Orbital stability index for stochastic systems. (English) Zbl 0970.60067
Suppose a stochastic differential equation has an invariant orbit, which is a closed smooth curve such that every solution starting on this curve remains on the curve. An “orthogonal linearisation” along the orbit of such a system is calculated explicitly. Almost sure and \(p\)th mean Lyapunov exponents, and the stability index (the second zero of the \(p\)th mean exponent) for the orthogonal linearisation are introduced, and their properties are stated. Then large deviations estimates for the convergence (or divergence, resp.) of solutions of the original nonlinear equation to the invariant orbit are derived, where the rate function’s exponent is the stability index. This generalises results known for equilibrium points of stochastic differential equations. An additional problem arises here from the fact that nondegeneracy conditions need not be satisfied, e.g., if the invariant orbit is a periodic solution without diffusion. For systems on the plane explicit calculations of the Lyapunov exponents are given, and this is applied to obtain expansions of the exponents for planar systems with small noise.
Reviewer: Hans Crauel (Ilmenau)
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 93E15 | Stochastic stability in control theory |
References:
| [1] | DOI: 10.1137/0144057 · Zbl 0561.93063 · doi:10.1137/0144057 |
| [2] | Arnold, L. and Crauel, H. Lyapunov exponents Proc. Proc., Oberwolfach. 1990. Edited by: Eckmann, J.P. Vol. 1486, Berlin: Springer. Lecture Notes in Math |
| [3] | Arnold, L. and Khasminskii, R.Z. Stability index for nonlinear stochastic differential equations. Proc.of Symposia in Pure Math. Vol. 57, pp.543–551. · Zbl 0827.93065 |
| [4] | Arnold L., Lecture Notes in Math 1186 pp 129– (1986) |
| [5] | DOI: 10.1016/0021-8928(82)90099-5 · Zbl 0522.34053 · doi:10.1016/0021-8928(82)90099-5 |
| [6] | Baxendale, P. Moment stability and large deviations for linear stochastic differential equations. Proc. Taniguchi Symp. on Probab. Meth. in Math. Physics. 1985, Katata and Kyoto. Edited by: Ikeda, N. pp.31–54. Tokyo: Kinokuniya. |
| [7] | Ikeda N., Stochastic differential equations and diffusion processes (1981) · Zbl 0495.60005 |
| [8] | DOI: 10.1137/1112019 · doi:10.1137/1112019 |
| [9] | Khasminskii R.Z., Stochastic stability of differential equations (1969) |
| [10] | Khasminskii R., SIAM J. on Appl. Math 158 (1998) |
| [11] | Milstein G.N., Random & Comp.Dyn 4 pp 301– (1996) |
| [12] | Milstein G.N., Random & Comp.Dyn 5 (1997) |
| [13] | DOI: 10.1137/0148024 · Zbl 0641.60065 · doi:10.1137/0148024 |
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