бифуркации

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.

Bifurcations were studied numerically in the system of partial differential equations, which is  a one variant of predator-prey models. The mathematical model takes into account spatial  distribution in habitat, active directed predator movements, birth and death process in prey  population. The analysis of possible population dynamics development was performed by two  qualitatively different discrete sampling techniques (Bubnov–Galerkin’s method and grid method).

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.

In the article we offer to consider case of a flexural deformation shifts of a tool when they are essential for nonlinear dynamics of cutting process. This situation is observed for drill deep  holes, because a boring bar has a small values of a flexural stiffness. In that case an angle of cutting  edge reduces and cutting forces increase if the deformation shifts also increased in velocity  direction. The last circumstance becomes occasion for positive feedback that essentially changes  dynamics of the cutting process.

The problems of nonlinear dynamics of cutting metal are considered in the article. We offer mathematical model of dynamical system that includes a dynamical relation of the cutting process by using turning example. Basic positions of the dynamical relation are the forces dependence of cutting area, the force’s delay of elastic deformation shift of a tool by relative to workpiece, limitations of the cutting forces on clearance face of the tool, dependence of the cutting forces of the cutting velocity.

Nonlinear dynamics of the ensemble consisting of two phase­locked generators, which are coupled in a ring with feedback, is discovered. The conditions of stability of the synchronous regimes and appropriatenesses of excitation and progress of the non­synchronous regimes are examined within the bounds of the dynamic model with one and a half degrees of freedom. The extensive image of the dynamic regimes and bifurcating transitions, creating resources for the formation in the system of various types of oscillations, is discovered.

The condition of a bifurcation of a round-trip time of ultrashort pulses (USP) in the Nd-laser with short-time resonant modulation of loss is found out experimentally. This condition is exhibited in the period doubling in the area of a small detunings of the modulation frequency and an intermodal interval. It is revealed, that the pumping level (amplification in the active medium) is a major factor, influencing this effect.

In this paper we study the bifurcational mechanisms of synchronization and multistability formation in a system of two interacting van der Pol oscillators, one of which is under external harmonic forcing. We draw a two­parametric bifurcation diagram for phasereduced system and study its evolution in transition from symmetrical to asymmetrical repulsive interaction. Relying on the results of bifurcation analysis of non­reduced system we conclude that the synchronization scenarios found in the phase­reduced system correspond to the ones in the non­reduced system.

The work is dedicated to the history of the theory of nonlinear integral equations, covering a period before the start of the 1930s. By analyzing the specifics of the initial period, authors emphasize that the integral equations (in particular, nonlinear equations) is independent object of research with their own problems, requiring its own system of concepts and own language. As a starting point here A.M.

The paper is devoted to some historical aspects of the rapidly developing field of modern mathematics – nonlinear functional analysis, which is presented as the basis of the mathematical apparatus of nonlinear dynamics. Its methods are demonstrated on the example of bifurcation. The first bifurcations problem – Euler problem on elastic instability rod under longitudinal compressive forces is considered. The formation of Voronezh school of functional analysis and its role in the development of nonlinear analysis in general is also discussed.