Lenin Avenue 51, Ekaterinburg, Russia
Ural State University
Bashkirtseva I. A., Ryashko L. B., Slepukhina E. S.
Splitting bifurcation of stochastic cycles in the FitzHugh–Nagumo model
2013, Vol. 9, No. 2, pp. 295-307
We study the stochastic dynamics of FitzHugh–Nagumo model in the zone of limit cycles. For weak noise, random trajectories are concentrated in a small neighborhood of the initial deterministic unperturbed orbit of the limit cycle. As noise increases, in the zone of Canard cycles of the FitzHugh–Nagumo model, the bundle of random trajectories begins to split into two parts. This phenomenon is investigated using the density distribution of random trajectories. It is shown that the threshold noise intensity corresponding to the splitting bifurcation depends essentially on the degree of the stochastic sensitivity of the cycle. Using the stochastic sensitivity functions technique, a critical value corresponding to the supersensitive cycle is found and comparative parametric analysis of the effect of the stochastic cycle splitting in the vicinity of the critical value is carried out.
Bashkirtseva I. A., Ryashko L. B., Fedotov S. P., Tsvetkov I. N.
Backward stochastic bifurcations of the discrete system cycles
2010, Vol. 6, No. 4, pp. 737-753
We study stochastically forced limit cycles of discrete dynamical systems in a perioddoubling bifurcation zone. A phenomenon of a decreasing of the stochastic cycle multiplicity with a noise intensity growth is investigated. We call it by a backward stochastic bifurcation (BSB). In this paper, for the BSB analysis we suggest a stochastic sensitivity function technique. As a result, a method for the estimation of critical values of noise intensity corresponding to BSB is proposed. The constructive possibilities of this general method for the detailed BSB analysis of the multiple stochastic cycles of the forced Verhulst and Ricker systems are demonstrated.
Bashkirtseva I. A., Zubarev A. Y., Iskakova L. Y., Ryashko L. B.
Regular and stochastic auto-oscillations in the reological model
2009, Vol. 5, No. 4, pp. 603-620
This paper is devoted to research of mathematical model of a suspension flow. For these flows, the transitions from stationary to the oscillatory regimes have been observed in experiments. Bifurcation analysis allows us to divide the space of parameters onto steady equilibria and limit cycles zones. Details of Hopf bifurcation depending on degree of system stiffness are investigated. On the basis of the stochastic sensitivity function technique, the parametrical analysis of influence of random disturbances on the system attractors is carried out. It is shown that as a system stiffness increases, the stochastic sensitivity of oscillations rises sharply. The narrow zone of super-sensitivity of oscillations was found. In this zone, even small background disturbances result in the essential fluctuations of their amplitude.