Bifurcations in Dynamical Systems

Changes of the parameter plane of driven auto-oscillatory system caused by delayed modulation of the parameter

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.

Optimum external impulse of low power for activation of one- dimensional dynamic system

In this paper we study the optimum form of external influence with low power necessary for activation of one-dimensional dynamical system. The Lagrange multipliers method is used. Optimum influence with low power and optimum law of change of a dynamical system state are determined analytically for linear dynamical system and numerically for nonlinear dynamical system. The opportunity of influence power reduction vs its duration is investigated.

Influence of fluctuations on evolution of three-dimensional torus in nonautonomous system

The transition to chaos through the destruction of three-dimensional torus is studied in a nonautonomous system with quasi-periodic impact as example. Analysis is carried out of the influence both of additive noise and frequency fluctuations impact on the stability of three-dimensional torus. It is shown that under the influence of additive noise and frequency fluctuations impact Lyapunov exponent remains negative. This allows to conclude that in this model three-dimensional torus is structurally stable in contrast to the autonomous system.

Local dynamics of difference and difference-differential equations

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.

Regular and chaotic dynamics of two-ring phase locked system. Part 1 - Dynamics of frequency-phase system with identical first-order filters in control circuits

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.

Backward stochastic bifurcations of the henon map

We study the stochastically forced limit cycles of discrete dynamical systems in a period-doubling 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 two-dimensional Henon model.

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Bifurcations in the problem of thermal convection of viscoelastic fluid in a closed cavity with free boundaries heated from below

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 weakly-nonlinear analysis explicit formula is obtained for the boundary separating the rheological parameter space into domains with different type of bifurcations (super- and subcritical).

Period doubling bifurcations and noise excitation effects in a multistable self-sustained oscillatory medium

The model of a self-oscillatory medium composed from the elements with complex self-oscillatory behavior is studied. Under periodic boundary conditions the stable selfoscillatory 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.

Dynamic regimes and multistability in the system of non-symmetrically coupled two-dimensional maps with period-doubling and Neimark–Sacker bifurcations

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 asymmetry are exposed.

Formation and breakdown of a multilayered closed curve in noninvertible maps

The paper describes the mechanism for the formation of closed invariant curves that are formed as layered structures of several sets of interlacing manifolds each with their associated stable or unstable resonance modes. Such invariant curves can arise, for instance, if the saddle cycle on a «simple resonance curves» undergoes period-doubling or pitchfork bifurcations transversely to the circumference of the closed curve.

Bifurcations in Dynamical Systems

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