# stability

## Periodic modes of group dominance in fully coupled neural networks

Nonlinear systems of differential equations with delay, which are mathematical models of fully connected networks of impulse neurons, are considered. Purpose of this work is to study the dynamic properties of one special class of solutions to these systems. Large parameter methods are used to study the existence and stability in сonsidered models of special periodic motions – the so-called group dominance or *k*-dominance modes, where *k* ∈ N. Results.

## Equations with the Fermi–Pasta–Ulam and dislocations nonlinearity

Issue. The class of Fermi–Pasta–Ulam equations and equations describing dislocations are investigated. Being a bright representative of integrable equations, they are of interest both in theoretical constructions and in applied research. Investigation methods. In the present work, a model combining these two equations is considered, and local dynamic properties of solutions are investigated. An important feature of the model is the fact that the inﬁnite set of characteristic numbers of the equation linearized at zero consists of purely imaginary values.

## Destruction of the coherent mode in system of two oscillators at the strong resonant mutual couplings

The hypothesis about destruction of a coherent mode in system of two mutual couplings microwave oscillators is examine, each of which in a stand-alone mode generates stable unifrequent oscillations. It is experimentally shown, that at strong resonant couplings synchronous oscillations are unstable, therefore the system go over in in a mode of dynamic chaos.

## Nonlinear effects in autooscillatory system with frequency-phase control

Dynamical modes and nonlinear phenomena in the models of oscillatory system with frequency-phase control in the case of periodic nonlinear characteristics of frequency discriminator are investigated. Stability of synchronous mode is analyzed. The existences of a great number various periodic and chaotic nonsynchronous modes are established. Peculiarities of the system dynamics caused by parameters of frequency control loop are considered.

## The modes of genetic structure and population size dynamics in evolution model of two-aged population

The modes of genetic structure and size dynamics of structured population are investigated in this work. The reproductive potential and survival rate of reproductive part of population in following years of life are determined on genetic level. It has been shown that evolutional increasing of average population fitness is followed by arising of complicated dynamics of population size and of genetic structure.

## Stability of a stationary critical state in a model of cluster formation

The paper considers a self-organized critical process of clasterization. The stability of the equilibrium for infinite system of the differential equations approximating this process is proved.

## Regular and chaotic dynamics of two-ring phase locked system. Part 1 - Dynamics of frequency-phase system with identical first-order filters in control circuits

We present the results of investigation of dynamical modes in the model of oscillatory system with frequency-phase control using multi-frequency discriminator inversely switched in the chain of frequency control. The study was carried out on the basis of mathematical model of the system with one degree of freedom with the use of qualitative and numerical methods of nonlinear dynamics. It is shown that in such a system may be realized both synchronous and great number of non-synchronous periodic modes.

## Bifurcations and oscillatory modes in complex system with phase control

The results are produced of research of dynamical modes and bifurcation in a complex system with phase control, based on mathematical model with two degrees of freedom in the cylindrical phase space. The location of domains corresponding to different dynamical states of the system is established. The processes developing in the system as a result of loss stability of the synchronous mode, and scenarios of evolution of nonsynchronous modes under variation of system parameters are investigated.

## Dynamic modes of two-age population model

In this paper we research a mathematical model of dynamics for the population number. We considered the population of the two-age classes by the beginning of the next season: the younger, one including not reproductive individuals, and the senior class, consisting of the individuals participating in reproduction. The model parameters (birth rate and survival rates) represent the exponential functions of the both age groups numbers. According to this supposition the density-dependent factors restrict the development of population.

## Regular and chaotic dynamics of two-ring phase locked system. Part 2 - Peculiarities of nonlinear dynamics of frequency-phase system with identical third-order filters in control circuits

The results of investigation of dynamical modes in the model of oscillatory system with frequency-phase control using multi-frequency discriminator inversely switched inthe chain of frequency control are presented. The study was carried out on the basis of mathematical model of the system with two degrees of freedom with the use of qualitative and numerical methods of nonlinear dynamics. It is shown that in such a system may be realized both synchronous and great number of non-synchronous periodic and chaotic modes of different complexity.