bifurcation

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.

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.

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:

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 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).

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.

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.

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.

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.

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.