Tatyana Vadivasova

This paper is concerned with the spatiotemporal dynamics of the 2D lattice of cubic maps with nonlocal coupling. Different types of chimera structures have been found. Also, the underexplored regime of solitary states has been found. It is shown that the solitary states are typical of a large coupling radius. The possibility of detecting such a regime increases with the transition to global interaction, while chimera states disappear.

Complex spatial structures, called chimeras, are the subject of considerable recent interest. They consist of stationary areas with coherent and incoherent behavior of neighboring elements. A number of problems related to similar structures have not been solved yet. One of these problems concerns the element interaction in ensembles, when stable chimera structures can be observed. Until quite recently it was assumed that one of the most important conditions for the existence of chimeras is the nonlocal character of interaction. However, this assumption is not exactly correct. Chimeras can be realized for special types of local coupling. So, the chimera examples were obtained in ensembles with inertial local coupling. The additional variable is introduced for a coupling specification. It is given by a linear differential equation. Also, the so-called virtual chimeras exist in oscillators with delayed feedback. This allows one to assume that chimera states can be obtained in a ring of local coupling oscillators with unidirectional interaction, which is inertialess, but has a nonlinear character. This assumption is based on a qualitative similarity between the behaviors of an oscillator with delay feedback and a ring of the same oscillators with local unidirectional coupling.
The basis of this work is the system with delay feedback, which demonstrates the existence of a virtual chimera. The distributed analog is investigated. It is an oscillator ring with unidirectional nonlinear local coupling.
The existence of chimera structures in the ring were found in the special area of parameter changing via computing simulation. This chimera moves in a ring with constant velocity and is similar to the chimera in the system with delay feedback. The area of chimera existence of parameter variations was studied. Regime diagrams were plotted on the plane of control parameters. The scenario of chimera destruction for the coupling increase was shown.

Effects of noisy influence on oscillators near oscillation threshold are studied by means of numerical simulation and natural experiments. Two qualitative different models (Van derPol and Anishchenko—Astakhov self-sustained oscillators) are considered. Evolution laws of probabilistic distribution with increase of noise intensity are established for two cases: addition of additive and parametric white gaussian noise in researched systems. It is shown that the noise destroys the distribution form, which is typical for self-oscillations, that leads to shift of bifurcation to direction of excitation parameter increase. The existence of bifurcation interval, which corresponds with gradual transition to regime of self-oscillation, was detected from experiments with additive noise.

The model of an active medium with periodical boundary conditions is studied. The elementary cell is chosen to be FitzHugh–Nagumo oscillator. According to the values of parameters the elementary cell is able to be either in self-sustained regime or in excitable one. In both cases there are sustained oscillations in each elementary cell of the medium, but the causes of its initiation are different. In case of the former each cell in itself is auto-oscillator, in case of the latter the oscillations appear because of feedback which is provided by the periodical boundary conditions. In both cases the phenomenon of multistability is observed. The comparative analysis of the regimes mentioned above is carried out. There are shown that the dependencies of oscillations characteristics from the system parameters in either cases significantly differ from one another. The bifurcational type of the transition from one cell regime to another is ascertained for some modes. The influence of spatial-uncorrelated noise on the active medium behavior is considered. The average period of oscillations versus noise intensity relation is obtained.

The model of a self-oscillatory medium whose cells represent Anishchenko-Astakhov self-sustained oscillators is studied. Under periodic boundary conditions the phenomenon of multistability is observed in the medium — the stable self-sustained oscillatory modes with different spatial structures coexist and can be realized by means of appropriately chosen initial conditions. The study of the time period doubling bifurcations is performed for different modes. It is shown that the evolution of the modes between two successive bifurcations leads to the complexification of instantaneous spatial profile and to the appearance of small-scale spatial oscillations. The distribution of the instantaneous phase shift along the medium is studied in different regimes. The influence of local noise source on the spatial structures is considered. It is demonstrated that noise can induce switchings between different regimes. The mechanism of such switchings is explored.

In the present paper possibilities of parameters estimation are considered in dynamical systems (DS) with additive noise. Simple and effective algorithms, optimal parameter values of numeric simulation and data filtration methods are proposed that enable one to find the controlling parameter value of a noisy DS with a high accuracy. Different DS are studied, and the accuracy of parameter estimation is examined for various dynamical modes and for different noise intensities.

In the present paper autonomous and nonautonomous oscillations of dynamical and stochastic systems are analyzed in the framework of common concepts. The definition of an attractor is introduced for a nonautonomous system. The definitions of self-sustained oscillations and a self-sustained oscillatory system is proposed, that generalize A.A.Andronov’s concept introduced for autonomous systems with one degree of freedom.

The effect of synchonization has been studied in a system of two coupled Van der Pol oscillators under external harmonic force. The bifurcation analysis has been carried out using the phase approach. The mechanisms of complete and partial synchronization have been established. The main type of bifurcation described in the paper is the saddle-node bifurcation of invariant curves that corresponds to the saddle-node bifurcation of two-dimensional tori in the complete system of differential equations for the dynamical system under study. We illustrate the bifurcational mechanisms obtained from numerical experiment by the results of physical experiment. The synchronization phenomenon in the vicinity of resonances on a torus with winding numbers 1 : 1 and 1 : 3 is considered in the physical experiment.

We investigate effective diffusion coefficient of instantaneous phase of chaotic self-sustained oscillations and its connection with synchronization threshold. It is showed that effective phase diffusion coefficient in contrast to maximal Lyapunov exponent allows to distinguish the regions of spiral and funnel attractor. We ascertain that synchronization threshold of chaos is in order-of-magnitude agreement with the value of diffusion coefficient divided by the mean frequency of self-sustained oscillations.