Asymptotic expansion of the quasipotential for a stochastically perturbed nonlinear oscillator (Q2703888)

Let \(dx=f(x)dt\) (with \(x(t)\) an \(n\)-vector and \(f\) sufficiently smooth) have an exponentially stable limit cycle \(x=\xi (t)\) (with phase curve \(\Gamma \)). The authors consider a randomly perturbed system \(dx=f(x)dt+\varepsilon \sigma (x)dw(t)\), where \(w\) is an \(n\)-dimensional standard Wiener process, \(\sigma \) is sufficiently smooth, and \(\varepsilon \) is a small scalar specifying the intensity of the perturbations. The paper suggests the use of a quasipotential approach [\textit{A. D. Ventsel'} and \textit{M. I. Freidlin}, Fluctuations in dynamical systems subject to small random perturbations (Russian) (1979; Zbl 0499.60053)] as a way of studying the asymptotic behaviour of the exit of random trajectories from a neighbourhood of \(\Gamma \) and reports results on how to approximate the quasipotential [\textit{G. N. Mil'shtein} and \textit{L. B. Ryashko}, J. Appl. Math. Mech. 59, No. 1, 47-56 (1995; Zbl 0880.34059)]. In the two-dimensional case, the authors suggest a local description of the quasipotential by a risk (scalar) function allowing the distinction between more and less dangerous parts of the orbit under stochastic perturbations of the motion. NEWLINENEWLINENEWLINEThese ideas are applied to a classical nonlinear oscillator NEWLINE\[NEWLINE\begin{aligned} dx_{1}&=x_{2}dt, \\ dx_{2}&=(-x_{1}+\delta f(x_{1},x_{2}))dt+\varepsilon \sigma (x_{1},x_{2})dw(t),\end{aligned}NEWLINE\]NEWLINE excited by a small additive noise and to construct an asymptotic expansion of the risk function. A numerical example of various approximations to the risk function is presented for the stochastic Van der Pol oscillator (\(f(x_{1},x_{2})=x_{2}(1-x_{1}^{2})\), \(\sigma (x_{1},x_{2})\equiv 1\)). The same example is used to study how well the risk function describes the effect of random perturbations.