Shadowing orbits of stochastic differential equations

Volume 9, Issue 5, pp 2006--2018 http://dx.doi.org/10.22436/jnsa.009.05.07

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1700 Downloads
• 3025 Views

Authors

Qingyi Zhan - College of Computer and Information Science, Fujian Agriculture and Forestry University, Fuzhou, Fujian 350002, P. R. China.

Abstract

This paper is devoted to the existence of a true solution near a numerical approximate solution of stochastic differential equations. We prove a general shadowing theorem for finite time of stochastic differential equations under some suitable conditions and provide an estimate of shadowing distance by computable quantities. The practical use of this theorem is demonstrated in the numerical simulations of chaotic orbits of the stochastic Lorenz system.

Share and Cite

Keywords

• Stochastic differential equations
• random dynamical system
• shadowing
• multiplicative ergodic theorem
• stochastic Lorenz system.

MSC

•  65C20
•  65P20
•  37H10
•  37C50

References

• [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin (2003)


• [2] L. Arnold, B. Schmalfuss , Lyapunov's second method for random dynamical systems, J. Differential Equations, 177 (2001), 235-265.


• [3] J. Duan , An introduction to stochastic dynamics, Cambridge University Press, New York (2015)


• [4] A. Fakhari, A. Golmakani , Shadowing properties of random hyperbolic sets , Internat. J. Math., 23 (2012), 10 pages.


• [5] G. H. Golub, C. F. Van Loan , Matrix computations , Johns Hopkins University Press, Baltimore (2013)


• [6] J. Hong, R. Scherer, L. Wang, Midpoint rule for a linear stochastic oscillator with additive noise , Neural Parallel Sci. Comput., 14 (2006), 1-12.


• [7] L. V. Kantorovich, G. P. Akilov, Functional analysis, Pergamon Press, Oxford (1982)


• [8] H. Keller , Attractors and bifurcations of stochastic Lorenz system, in ''Technical Report 389'',Institut fur Dynamische Systeme, Universitat Bremen (1996)


• [9] Y. Li, Z. Brzeźniak, J. Zhou , Conceptual analysis and random attractor for dissipative random dynamical systems, Acta Math. Sci. Ser. B Engl. Ed., 28 (2008), 253-268.


• [10] G. N. Milstein , Numerical integration of stochastic differential equations, Kluwer Academic Publishers, Dordrecht (1995)


• [11] K. J. Palmer, Shadowing in dynamical systems, Theory and applications, Kluwer Academic Publishers, Dordrecht (2000)


• [12] H. J. Sussmann, An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point , Bull. Amer. Math. Soc., 83 (1977), 296-298.


• [13] T. Wang , Maximum error bound of a linearized difference scheme for coupled nonlinear Schrodinger equation, J. Comput. Appl. Math., 235 (2011), 4237-4250.


• [14] X. Xie, Q. Zhan , Uniqueness of limit cycles for a class of cubic system with an invariant straight line , Nonlinear Anal., 70 (2009), 4217-4225.