Semi-analytical expression of stochastic closed curve attractors in nonlinear dynamical systems under weak noise

•Distribution of stochastic closed curve attractor is decomposed into two low-dimensional distributions.•Distribution of stochastic quasi-periodic closed curve attractor is determined by rational approximation method.•A simple but accurate semi-analytical method to approximate stochastic closed curve attractors is proposed.

In this paper, a simple but accurate semi-analytical method to approximate probability density function of stochastic closed curve attractors is proposed. The expression of distribution applies to systems with strong nonlinearities, while only weak noise condition is needed. With the understanding that additive noise does not change the longitudinal distribution of the attractors, the high-dimensional probability density distribution is decomposed into two low-dimensional distributions: the longitudinal and the transverse probability density distributions. The longitudinal distribution can be calculated from the deterministic systems, while the probability density in the transverse direction of the curve can be approximated by the stochastic sensitivity function method. The effectiveness of this approach is verified by comparing the expression of distribution with the results of Monte Carlo numerical simulations in several planar systems.