Noise-Induced Transitions for Limit Cycles of Nonlinear Systems

Abstract

An analysis of noise-induced transitions is challenging and fundamental problem of nonlinear engineering. We consider noise-induced transitions for limit cycles in Hopf bifurcation zone and period-doubling bifurcation zone. Near bifurcation values, even small external disturbances may essentially change behavior of dynamic system. An underlying reason of the unexpected chaotic-looking response of regular oscillations under the small perturbations is a high sensitivity of the nonlinear dynamic system. We present the results of the stochastic analysis for the classical Hopf differential system. The difference in the response of Hopf systems perturbed by additive and multiplicative random noises is demonstrated. A delaying shift of the Hopf bifurcation point induced by multiplicative noise is observed. We show that significant sensitivity of the cycle can cause the noise-induced transitions both from an attractor to another one and between different parts of single attractor. For Lorenz model, the growth of stochastic sensitivity near chaos bifurcations analysis is studied. We demonstrate noise-induced transitions and qualitative transformation of stochastic attractors.