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Two models of semiconductor laser with delayed optical feedback are studied. We consider singularly perturbed problem because of the large parameter presence. We construct and discuss quasinormal forms of models in trancritical cases.

The results of investigation of virtual cathode mechanisms forming and its dynamics in the context of 2-dimensional model are presented. There were considered entire and tubular electron beams in external axial magnetic field. Two different types of virtual cathode dynamics were discovered. The value of external magnetic field determines a dominant type of dynamics. Therefore the current critical value (when virtual cathode arises in a beam) depends on external magnetic field value.

We present a detailed description of the group-theoretical method which has been published in 2006 by the authors. This method can frequently simplify the study of the stability of different dynamical regimes in nonlinear physical systems with discrete symmetry since it allows one to split the set of the linearized (near a considered regime) nonlinear differential equations into a number of independent subsets of small dimensions. The above method is illustrated with the case of stability analysis of some dynamical regimes in the simple octahedral structure.

The local dynamics is considered of differential equations with two delays in the case of one delay is asymptotically large. Under this condition, critical cases have infinite dimension. As the normal form equations the Ginzburg–Landau equations have been. Their nonlocal dynamics defines local behavior of solutions of initial equations.

Synchronization in the system of coupled nonidentical and nonisochronous van der Pol oscillators with dissipative and inertial type of coupling is discussed. Generalized Adler equation is obtained and investigated in the presence of all factors. Basic symmetry of the equation, with leads to equivalence of some physical factors, is displayed. Numerical investigation of parameters space of initial differential system is realized. Results of two methods are compared and discussed.

By means of modeling and numeric simulation we consider, how the rise of extracellular potassium concentration due to the neuronal activity can affect the firing patterns of the neighboring neurons. To take into account mentioned above effects, we suggest simple extension of Hodgkin-Huxley model. We consider the behavior of 2, 4, and 8 excitable neurons being forced by external noisy stimulus. We reveal the main effects being the attributes of ionic coupling that are include the emergence of new time scales and spatially-ordered firing patterns.

In the paper an effect of phase space trajectories mixing in chaotic systems is considered. Approximate analytic estimations are given of mixing dynamics in discrete and continuous chaotic systems. Easy algorithm is developed for experimental calculation of mixing degree and mixing velocity, both local and average over the attractor. Results of this algorithm application to Henon map and to Chua system are discussed.

We study external synchronization of periodic oscillations in a ring of oscillators driven by periodic force. It is shown that each multistable state that co-exists in the system possesses its own synchronization region. We find that the periodic force with a certain frequency applied to one of the oscillators enables to switch the ring to another stable regime.

The pulse-driven van der Pol oscillator with the external pulse amplitude depending on the system variables is considered. The discrete map for values of the system variables just before the pulse moment was obtained by the slow-varying-amplitude method. Further the parameter space of this map was analyzed, and the existence of the Hamiltonian critical behavior in this system was shown. The remarkable fact is that our system is the system with the dissipation depending not only on the parameter values, but on the variable values too.

In this work the nonlinear dynamics in a chain of unidirectionally coupled gyrobackward wave oscillators is studied. In coupled system, when the control parameters of each distributed system are changed, it is possible to show both a developed chaos dynamics and the regimes of stationary oscillations with one frequency.