стохастическая бифуркация

In the present paper we study an unstable nonlinear oscillator in which the growth of amplitude of oscillations is limited by noise influence. We calculate the characteristics of noise-stabilized fluctuations. It is shown when the noise intensity changes, the system can demonstrate different effects such as the suppression of exponential instability of trajectories and the narrowing of the spectrum of fluctuations.

The effect of noise on the self­sustained oscillator near subcritical Andronov–Hopf bifurcation is studied in numerical and full­scale experiments. Van der Pol oscillator is chosen as base model for investigation. The influence of both additive and multiplicative Gaussian white noise is considered. The regularities of evolution of the probability distribution in the self­sustained oscillator are analyzed with increase of the noise intensity for the cases of additive and parametric noise.

We investigate bistable oscillator under the influence of additive, white and colored, noise. We have found noise-induced bifurcations that consist in a qualitative change of stationary distribution of oscillations amplitude. In the region of bimodal distribution the effect of coherent resonance takes place both for white and colored noise.

The modern knowledges of bifurcations of dynamical systems in the presence of noise are presenred. The main definitions are given and certain typical examples of the bifurcations in the presence of additive and multiplicative noise are considered.

The appearance of the instability of oscillator equilibrium state in a case of noisy modulation of the natural frequency is considered in the work. The threshold of instability and the properties of stochastic oscillations arising over the threshold are studied for the different noise characteristics.

The model of a self-oscillatory medium composed from the elements with complex self-oscillatory behavior is studied. Under periodic boundary conditions the stable self-oscillatory regimes in the form of traveling waves with different phase shifts are coexisted in medium. The study of mechanisms of the oscillations period doubling in time is performed for different coexisted modes. For all observed spatially-non-uniform regimes (traveling waves) the period doubling occurs through the appearance of time-quasiperiodic oscillations and their further evolution.

The multiplicative noise influence on the self­sustained oscillatory medium near the oscillation threshold is studied. The chain of the identical quasi­harmonic self­sustained oscillators with the periodic boundary conditions is taken as a simplest model of the oscillatory medium. The parameters of the oscillators are modulated with the white Gaussian noise. The stochastic bifurcations are analyzed for the cases of homogenous and spatially­nonhomogenous noise.