The Hopf bifurcation of twodimensional systems under the influence of one external noise source

The behaviour of the Hopf bifurcation under the influence of external noise is investigated by means of a twodimensional model which uses Gaussian white noise as input. The model includes the case of multiplicative and/or additive noise. Applying the Birkhoff transformation the model is transformed to the coordinates normally used to discuss the deterministic Hopf bifurcation. Then the stationary solution of the model is calculated as an expansion for weak noise: The Hopf bifurcation under the influence of noise exhibits a bifurcation interval with width and position depending on the noise power. Moreover, a class of the systems described by the model can perform noise driven bifurcations.