Показатели Ляпунова

Aim. The aim of the work is to study the scientific heritage of A.M. Lyapunov from the standpoint of nonlinear physics. Fundamental importance Lyapunov’s contribution is determined not only by the methods he created, which became the basis of the mathematical apparatus in the study of nonlinear phenomena, but his ideas and concepts introduced by him contributed to the formation of concepts and principles of nonlinear dynamics. Method.

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.

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.

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.

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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.