Resonance in isochronous systems with decaying oscillatory and stochastic perturbations

The combined influence of oscillatory excitations and multiplicative stochastic perturbations of white noise type on isochronous systems in the plane is investigated. It is assumed that the intensity of perturbations decays with time and the excitation frequency satisfies a resonance condition. The occurrence and stochastic stability of solutions with an asymptotically constant amplitude are discussed. By constructing an averaging transformation, we derive a model truncated deterministic system that describes possible asymptotic regimes for perturbed solutions. The persistence of resonant solutions in the phase locking and the phase drifting modes is justified by constructing suitable Lyapunov functions for the complete stochastic system. In particular, we show that decaying stochastic perturbations can shift the boundary of stability domain for resonant solutions.