# Bifurcations in Dynamical Systems

## Four-dimensional system with torus attractor birth via saddle-node bifurcation of limit cycles in context of family of blue sky catastrophes

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. Also the torus birth through the Neimark-Sacker bifurcation occurs in some other region of the parameter space.

## Effective criteria for the existence of homoclinic bifurcations in dissipative systems

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.

## Bifurcation of universal regimes at the border of chaos

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 κ 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 κ.

## Features of the parameter plane of two nonidentical coupled Van der Pol – Duffing oscillators

The system of two nonidentical dissipative coupled Van der Pol – Duffing oscillators is considered. A possibility of Adler equation application to describe the synchronization areas is shown due to transition to the closed equations. There is a nontrivial form of the main synchronization tongue on the plane of the control parameters. The view of synchronization tongues system of the original differential model and the influence of the phase nonlinearity on its configuration are discussed. The case of the nonsymmetrical nonlinearity in oscillators is also considered.

## Noise-induced backward bifurcations in stochastic Roessler system

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.

## Two-parametric bifurcational analysis of formation and destruction of regimes of partial synchronization of chaos in ensemble of three discrete-time oscillators

We invetsigate mechanisms of appearance and disappearance of regimes of partial synchronization of chaos in a ring of three logistic maps with symmetric diffusive coupling. Two-parametric bifurcational analysis is carried out and typical oscillating regimes and transitions between them are considered. Partial chaotic synchronization is revealed to lead to generalized synchronization.

## Birth of a stable torus from the critical closed curve and its bifurcations in a laser system with frequency detuning

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.

## Critical dynamics for one-dimensional maps part 1: feigenbaum's scenario

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.