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The article discusses the emergence of chaotic attractors in the system of three ordinary differential equations arising in the theory of «reaction-diffusion» systems. The dynamics of the corresponding one- and two-dimensional maps and Lyapunov exponents of such attractors are studied. It is shown that the transition to chaos is in accordance with a non-traditional scenario of repeated birth and disappearance of chaotic regimes, which had been previously studied for one-dimensional maps with a sharp apex and a quadratic minimum. Some characteristic features of the system — zones of bistability and hyperbolicity, the crisis of chaotic attractors — are studied by means of numerical analysis.

The work outlines the key ideas of Kurdyumov S.P., an outstanding specialist in applied mathematics, self-organization theory, transdisciplinary research. It considers the development of his scientific ideas in the last decade and formulates a set of open problems in synergetics which will probably stimulate the development of this approach. The article is an engaged version of the report made at Xth Kurdyumov readings held in Tver State University in 2015.

In the last decades, universal scenarios of the transition to chaos in dynamic systems have been well studied. The scenario of the transition to chaos is defined as a sequence of bifurcations that occur in the system under the variation one of the governing parameters and lead to a qualitative change in dynamics, starting from the regular mode and ending with chaotic behavior. Typical scenarios include a cascade of period doubling bifurcations (Feigenbaum scenario), the breakup of a low-dimensional torus (Ruelle–Takens scenario), and the transition to chaos through the intermittency (Pomeau–Manneville scenario). In more complicated spatially distributed dynamic systems, the complexity of dynamic behavior growing with a parameter change is closely intertwined with the formation of spatial structures. However, the question of whether the spatial and temporal axes could completely exchange roles in some scenario still remains open. In this paper, for the first time, we propose a mathematical model of convection–diffusion–reaction, in which a spatial transition to chaos through the breakup of the quasi–periodic regime is realized in the framework of the Ruelle–Takens scenario. The physical system under consideration consists of two aqueous solutions of acid (A) and base (B), initially separated in space and placed in a vertically oriented Hele–Shaw cell subject to the gravity field. When the solutions are brought into contact, the frontal neutralization reaction of the second order A + B $\to$ C begins, which is accompanied by the production of salt (C). The process is characterized by a strong dependence of the diffusion coefficients of the reagents on their concentration, which leads to the appearance of two local zones of reduced density, in which chemoconvective fluid motions develop independently. Although the layers, in which convection develops, all the time remain separated by the interlayer of motionless fluid, they can influence each other via a diffusion of reagents through this interlayer. The emerging chemoconvective structure is the modulated standing wave that gradually breaks down over time, repeating the sequence of the bifurcation chain of the Ruelle–Takens scenario. We show that during the evolution of the system one of the spatial axes, directed along the reaction front, plays the role of time, and time itself starts to play the role of a control parameter.

This article presents the results of an analytical and computer study of the collective dynamic properties of a chain of self-oscillating systems (conditionally — oscillators). It is assumed that the couplings of individual elements of the chain are non-reciprocal, unidirectional. More precisely, it is assumed that each element of the chain is under the influence of the previous one, while the reverse reaction is absent (physically insignificant). This is the main feature of the chain. This system can be interpreted as an active discrete medium with unidirectional transfer, in particular, the transfer of a matter. Such chains can represent mathematical models of real systems having a lattice structure that occur in various fields of natural science and technology: physics, chemistry, biology, radio engineering, economics, etc. They can also represent models of technological and computational processes. Nonlinear self-oscillating systems (conditionally, oscillators) with a wide “spectrum” of potentially possible individual self-oscillations, from periodic to chaotic, were chosen as the “elements” of the lattice. This allows one to explore various dynamic modes of the chain from regular to chaotic, changing the parameters of the elements and not changing the nature of the elements themselves. The joint application of qualitative methods of the theory of dynamical systems and qualitative-numerical methods allows one to obtain a clear picture of all possible dynamic regimes of the chain. The conditions for the existence and stability of spatially-homogeneous dynamic regimes (deterministic and chaotic) of the chain are studied. The analytical results are illustrated by a numerical experiment. The dynamical regimes of the chain are studied under perturbations of parameters at its boundary. The possibility of controlling the dynamic regimes of the chain by turning on the necessary perturbation at the boundary is shown. Various cases of the dynamics of chains comprised of inhomogeneous (different in their parameters) elements are considered. The global chaotic synchronization (of all oscillators in the chain) is studied analytically and numerically.

We present the results of a mathematical model for the aquatic communities which include zooplankton, planktivorous fish and predator fish. The aquatic populations are considered to be body mass- and agestructured, while the trophic relations between the populations to be correspondingly status-specific. The model reproduces diverse dynamic regimes as such steady states and oscillations in the population size. Oscillations in the fish population size are shown to be both regular and irregular. We show that the period of the regular oscillations can be up to decades. The irregular oscillations are shown to be both chaotic and non-chaotic. Analyzing the dynamics in the model parameter space has enabled us to conclude that predictability of fish population dynamics can face difficulties both due to dynamical chaos and to the competition between various dynamical regimes caused by variations in the model parameters, specifically in the zooplankton growth rate.

The propagation of stable coherent entities of an electromagnetic field in nonlinear media with parameters varying in space can be described in the framework of iterations of nonlinear integral transformations. It is shown that for a set of geometries relevant to typical problems of nonlinear optics, numerical modeling by reducing to dynamical systems with discrete time and continuous spatial variables to iterates of local nonlinear Feigenbaum and Ikeda mappings and nonlocal diffusion-dispersion linear integral transforms is equivalent to partial differential equations of the Ginzburg–Landau type in a fairly wide range of parameters. Such nonlocal mappings, which are the products of matrix operators in the numerical implementation, turn out to be stable numerical- difference schemes, provide fast convergence and an adequate approximation of solutions. The realism of this approach allows one to take into account the effect of noise on nonlinear dynamics by superimposing a spatial noise specified in the form of a multimode random process at each iteration and selecting the stable wave configurations. The nonlinear wave formations described by this method include optical phase singularities, spatial solitons, and turbulent states with fast decay of correlations. The particular interest is in the periodic configurations of the electromagnetic field obtained by this numerical method that arise as a result of phase synchronization, such as optical lattices and self-organized vortex clusters.

This paper provides an overview of studies on nonlinear supratransmission and related phenomena. This effect consists in the transfer of energy at frequencies not supported by the systems under consideration. The supratransmission does not depend on the integrability of the system, it is resistant to damping and various classes of boundary conditions. In addition, a nonlinear discrete medium, under certain general conditions imposed on the structure, can create instability due to external periodic influence. This instability is the generative process underlying the nonlinear supratransmission. This is possible when the system supports nonlinear modes of various nature, in particular, discrete breathers. Then the energy penetrates into the system as soon as the amplitude of the external harmonic excitation exceeds the maximum amplitude of the static breather of the same frequency.

The effect of nonlinear supratransmission is an important property of many discrete structures. A necessary condition for its existence is the discreteness and nonlinearity of the medium. Its manifestation in systems of various nature speaks of its fundamentality and significance. This review considers the main works that touch upon the issue of nonlinear supratransmission in various systems, mainly model ones.

Many teams of authors are studying this effect. First of all, these are models described by discrete equations, including sin-Gordon and the discrete Schr¨odinger equation. At the same time, the effect is not exclusively model and manifests itself in full-scale experiments in electrical circuits, in nonlinear chains of oscillators, as well as in metastable modular metastructures. There is a gradual complication of models, which leads to a deeper understanding of the phenomenon of supratransmission, and the transition to disordered structures and those with elements of chaos structures allows us to talk about a more subtle manifestation of this effect. Numerical asymptotic approaches make it possible to study nonlinear supratransmission in complex nonintegrable systems. The complication of all kinds of oscillators, both physical and electrical, is relevant for various real devices based on such systems, in particular, in the field of nano-objects and energy transport in them through the considered effect. Such systems include molecular and crystalline clusters and nanodevices. In the conclusion of the paper, the main trends in the research of nonlinear supratransmission are given.

The dynamics of coupled elements’ ensembles are investigated in the context of description of spatio-temporal processes in the myocardium. Basic element is map-based model constructed by simplification and reduction of Luo-Rudy model. In particular, capabilities of the model in replication of different regimes of cardiac activity are shown, including excitable and oscillatory regimes. The dynamics of 1D and 2D lattices of coupled oscillatory elements with a random distribution of individual frequencies are considered. Effects of cluster synchronization and transition to global synchronization by increasing of coupling strength are discussed. Impulse propagation in the chain of excitable cells has been observed. Analysis of 2D lattice of excitable elements with target and spiral waves have been made. The characteristics of the spiral wave has been analyzed in depending on the individual parameters of the map and coupling strength between elements of the lattice. A study of mixed ensembles consisting of excitable and oscillatory elements with a gradient changing of the properties have been made, including the task for description of normal and pathological activity of the sinoatrial node.

The paper reviews possibilities to predict buckling of thin cylindrical shells with non-destructive techniques during operation. It studies shallow shells made of high strength materials. Such structures are known for surface displacements exceeding the thickness of the elements. In the explored shells relaxation oscillations of significant amplitude can be generated even under relatively low internal stresses. The problem of the cylindrical shell oscillation is mechanically and mathematically modeled in a simplified form by conversion into an ordinary differential equation. To create the model, the researches of many authors were used who studied the geometry of the surface formed after buckling (postbuckling behavior). The nonlinear ordinary differential equation for the oscillating shell matches the well-known Duffing equation. It is important that there is a small parameter before the second time derivative in the Duffing equation. The latter circumstance enables making a detailed analysis of the obtained equation and describing the physical phenomena — relaxation oscillations — that are unique to thin high-strength shells.

It is shown that harmonic oscillations of the shell around the equilibrium position and stable relaxation oscillations are defined by the bifurcation point of the solutions to the Duffing equation. This is the first point in the Feigenbaum sequence to convert the stable periodic motions into dynamic chaos. The amplitude and the period of relaxation oscillations are calculated based on the physical properties and the level of internal stresses within the shell. Two cases of loading are reviewed: compression along generating elements and external pressure.

It is highlighted that if external forces vary in time according to the harmonic law, the periodic oscillation of the shell (nonlinear resonance) is a combination of slow and stick-slip movements. Since the amplitude and the frequency of the oscillations are known, this fact enables proposing an experimental facility for prediction of the shell buckling with non-destructive techniques. The following requirement is set as a safety factor: maximum load combinations must not cause displacements exceeding specified limits. Based on the results of the experimental measurements a formula is obtained to estimate safety against buckling (safety factor) of the structure.

This work is devoted to the study of the problem of modeling and analyzing complex oscillatory modes, both regular and chaotic, in systems of interacting populations in the presence of random perturbations. As an initial conceptual deterministic model, a Volterra system of three differential equations is considered, which describes the dynamics of prey populations of two competing species and a predator. This model takes into account the following key biological factors: the natural increase in prey, their intraspecific and interspecific competition, the extinction of predators in the absence of prey, the rate of predation by predators, the growth of the predator population due to predation, and the intensity of intraspecific competition in the predator population. The growth rate of the second prey population is used as a bifurcation parameter. At a certain interval of variation of this parameter, the system demonstrates a wide variety of dynamic modes: equilibrium, oscillatory, and chaotic. An important feature of this model is multistability. In this paper, we focus on the study of the parametric zone of tristability, when a stable equilibrium and two limit cycles coexist in the system. Such birhythmicity in the presence of random perturbations generates new dynamic modes that have no analogues in the deterministic case. The aim of the paper is a detailed study of stochastic phenomena caused by random fluctuations in the growth rate of the second population of prey. As a mathematical model of such fluctuations, we consider white Gaussian noise. Using methods of direct numerical modeling of solutions of the corresponding system of stochastic differential equations, the following phenomena have been identified and described: unidirectional stochastic transitions from one cycle to another, trigger mode caused by transitions between cycles, noise-induced transitions from cycles to the equilibrium, corresponding to the extinction of the predator and the second prey population. The paper presents the results of the analysis of these phenomena using the Lyapunov exponents, and identifies the parametric conditions for transitions from order to chaos and from chaos to order. For the analytical study of such noise-induced multi-stage transitions, the technique of stochastic sensitivity functions and the method of confidence regions were applied. The paper shows how this mathematical apparatus allows predicting the intensity of noise, leading to qualitative transformations of the modes of stochastic population dynamics.