Hopf bifurcation with fluctuating control parameter

Abstract

Systems are considered that undergo a continuous forward Hopf bifurcation into an oscillatory steady state, breaking a continuous $U\mathrm{}\left(1\right)\mathrm{}$ symmetry. The control parameter is assumed to consist of a systematic constant part and a superimposed Gaussian white-noise part. The Fokker-Planck equation describing the dynamics of fluctuations near instability is solved exactly, generalizing earlier results for nonoscillatory bifurcation. A check of the completeness of the eigenfunctions by a sum rule is presented. Correlation functions and transient moments are evaluated from the Fokker-Planck solution on both sides of the bifurcation point and their asymptotic long-time behavior is discussed. Below the bifurcation point transient moments are found to exist whose decay constants are smaller than the smallest eigenvalue in the Fokker-Planck spectrum. Two methods for calculating such moments from the Fokker-Planck spectrum are presented and exemplified.