On noise induced Poincaré–Andronov–Hopf bifurcation

Abstract

It has been numerically seen that noise introduces stable well-defined oscillatory state in a system with unstable limit cycles resulting from subcritical Poincaré–Andronov–Hopf (or simply Hopf) bifurcation. This phenomenon is analogous to the well known stochastic resonance in the sense that it effectively converts noise into useful energy. Herein, we clearly explain how noise induced imperfection in the bifurcation is a generic reason for such a phenomenon to occur and provide explicit analytical calculations in order to explain the typical square-root dependence of the oscillations' amplitude on the noise level below a certain threshold value. Also, we argue that the noise can bring forth oscillations in average sense even in the absence of a limit cycle. Thus, we bring forward the inherent general mechanism of the noise induced Hopf bifurcation naturally realisable across disciplines.