# Bifurcations in Dynamical Systems

## An example of hard turbulence in the channels and jokers system

Hard turbulence, a chaotic mode distinguished by infrequent catastrophic outbreaks on the weak irregular space oscillation background, is considered. On-off intermittency as one of the possible ways of simplified qualitative definition of hard turbulence is discussed.

## Hopf bifurcations of cycles of period two of two-dimensional logistic map

Maps having cycles of period two in which Hopf bifurcations of new cycles occur are localized in the family of two-dimensional logistic maps. For the purposes of illustration of the bifurcation property one-dimensional sections of bifurcation diagrams with one fixed parameter for two-parameters’ first-return maps of two-dimensional logistic maps are given.

## Dynamics of the sprott’s coupled oscillators with nonidentical control parameters

The structure of the plane of period-doubling control parameters is discussed for the set of coupled differential systems proposed by J. Sprott. It is shown, that the behavior of these systems may be both similar to one of the popular coupled Ressler system and different from it.

## About scaling properties in the noisy circle map at the golden-mean winding number

Scaling regularities are examined associated with effect of additive noise upon a critical circle map at the golden-mean winding number. On a basis of the RG approach of Hamm and Graham [1] we present an improved numerical estimate for the scaling constant responsible for the effect of noise, γ = 2.3061852653... Decrease of the noise amplitude by this number ensures possibility of observation for one more level of fractal-like structure associated with increase of characteristic time scale by factor ( √ 5 + 1)/2.

## Dynamics of two nonidentical coupled self-sustained systems with period doublings on the example of Rossler oscillators

The system of two coupled Rossler oscillators is considered. Detailed investigation is carried out on the plane of parameters which control the period-doubling bifurcations in the subsystems. Dynamical regimes in different points of the control parameter plane are determined using the methods of the bifurcation plot and the highest nonzero Lyapunov exponent plot computation. The synchronization picture of two coupled Rossler oscillators is compared with synchronization pictures of more simple systems: two coupled Van der Pol oscillators and coupled logistic maps.

## Automodel periodic solutions and bifurcations from them in the problem of the interaction of two weakly coupled oscillators

The problem of the interaction of two identical weakly coupled van der Pol – Duffing oscillations has been considered. The method of Poincare – Dulak normal forms has been used for its solution. All automodel periodic solutions have been found analytically. The problem of local bifurcations of these periodic solutions has been studied.

## Bifurcational mechanisms of destruction of antiphase chaotic synchronization in coupled discrete-time systems

Bifurcational mechanisms responsible for destruction of antiphase synchronization of chaos are studied. Two cubic discrete maps with symmetric diffusive coupling and additional control term are used as a model. Phenomenon of synchronization formation and destruction are explored in connection with bifurcations of principal periodic orbits embedded in the chaotic attractor.

## Antiphase synchronization and multistability formation in symmetrically coupled bistable systems

Bifurcational mechanizms of multistability formation on base of regimes of antiphase synchronization in diffusivelly coupled cubic maps are considered. Bifurcations of periodic orbits inside symmetric invariant subspace, which containes attractors of synchronous oscillations, are studied.

## Bifurcation theory inverse problem in a noisy dynamical system. Example solution

Bifurcations in nonlinear systems with weak noise are considered. The local bifurcations are discussed: the saddle-node bifurcation, the transcritical bifurcation, the supercritical and subcritical pitchfork bifurcations. Basing on the known prebifurcational noise rise and saturation phenomenon, the inverse problem is introduced: the problem of the bifurcation detection and determining it’s type by the observed noise change (noise deviation growth fashion, saturation level, probability density). The inverse problem solution algorithm is suggested.

## Two-parametric bifurcational analysis of regimes of complete synchronization in ensemble of three discrete-time oscillators

We invetsigate mechanisms of appearance and disappearance of regimes of complete 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.