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Stochastic averaging and asymptotic behavior of the stochastic Duffing-van der Pol equation

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  • Baxendale, Peter H.

Abstract

Consider the stochastic Duffing-van der Pol equationwith A[greater-or-equal, slanted]0 and B>0. If [beta]/2+[sigma]2/8[omega]2>0 then for small enough [var epsilon]>0 the system is positive recurrent in R2[-45 degree rule]{0}. Let denote the top Lyapunov exponent for the linearization of this equation along trajectories. The main result asserts thatwhere is the top Lyapunov exponent along trajectories for a stochastic differential equation obtained from the stochastic Duffing-van der Pol equation by stochastic averaging. In the course of proving this result, we develop results on stochastic averaging for stochastic flows, and on the behavior of Lyapunov exponents and invariant measures under stochastic averaging. Using the rotational symmetry of the stochastically averaged system, we develop theoretical and numerical methods for the evaluation of . We see that the sign of , and hence the asymptotic behavior of the stochastic Duffing-van der Pol equation, depends strongly on [omega]B/A. This dimensionless quantity measures the relative strengths of the nonlinear dissipation and the nonlinear restoring force Ax3.

Suggested Citation

  • Baxendale, Peter H., 2004. "Stochastic averaging and asymptotic behavior of the stochastic Duffing-van der Pol equation," Stochastic Processes and their Applications, Elsevier, vol. 113(2), pages 235-272, October.
    • Handle: RePEc:eee:spapps:v:113:y:2004:i:2:p:235-272
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    Cited by:

    1. Zhang, Ruiting & Hou, Zhonghuai & Xin, Houwen, 2011. "Effects of non-Gaussian noise near supercritical Hopf bifurcation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(2), pages 147-153.
    2. Sowers, Richard B., 2009. "Averaging of stochastic flows: Twist maps and escape from resonance," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3549-3582, October.
    3. H. A. Mardones & C. M. Mora, 2020. "First-Order Weak Balanced Schemes for Stochastic Differential Equations," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 833-852, June.

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