## The Annals of Probability

### Lyapunov Exponents for Small Random Perturbations of Hamiltonian Systems

#### Abstract

Consider the stochastic nonlinear oscillator equation

$\ddot{x} = -x -x^3 + \varepsilon^2 \beta \dot{x} + \varepsilon \sigma x \dot{W}_t$

with $\beta < 0$ and $\sigma \neq 0$. If $4\beta + \sigma^2 > 0$ then for small enough $\varepsilon > 0$ the system $(x,\dot{x})$ is positive recurrent in ${\bf R}^2 \setminus \{(0,0\}$. Now let $\overline{\lambda}(\varepsilon)$ denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts that

$\overline{\lambda}(\varepsilon) = \varepsilon^{2/3}\overline{\lambda} + O(\varepsilon^{4/3}) \qquad\mbox{as}\ \varepsilon \to 0$

with $\overline{\lambda} > 0$. This result depends crucially on the fact that the system above is a small perturbation of a Hamiltonian system. The method of proof can be applied to a more general class of small perturbations of two-dimensional Hamiltonian systems. The techniques used include (i) an extension of results of Pinsky and Wihstutz for perturbations of nilpotent linear systems, and (ii) a stochastic averaging argument involving

#### Article information

Source
Ann. Probab., Volume 30, Number 1 (2002), 101-134.

Dates
First available in Project Euclid: 29 April 2002

https://projecteuclid.org/euclid.aop/1020107762

Digital Object Identifier
doi:10.1214/aop/1020107762

Mathematical Reviews number (MathSciNet)
MR1894102

Zentralblatt MATH identifier
1037.60053

#### Citation

Baxendale, Peter H.; Goukasian, Levon. Lyapunov Exponents for Small Random Perturbations of Hamiltonian Systems. Ann. Probab. 30 (2002), no. 1, 101--134. doi:10.1214/aop/1020107762. https://projecteuclid.org/euclid.aop/1020107762

#### References

• [1] ARIARATNAM, S. T. and XIE, W. C. (1990). Lyapunov exponent and rotation number of a two-dimensional nilpotent stochastic system. Dynam. Stability Systems 5 1-9.
• [2] ARNOLD, L. (1998). Random Dynamical Systems. Springer, Berlin.
• [3] ARNOLD, L., PAPANICOLAOU, G. and WIHSTUTZ, V. (1986). Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and applications. SIAM J. Appl. Math. 46 427-450.
• [4] ARNOLD, L., SRI NAMACHCHIVAYA, N. and SCHENK-HOPPÉ, K. (1996). Toward an understanding of the stochastic Hopf bifurcation: a case study. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 6 1947-1975.
• [5] AUSLENDER, E. and MIL'SHTEIN, G. (1982). Asymptotic expansions of the Liapunov index for linear stochastic systems with small noise. J. Appl. Math. Mech. 46 277-283.
• [6] BAXENDALE, P. H. (1991). Statistical equilibrium and two-point motion for a stochastic flow of diffeomorphisms. In Spatial Stochastic Processes (K. Alexander and J. Watkins, eds.) 189-218. Birkhäuser, Boston.
• [7] BAXENDALE, P. H. (1994). A stochastic Hopf bifurcation. Probab. Theory Related Fields 99 581-616.
• [8] BAXENDALE, P. H. (1999). Stability along trajectories at a stochastic bifurcation point. In Stochastic Dynamics (H. Crauel and M. Gundlach, eds.) 1-25. Springer, Berlin.
• [9] BAXENDALE, P. H. and GOUKASIAN, L (2001). Lyapunov exponents of nilpotent Itô systems with random coefficients. Stochastic Process. Appl. 95 219-233.
• [10] BAXENDALE, P. H. and STROOCK, D. W. (1988). Large deviations and stochastic flows of diffeomorphisms. Probab. Theory Related Fields 80 169-215.
• [11] FREIDLIN, M. and WEBER, M. (1998). Random perturbations of nonlinear oscillators. Ann. Probab. 26 925-967.
• [12] FREIDLIN, M. and WEBER, M. (1999). A remark on random perturbations of the nonlinear pendulum. Ann. Appl. Probab. 9 611-628.
• [13] FREIDLIN, M. and WENTZELL, A. (1994). Random Perturbations of Hamiltonian Systems. Amer. Math Soc., Providence, RI.
• [14] IMKELLER, P. and LEDERER, C. (1999). An explicit description of the Lyapunov exponents of the noisy damped harmonic oscillator. Dynam. Stability Systems 14 385-405.
• [15] KELLER, H. and OCHS, G. (1999). Numerical approximation of random attractors. In Stochastic Dynamics (H. Crauel and M. Gundlach, eds.) 93-115. Springer, Berlin.
• [16] KHAS'MINSKII, R.(1964). The behavior of a conservative system under the action of slight friction and slight random noise. J. Appl. Math. Mech. 28 1126-1130.
• [17] KHAS'MINSKII, R.(1980). Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen aan den Rijn.
• [18] LIANG, Y. and SRI NAMACHCHIVAYA, N. (1999). P -bifurcations in the noisy Duffing-van der Pol equation. In Stochastic Dynamics (H. Crauel and M. Gundlach, eds.) 49-70. Springer, Berlin.
• [19] MEYN, S. and TWEEDIE, R. (1993). Stability of Markovian processes III: Foster-Lyapunov criteria for continuous time processes. Adv. Appl. Prob. 25 518-548.
• [20] PINSKY, M. and WIHSTUTZ, V. (1988). Lyapunov exponents of nilpotent Itô systems. Stochastics 25 43-57.
• [21] SAN MARTIN, L. and ARNOLD, L. (1986). A control problem related to the Lyapunov spectrum of stochastic flows. Mat. Apl. Comput. 5 31-64.
• [22] SCHENK-HOPPÉ, K. (1996). Bifurcation scenarios of the noisy Duffing-van der Pol oscillator. Nonlinear Dynamics 11 255-274.
• [23] SOWERS, R. (2001). On the tangent flow of a stochastic differential equation with fast drift. Trans. Amer. Math. Soc. 353 1321-1334.
• [24] WEDIG, W. (1991). Lyapunov exponents and invariant measures of equilibria and limit cycles. Lyapunov Exponents. Lecture Notes in Math. 1486 308-321. Springer, Berlin.
• [25] WHITTAKER, E. and WATSON, G. (1927). A Course of Modern Analysis, 4th ed. Cambridge Univ. Press.