Bifurcations in Dynamical Systems

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.

A review of main results is given, concerning the Feigenbaum's scenario in the context of critical phenomena theory. Computer-generated illustrations of scaling are presented. Approximate renormalization group (RG) analysis is considered, allowing to obtain RG transformation in an explicit form. Examples of nonlinear systems are discussed, demonstrating this type of critical behaviour.

Realization of stable two­frequency oscillations is shown in the Maxwell–Bloch model. Birth of a stable ergodic two­dimensional torus from the critical closed curve is observed. The conditions of the passage to chaos via a cascade of torus doubling bifurcations are obtained. It is established that at bifurcations points a structurally unstable three­dimensional torus is produced, which gives rise to a stable doubled ergodic torus. Analytical approximation describing dynamics of the system near a point of torus birth is found.

A new four-dimensional model with quasi-periodic dynamics is suggested. The torus attractor originates via the saddle-node bifurcation, which may be regarded as a member of a bifurcation family embracing different types of blue sky catastrophes.

It is shown that a fixed point of the renormalization group transformation for a system of two subsystems with unidirectional coupling, one represented by a unimodal map with extremum of degree k and another by a map accumulating a sum of terms expressed as a function of a state of the first subsystem, undergoes a period-doubling bifurcation in a course of increase of the parameter k. At k = 2 the respective solution (period-2 cycle of the renormalization group equation) corresponds to a situation at the chaos threshold designated as the C-type critical behavior (Kuznetsov and Sataev, Phys.

The path bifurcation problem is formulated. The application of it for the classical result of F. Tricomi on the existence of homoclinic bifurcations in a dissipative pendulum system is discussed. The survey of results concerning to the solving of the path homoclinic bifurcation problems for Lorenz system is given.

Noise essentially influences the behavior of deterministic cycles of dynamical systems. Backward bifurcations of stochastic cycles for nonlinear Roessler model are investigated. Two approaches are demonstrated. In empirical approach the distribution densities of intersection points in intersecting planes are used. Theoretical analysis is based on stochastic sensitivity functions. This approach allows to achieve rather simple approximation of distribution densities in planes. Вifurcational values for noise intensities are found.