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It is shown that a fixed point of the renormalization group transformation for a system of two subsystems with unidirectional coupling, one represented by a unimodal map with extremum of degree k and another by a map accumulating a sum of terms expressed as a function of a state of the first subsystem, undergoes a period-doubling bifurcation in a course of increase of the parameter k. At k = 2 the respective solution (period-2 cycle of the renormalization group equation) corresponds to a situation at the chaos threshold designated as the C-type critical behavior (Kuznetsov and Sataev, Phys.

The path bifurcation problem is formulated. The application of it for the classical result of F. Tricomi on the existence of homoclinic bifurcations in a dissipative pendulum system is discussed. The survey of results concerning to the solving of the path homoclinic bifurcation problems for Lorenz system is given.

On the basis of a simple mathematical model «producers-products-managers», reason, substance, and possible ways of overcoming of a centuries-old conflict in humanity life are considered.

Complication of fractal structure of ammonite’s shells has been considered during ontogeny and phylogeny. Analogy to Koh curves has been presumed. Fractality among ammonites may be considered as adaptive and evolutionary advantage that explains their high rate of evolution and certain causes of extinctions.

This work presents a dynamical phenomenon strongly related with the problems of synchronization and control of chaotic dynamical systems. Considering externally driven homoclinic chaotic systems, it is shown experimentally and theoretically that they tend to synchronize with signals strongly correlated with the saddle cycles of their skeleton; furthermore, when they are perturbed with a generic signal, uncorrelated with their skeleton, their chaotic behavior is reinforced.

The processes of oscillatory pattern formation in autooscillatory neuronal models are investigated. Such patterns play a key role in the information processes used in higher brain functions. The effect of pulse-induced phase autoreset in the model of neurons with subthreshold oscillations is studied. As a result of this effect the reset phase value does not depend on the initial phase. It is defined only by the stimulus parameters. The autoreset effect can be used for phase synchronization and phase cluster formation in ensembles of autooscillatory units.

The work is devoted to investigation of dynamics of running waves in the ring of Van-der-Pol oscillators with diffusive coupling. Regions of existence and stability are built in the parameters space. Typicalness of appearance of regimes with different wavelengths and regularities of their disappearance are considered. Influence of anharmonicity on multistability of spatio-periodic regimes is studied.

Noise essentially influences the behavior of deterministic cycles of dynamical systems. Backward bifurcations of stochastic cycles for nonlinear Roessler model are investigated. Two approaches are demonstrated. In empirical approach the distribution densities of intersection points in intersecting planes are used. Theoretical analysis is based on stochastic sensitivity functions. This approach allows to achieve rather simple approximation of distribution densities in planes. Вifurcational values for noise intensities are found.

The system of two nonidentical dissipative coupled Van der Pol – Duffing oscillators  is considered. A possibility of Adler equation application to describe the synchronization areas is shown due to transition to the closed equations. There is a nontrivial form of the main synchronization tongue on the plane of the control parameters. The view of synchronization tongues system of the original differential model and the influence of the phase nonlinearity on its configuration are discussed. The case of the nonsymmetrical nonlinearity in oscillators is also considered.

We invetsigate mechanisms of appearance and disappearance of regimes of partial synchronization of chaos in a ring of three logistic maps with symmetric diffusive coupling. Two-parametric bifurcational analysis is carried out and typical oscillating regimes and transitions between them are considered. Partial chaotic synchronization is revealed to lead to generalized synchronization.