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.

We study the effects of noise and diffusion in an excitable slow–fast population system of the Leslie–Gower type. The phenomenon of noise-induced excitement is investigated in the zone of stable equilibria near the Andronov–Hopf bifurcation with the Canard explosion. The stochastic generation of mixed-mode oscillations is studied by numerical simulation and stochastic sensitivity analysis. Effects of the diffusion are considered for the spatially distributed variant of this slow–fast population model. The phenomenon of the diffusion-induced generation of spatial patterns-attractors in the Turing instability zone is demonstrated. The multistability and variety of transient processes of the pattern formation are discussed. This article is part of the theme issue ‘Patterns in soft and biological matters’.

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Motivated by an important geophysical application, we analyze the nonlinear dynamics of the number of earthquakes per unit time in a given Earth's surface area. At first, we consider a dynamical model of earthquakes describing their rhythmic behavior with time delays. This model comprises different earthquake scenarios divided into three types (A, B, and C) accordingly to various system dynamics. We show that the deterministic system contains stable equilibria and a limit cycle whose size drastically depends on the production rate of earthquakes and their time delay effect. As this takes place, the frequency of earthquakes possesses an oscillatory behavior dependent on . To study the role of in more detail, we have introduced a white Gaussian noise in the governing equation. First of all, we have shown that the dynamical system is stochastically excitable, that is, it excites larger‐amplitude noise‐induced fluctuations in the frequency of earthquakes. In addition, these large‐amplitude stochastic fluctuations can alternate with small‐amplitude fluctuations over time. In other words, the frequency of earthquakes can change its amplitude in an irregular manner under the influence of white noise. Another important effect is how close the current value of is to its bifurcation point. The closer this value is, the less noise generates large‐amplitude fluctuations in the earthquake frequency.