Bifurcations in Dynamical Systems

We consider the Hopf system as a classical model of a stiff birth of a cycle. In the presence of parametrical and additive random disturbances, various types of the stochastic attractors are observed. The solution of the corresponding Fokker–Planck–Kolmogorov equation is found. The qualitative changes of the form for stochastic attractors under multiplicative noise are shown. The phenomenon of backward stochastic bifurcations is described in details.

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.

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. Location parameters domains are established with different dynamic modes of the system.

We study local dynamics of difference and singular perturbed difference-differential systems in the neighborhood of zero equilibrium state. All critical cases in this problem have infinite dimension. We construct special nonlinear equations that play the role of normal form. Their nonlocal dynamics describes the behavior of solution of initial system.

We study the stochastically forced limit cycles of discrete dynamical systems in a period­doubling 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 two­dimensional Henon model.  ́

Bifurcations in the problem of thermal convection of viscoelastic fluid in a square cavity with free boundaries heated from below are studied. General Odroyd model is used for the description of rheological behaviour of the fluid. In the frame of weakly­nonlinear analysis explicit formula is obtained for the boundary separating the rheological parameter space into domains with different type of bifurcations (super­ and subcritical).

The phenomenon of multistability in the system of coupled universal two-dimensional maps which shows period-doubling and Neimark–Sacker bifurcations is investigated. The decreasing of possible coexisting attractors number, the evolution of the attractor basins, the disappearance of hyperchaos and three-dimensional torus while putting coupling asymmetryare exposed.

The model of a self-oscillatory medium composed from the elements with complex self-oscillatory behavior is studied. Under periodic boundary conditions the stable self-oscillatory regimes in the form of traveling waves with different phase shifts are coexisted in medium. The study of mechanisms of the oscillations period doubling in time is performed for different coexisted modes. For all observed spatially-non-uniform regimes (traveling waves) the period doubling occurs through the appearance of time-quasiperiodic oscillations and their further evolution.

The driven auto-oscillatory system with the delayed modulation of driving amplitude was investigated. It was shown that synchronous regime destructs in different ways at small and large modulation amplitudes. The changes in the «driving amplitude–driving frequency» plane were revealed.