Bifurcations in Dynamical Systems

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.

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.

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, g = 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 (p5 + 1)=2.

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.

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.

The system of two coupled R¨ ossler 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 R¨ ossler oscillators is compared with synchronization pictures of more simple systems: two coupled Van der Pol oscillators and coupled logistic maps.

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 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.

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.

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.