ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


bifurcation

Synchronization and multi-frequency quasi-periodicity in the dynamics of coupled oscillators

The dynamics of ensembles of oscillators containing a small number of bibitemlits is discussed. The possible types of regimes and pecularities of bifurcations of regular and quasi-periodic attractors are analyzed. By using the method of Lyapunov exponents charts the picture of  embedding of quasi-periodic regimes of different dimension in the parameter space is revealed. Dynamics of ensembles of van der Pol and phase oscillators are compared.

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.

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.

Regular and chaotic oscillations in astrocyte model with regulation of calcium release kinetics

The dynamics of an astrocyte model is investigated. The astrocytes represent a type of glial cells regulating oscillations of major signaling cells, e.g. neurons. Subserved by complex molecular mechanisms the astrocytes generate calcium auto-oscillations which, in turn, are associated with the release of neuroactive chemicals into extracellular space. At variance with classical astrocyte models the three-component model considered takes into account a regulation of calcium release due to nonlinear dynamics of inositol-1,4,5 trisphosphate (IP3).

The averaging method, a pendulum with a vibrating suspension: N.N. Bogolyubov, A. Stephenson, P.L. Kapitza and others

  The main moments of the historical development of one of the basic methods of nonlinear systems investigating (the averaging method) are traced. This method is understood as a transition from the so-called exact equation: dx/dt = ?X(t, x),     ? ? is small parameter, to the averaging equation d?/dt= ?X0(?) + ?2P2(?) + ...?mPm(?) by corresponding variable substitution. Bogolyubov–Krylov’s approach to the problem of justifying the averaging method, based on the invariant measure theorem, is analyzed.

Bifurcations of attracting sets of deformation displacement of cutting tool depending on the spindle group beats

Subject of the study. The bifurcations of the attracting sets of the deformational displacement of the tool in the dynamic system of the turning machine depending on the beats periodic trajectory of the spindle group are considered in the article. The dynamic system is represented by the two interact mechanical subsystems through the dynamic link formed by the cutting process. Through the link is represented by the forces model in the coordinate condition, the trajectories of the executive elements and the trajectories of the spindle beats group.

Traveling waves solution in parabolic problem with a rotation

Optical systems with two-dimensional feedback demonstrate wide possibilities for emergence of dissipative structures. Feedback allows to influence on dynamics of the optical system by controlling the transformation of spatial variables performed by prisms, lenses, dynamic holograms and other devices. Nonlinear interferometer with mirror reflection of a field in two-dimensional feedback is one of the simplest optical systems in which the nonlocal interaction of light fields is realized.