Backward stochastic bifurcations of the discrete system cycles

Abstract

В работе рассматриваются стохастически возмущенные предельные циклы дискретных динамических систем в зоне удвоения периода. Исследуется явление обратных стохастиче-ских бифуркаций (ОСБ) -уменьшения кратности цикла при увеличении интенсивности шума. Предлагается метод анализа ОСБ на основе техники функции стохастической чув-ствительности. Конструктивные возможности данного метода демонстрируются на примере анализа ОСБ стохастических циклов систем Ферхюльста и Риккера.Ключевые слова: бифуркации, дискретные системы, модель Ферхюльста, стохастиче-ская чувствительность I. A. Bashkirtseva, L. B. Ryashko, S. P. Fedotov, I. N. Tsvetkov Backward stochastic bifurcations of the discrete system cycles 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.

Analysis of noise-induced transitions in a generalized logistic model with delay near Neimark–Sacker bifurcation

“…Under the random disturbances, a stationary probabilistic distribution p(x, ε) of random states of system (A.2) is formed around attractors of the deterministic system (A.1). For the approximation of this distribution in the presence of small noise, a method of stochastic sensitivity functions was proposed in [5].…”

“…In these circumstances, semi-analytical approximations are very useful. In [5], the stochastic sensitivity functions (SSF) technique has been proposed for the analysis of randomly forced equilibria and discrete cycles. Recently, this technique has been developed for more complicated attractors, such as closed invariant curves [6] and chaotic attractors [7].…”

Analysis of stochastic phenomena in Ricker-type population model with delay

“…In the parametric analysis of the phenomenon of the noise-induced extinction, we will use an analytical approach based on the stochastic sensitivity function technique and the confidence domains method [17,18,19].…”

“…However, a direct use of these functional equations is very difficult technically, even in onedimensional case. For the constructive description of probabilistic distributions near deterministic attractors the new method based on the stochastic sensitivity functions technique [17,18,19] was proposed. In the present paper, we apply this technique to the analysis of noise-induced extinction in the Ricker model with delay and Allee effect.…”

Stochastic sensitivity analysis of the attractors for the randomly forced Ricker model with delay

“…Our approach is based on the stochastic sensitivity function technique that was set out earlier in some particular cases [31,32]. To demonstrate the universality and efficiency of this approach, we apply the elaborated theory to the stochastic Ricker model with delay.…”