verify the existence of random periodic orbits nearby a numerically computed
Utilizing the onestep numerical scheme (EM scheme [
Figure
Existence of RPLSO.
This section will provide numerical experiments to compute the shadowing distance of SDE (
Firstly, in order to show the influence of noise on the
Pseudoorbits for different sizes of noise:
Secondly, we focus on numerically performing the shadowing distance
Table
Summaries of the parameters for stochastic logistic equation.
EM 










0.5  0.3162  1.1800  0.2236  1.0148 
1.0  0.3162  1.2578  0.2236  1.1579 
1.5  0.3162  1.3334  0.2236  1.2097 
2.0  0.3162  1.4070  0.2236  1.2600 
Figure
Zoom in parts showing the results in Table
As can be seen from these numerical results, there is an explicit dependent relationship between the shadowing distance and the local error, and there exists a true random periodic orbit in the appropriate neighbourhood of a given
Finally, to check the convergence of numerical approximations to the shadowing distance, we plot the curves from different starting points at the time
Convergence of the shadowing distance
Finally, conclusions and future work are summarized. In this paper, the main result is
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors would like to express their gratitude to Mr. Jialin Hong for his discussion. This work is supported by the Science Research Projection of the Education Department of Fujian Province, no. JAT160182, and the Natural Science Foundation of Fujian Province, no. 2015J01019.