Bifurcations in Dynamical Systems

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.

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. The paper introduces an example solution of the inverse problem, constructing a deterministic-probabilistic system (a channels and jokers system) generating time series with characteristics similar to the ones of the time series generated by a simple system working in on-off intermittency mode (the Ershov mapping).

The effect of noise on the self­sustained oscillator near subcritical Andronov–Hopf bifurcation is studied in numerical and full­scale experiments. Van der Pol oscillator is chosen as base model for investigation. The influence of both additive and multiplicative Gaussian white noise is considered. The regularities of evolution of the probability distribution in the self­sustained oscillator are analyzed with increase of the noise intensity for the cases of additive and parametric noise.

We investigate bistable oscillator under the influence of additive, white and colored, noise. We have found noise-induced bifurcations that consist in a qualitative change of stationary distribution of oscillations amplitude. In the region of bimodal distribution the effect of coherent resonance takes place both for white and colored noise.

Considering a set of coupled nonautonomous differential equations with discontinuous right­hand sides, we discuss two different scenarios for torus birth bifurcations in piecewise­smooth dynamical systems. One scenario is the continuous transformation of the stable equilibrium into an unstable focus period­1 orbit surrounded by a resonant or ergodic torus. Another is the transition from a stable periodic orbit to an invariant torus through a border­collision bifurcation in which two complex­conjugate multipliers jump abruptly from the inside to the outside of the unit circle.

The response problem of equilibrium and cycles for stochastically forced Verhulst population model is considered. Theoretical and empirical approaches are used for stochastically sensitivity analysis. The theoretical approach is based on the firth approximation method and the empirical approach is based on direct numerical simulation. The correspondence between the two approaches for Verhulst population model is demonstrated. The increase of discrete system sensitivity to external noise in the period­doubling bifurcation zone under transition to chaos is shown.

The problem of the dynamics of three coupled self­oscillators and three coupled periodically driven self­oscillators is discussed, in the last case only one of the oscillators is directly exited by the external fore. The regions of complete synchronization, two­, three­and four­frequency tori and chaos are revealed. Three typical situations of synchronization of three self­oscillators by the external driving are found.

The results are produced of research of dynamical modes and bifurcation in a complex system with phase control, based on mathematical model with two degrees of freedom in the cylindrical phase space. The location of domains corresponding to different dynamical states of the system is established. The processes developing in the system as a result of loss stability of the synchronous mode, and scenarios of evolution of nonsynchronous modes under variation of system parameters are investigated.

The results of analysis of bifurcation transitions to synchronous regimes and amplitude death are discussed for two dissipatively coupled generators with inertial nonlinearity. It was determined that there are two types of synchronization regions in this system: first consists of both frequency lock and suppression areas, second has only frequency lock area. At the weakly non­identical excitement parameters the first­type synchronization regions merge together.

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 period­doubling or pitchfork bifurcations transversely to the circumference of the closed curve.