Noise-induced transitions and deformations of stochastic attractors for one-dimensional systems

The influence of additive and parametrical noise on attractors of the one-dimensional system governed by the stochastic differential Ito equation is investigated. It is shown that unlike additive, parametrical disturbances lead to the shift of extrema of probability density function. For the value of this shift, a decomposition on small parameter of noise intensity is obtained. It is shown that the influence of the parametrical noise can change not only the arrangement, but also the quantity of extrema of probability density function. The corresponding noise-induced phenomena are studied for three dynamical models in detail. An analysis of the error for the different order estimations of the shift of extrema for the probability density function is presented by the example of a linear model. Two scenarios of the transition between unimodal and bimodal forms of the stochastic attractor are investigated for systems with different types of cubic nonlinearity.