bifurcations

For a typical phase lock loop with the second-order filter and delayed feedback, conditions of appearance and characteristics of regular and chaotic automodulation regimes are studied.

We study the stochastically forced limit cycles of discrete dynamical systems in a period-doubling bifurcation zone. A phenomenon of a decreasing of the stochastic cycle multiplicity with a noise intensity growth is investigated. We call it by a backward stochastic bifurcation. In this paper, for such a bifurcation analysis we suggest a stochastic sensitivity function technique. The constructive possibilities of this method are demonstrated for analysis of the two-dimensional Henon model.

We study the process of competition in the two­component model of the immune T­cells ensemble that underpins the selection mechanism of the most efficient T­cell species (clonotypes). We demonstrate the absence of periodic oscillations, determine the regions of coexistence, partial and mutual extinction of clonotypes. Applicability of the mean field approximation is analyzed. The biological implications of the results are discussed.

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).

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.