Hostname: page-component-5c6d5d7d68-wbk2r Total loading time: 0 Render date: 2024-08-10T01:07:29.679Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic non-linear oscillators

Part of: Stochastic analysis Equations and systems with randomness

Published online by Cambridge University Press:  01 July 2016

Lawrence Markus  and
Ananda Weerasinghe
Lawrence Markus*
Affiliation:
University of Minnesota
Ananda Weerasinghe*
Affiliation:
Iowa State University
*
∗ Postal address: Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.
∗∗ Postal address: Department of Mathematics, Iowa State University, Ames, IA 50011, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Non-linear stochastic systems driven by white noise are analysed from the viewpoint of non-linear oscillation theory. Under various familiar hypotheses concerning dissipative and restorative dynamical forces, the existence and uniqueness, asymptotic growth, and oscillatory behavior of the solutions are demonstrated.

Keywords

STOCHASTIC DIFFERENTIAL EQUATIONSOSCILLATIONS

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 60H20: Stochastic integral equations 34F05: Equations and systems with randomness
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 649 - 666
Copyright
Copyright © Applied Probability Trust 1993 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Research partially supported by NSF grants DMS 88–02792 and DMS 88–03639.

References

Hale, J. (1963) Oscillations in Non-linear Systems. McGraw-Hill, New York.Google Scholar
Ikeda, N. and Watanabe, S. (1981) Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.Google Scholar
Khasminskii, R. Z. (1980) Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands.Google Scholar
Markus, L. and Weerasinghe, A. (1988) Stochastic oscillators. J. Differential Eqns 71, 288–314.Google Scholar
Markus, L. and Weerasinghe, A. (1989) Oscillation and energy bounds for non-linear dissipative stochastic differential systems. Math. Report 88-119, University of Minnesota.Google Scholar
Mao, X. and Markus, L. (1991) Energy bounds for nonlinear dissipative stochastic differential equations with respect to semimartingales. Stoch. Stoch. Rep. 37, 1–14.CrossRefGoogle Scholar
Mckean, H. P. (1963) A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2, 227–235.Google Scholar
Mckean, H. P. (1969) Stochastic Integrals. Academic Press, New York.Google Scholar
Narita, K. (1982a) Non-explosion criteria for stochastic differential equations. J. Math. Soc. Japan 34, 191–203.Google Scholar
Narita, K. (1982b) A priori estimate and asymptotic behavior of solutions of stochastic differential equations. Yokohama Math. J. 30, 91–101.Google Scholar
Narita, K. (1984) Explosion time of second order Itô processes. J. Math. Anal. Appl. 104, 418–427.Google Scholar
Narita, K. (1989) Stochastic Liénard equation with mean field interaction. SIAM J. Appl. Math. 49, 888–905.Google Scholar