ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

Детерминированный хаос

Experimental realization of lorenz model of liquid’s convective instability in vertical toroidal loop

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.

Verification of hyperbolicity conditions for a chaotic attractor in a system of coupled nonautonomous van der pol oscillators

 We present a method and results of numerical computations on veri?cation 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.

Statistical properties of the intermittent transition to chaos in the quasi-periodically forced system

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.

Influence of noise on chaotic self-sustained oscillations in the regime of spiral attractor

In the present paper we analyze the in?uence 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 in?uence of narrow-band noise signals with equal spectra and di?erent probability densities are compared.

Critical behavior of asymmetrically coupled noisy driven nonidentical systems with period-doublings

We investigated the in?uence 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.

Peculiarities of complex dynamics and transitions to chaotic regimes in the model of two interacting systems with phase control

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.

About scaling properties of identical coupled logistic maps with two types of coupling without noise and under influence of external noise

In this paper the in?uence 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.

Analytical solution of spectral problem for the perron – frobenius operator of piece-wise linear chaotic maps

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 di?erent for original and inverse maps having even and odd number of branches.

Synchronization in systems with bimodal dynamics

Considering model with bimodal dynamics we investigate the synchronization of di?erent 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.

Computation of Lyapunov exponents for spatially extended systems: advantages and limitations of various numerical methods

Problems emerging in computations of Lyapunov exponents for spatially extended systems are considered. We concentrate on the incorrect orthogonalization of high sized ill conditioned matrices appearing in course of the computation, and on large errors emerging under certain conditions if the finite difference numerical method is applied to solve equations. The practical guidelines helping to avoid the mentioned problems are represented.