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

Lyapunov exponents

Synchronization of coupled generators of quasi-periodic oscillations upon destruction of invariant curve

The purpose of this study is to describe the complete picture of synchronization of two coupled generators of quasi-periodic oscillations, to classify various types of synchronization, to study features of occurrence and destruction of multi-frequency quasi-periodic oscillations. Methods. The object of the research is systems of ordinary differential equations of various dimensions. The work uses the fourth-order Runge–Kutta method to solve a system of differential equations.

On methods for verification of the pseudohyperbolicity of strange attractors

The topic of the paper is strange attractors of multidimensional maps and flows. Strange attractors can be divided into two groups: genuine attractors, that keep their chaoticity under small perturbations, and quasi-attractors (according to Afraimovich–Shilnikov), inside which stable periodic orbits can arise under small perturbations. Main goal of this work is to construct effective criteria that make it possible to distinguish such attractors, as well as to verify these criteria by means of numerical experiments.

Complex dynamics in the system of two coupled discrete Rossler oscillators

We considered the discrete map with quasi-periodic dynamics in the wide band of the parameters and investigated the structure of the parameter plane of two coupled maps. We revealed the doublings of 3D-tori, the systems of 2D-tori and synchronization tongues and the resonance web. Also we revealed the attractors with complex structure and the largest Lyapunov exponent close to zero.

Method for calculation of lyapunov exponents spectrum from data series

The new method for the calculating of the spectrum of the Lyapunov exponents from data series is proposed. The already known methods of the same thematic are investigated. The Roessler system is given as an example for describing the proposed method. The results of numerical modeling are presented.

On the problem of computation of the spectrum of spatial lyapunov exponents for the spatially extended beam plasma systems

The behavior of the Pierce diode has been considered from the point of view of the spatial Lyapunov exponents. The method of calculation of the spectrum of the spatial Lyapunov exponents for the electron spatial extended systems has been proposed. The autonomous dynamics of the Pierce diode as well as the behavior of two unidirectionally coupled Pierce diodes when the generalized synchronization is taken place have been considered.

Hyperchaos in model nonautonomous system with a cascade excitation transmission through the spectrum

One of the key turbulence theory idea is a cascade energy transmission through the spectrum from large to small scales. It appears that this idea could be used for complex dynamics realization in a different-nature systems even when equations are knowingly differ from hydrodynamical. The system of four van der Pol oscillators is considered in this paper. Chaos generation is realized by cascade excitation transmission from one oscillator to another with frequency doubling.

Technique and results of numerical test for hyperbolic nature of attractors for reduced models of distributed systems

A test of hyperbolic nature of chaotic attractors, based on an analysis of statistics distribution of angles between stable and unstable subspaces, is applied to reduced finite­dimensional models of distributed systems which are the modifications of the Swift–Hohenberg equation and Brusselator model, as well as to the problem of parametric excitation of standing waves by the modulated pump.

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.

Autonomous systems with quasiperiodic dynamics examples and their properties: review

The paper is a review of well-known in nonlinear dynamics models with low dimensional of phase space and quasiperiodic behavior. Also new results related to analysis of many-frequencies quasiperiodic oscillations for models with external action and coupled oscillators are discussed. 

Largest Lyapunov exponent of chaotic oscillatory regimes computing from point processes in the noise presence

We propose a modified method for computing of the largest Lyapunov exponent of chaotic oscillatory regimes from point processes at the presence of measurement noise that does not influence on the system’s dynamics. This modification allow a verification to be made of the estimated dynamical characteristics precision. Using the Rossler system in the regime of a phase-coherent chaos we consider features of application of this method to point processes of the integrate-and-fire and the threshold-crossing models.