Sensitivity of a Hopf Bifurcation to External Multiplicative Noise

Abstract

It has been found experimentally that the properties of oscillatory physical and chemical systems are strongly sensitive to the influence of external multiplicative noise [1–2]. In the case of a well-understood experimental system, namely the electrical parametric oscillator studied by KABASHIMA and coworkers [1], the main effects reported are: (i) The onset of the oscillatory regime is still sharply localized at a well-defined value of the average of the control parameter in spite of the perturbations by the noise, (ii) The noise stabilizes the non-oscillatory regime. In first approximation the threshold of instability is shifted to higher values of the average control parameter by an amount proportional to the intensity of the noise. (iii) There is a slowing down of the relaxation rate of the state variables near the threshold. The theoretical explanation of these experimental findings is based on the properties of the stochastic process x_t. given by a stochastic differential equation of the form

$$ d{x_t}=\left[{\mu \left({{\lambda_t}} \right){x_t}-{x^3}_t} \right]dt $$

(1)

It has been shown indeed by KABASHIMA and coworkers that near the threshold at which the oscillatory regime sets in, the amplitude of the oscillatory current x_t is in first approximation given by (1) where

$$ \begin{gathered} \mu ({\lambda_t})={\lambda_t} \hfill \\=\lambda+\sigma {\xi_t} \hfill \\ \end{gathered} $$

is proportional to the fluctuating pumping current used as control parameter; λ is the constant average value and ξ_t is Gaussian white noise with intensity σ