# бифуркация

## Calcium oscillations in astrocytes part 1 astrocyte as generator of calcium oscillations

Bifurcation mechanisms of oscillatory dynamics in a biophysical model of chemically excitable brain cells (astrocytes) were analyzed. In contrast to neuronal oscillators widely studied in nonlinear dynamics the astrocytes do not possess electrical excitability but capable to generate chemical oscillations which modulate neuronal signaling. Astrocyte dynamics is described by third-order system of ordinary di?erential equations derived from biophysical kinetics.

## Calcium oscillations in astrocytes part 2 dynamics of interacting calcium oscillators

We investigated bifurcation mechanisms of oscillatory dynamics of interacting chemically excitable cells (astrocytes). In model of three interacting astrocytes we studied bifurcation transitions leading to generation of calcium oscillations induced by the intercellular di?usion. We analyzed basic mechanisms of limit cycle instabilities and destructions, typical transitions to chaotic oscillations and basic properties of intercellular synchronization.

## Chaos in radio device with squarelaw phase modulator and interference amplification of quasiharmonic signal: a model and simulation

The attempt is undertaken to define a class of oscillations or waves sources, the operation principle of which is based on interference amplification of feedback signal by an input signal. The precedent here is the optical Ikeda’s system. The radioelectronic analog of a nonlinear ring interferometer and it modification are offered, the block diagrams and mathematical models are constructed. The computer simulation is performed. An intermittency, chaos, regular, static modes are detected.

## Experiments with a source of chaos – a radioelectronic device with squarelaw phase modulator and interference amplification of quasiharmonic signal

A modified radioelectronic analog of the nonlinear ring cavity is realized in laboratory. The device represents a special class of oscillations or waves sources. An operation principle of the sources is based on interference amplification of feedback signal by an input signal. A laboratory experiments are performed, the likeness of their results and simulation data is shown. An intermittency, chaos, regular, static modes are detected. A thesis on controlled nonlinearity of dynamical systems is suggested.

## Synchronizing the period2 cycle in the system of symmetrical coupled populations with stock–recruitment based on the ricker population model

We investigated coupled map lattices based on the Ricker model that describes the spatial dynamics of heterogeneous populations represented by two connected groups of individuals with a migration interaction between them. Bifurcation mechanisms inphase and antiphase synchronization of multistability regimes were considered in such systems. To identify a synchronization mode we introduced the quantitative measure of synchronization.

## 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.

## 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.

## 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.