Deterministic Chaos

Considering model with bimodal dynamics we investigate the synchronization of different time scales. Transition between mode-locked and mode-unlocked chaotic attractors is investigated. It is shown that this transition involves a situation in which the synchronized chaotic attractor loses its band structure.

Spectral properties of the linear non-self-adjoint Perron – Frobenius operator of piece-wise linear chaotic maps having regular structure are investigated. Eigenfunctions of the operator are found in the form of Bernoulli and Euler polynomials. Corresponding eigenvalues are presented by negative powers of number of map brunches. The solution is obtained in general form by means of generating functions for eigenfunctions of the operator. Expressions for eigenfunctions and eigenvalues are different for original and inverse maps having even and odd number of branches.

In this paper the influence of noise in system of identical coupled logistic maps with two types of coupling – dissipative and inertial – is discussed.   The corresponding renormalization group analysis is presented. Scaling property in the presence of noise is considered, and necessary illustrations in «numerical experiment style» are given.

The work is devoted to investigation of complex dynamics in the model of two interacting systems with phase and delay control. Stability conditions of synchronous regime are determined. The processes of excitement of nonsynchronous regimes and transitions between them are considered. Scenarios of development of nonsynchronous regimes under variation of the systems parameters are determined. Routes to chaotic behavior of the model are discussed. Results are presented in the form of one-parameter bifurcation diagrams and phase portraits of the model attractors.

We investigated the influence of external noise on the critical behavior typical to nonidentical coupled systems with period-doubling. We obtained the numerical value of the scaling factor for noise amplitude by means of the renormalization group analysis. Also we demonstrated the selfsimilar structure of the parameter plane near the critical point in the model system of two noisy driven coupled logistic maps.

In the present paper we analyze the influence of white and colored noise on chaotic selfsustained oscillations in the regime of spiral attractor. We study characteristics of instantaneous phase and spectra of noisy chaotic oscillations. The phenomenon of chaos synchronization by external narrow-band noise has been estimated. Synchronization phenomena under the influence of narrow-band noise signals with equal spectra and different probability densities are compared.

By the example of the quasi-periodically forced logistic map we investigate statistical properties of the transition from strange nonchaotic attractor to chaos in the system with intermittent dynamics. The probability characteristics of laminar and chaotic phase distributions, as well as scaling laws for distributions of local Lyapunov exponents are studied at parameter values near the transition point.

 We present a method and results of numerical computations on verification of hyperbolic nature for the chaotic attractor in a system of two coupled nonautonomous van der Pol oscillators (Kuznetsov, Phys. Rev. Lett., 95, 2005, 144101). At selected parameter values, we indicate a toroidal domain in four-dimensional phase space of Poincar´ e map (topologically, a direct product of a circle and a three-dimensional ball), which is mapped into itself and contains the attractor we analyze.

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.

Stable and unstable regimes of glycerine convection in vertical toroidal loop are investigated experimentally. The results of Fourier-analysis, DFA, wavelet-, and correlation analysis of liquid’s motion peculiarities are presented. Chaotic attractor with Lorenz-attractor signs is constructed.