STOCHASTIC BIFURCATION IN HEREDITARY SYSTEMS

The paper examines some concepts of bifurcations in stochastically perturbed dynamical systems governed by integro-differential equations. The particular system considered for illustration is a Van der Pol oscillator with a hereditary restoring force term, whose linear damping parameter is perturbed by white noise. When certain parameters in the system can be regarded as being small, the method of stochastic averaging is employed to evaluate the largest Lyapunov exponent and the response probability density. From these expressions, the critical values of the damping parameter at which a dynamical (or D-) bifurcation and a further phenomeno-logical (or P-) bifurcation from the trivial state occur are determined. The results for the P-bifurcation are confirmed by evaluating the moment Lyapunov exponent of the linearized equation and using a result from the theory of the large deviations