Methodical Papers on Nonlinear Dynamics

The concept of spatial deterministic chaos is justified. An attempt to give its settheoretic definition is undertaken. Transition from the ordinary differential equations to discrete maps without use of an approximation of the instantaneous response is realized for mathematical description of spatial deterministic chaos. The developed theoretical theses are applied for deriving a dynamics model in terms of discrete maps of nonlinear phase shift in a ring interferometer.

Families of initial-final maps, bifucation lines, maps of Lyapunov’s characterictic exponents and fractal dimentionality D0 are constructed for a model of nonlinear pphase shift dynamics for ont- and two-frequency field in a ring interferometer. The influence of a spectrum form of two-frequency radiation to a structure of mentioned maps is clarified.Ways of maps quantitative analysis are suggested and realized. Two languages of nonlinear dynamics description in the ring interferometer are compared: with the help of ordinary differential equations and of the discrete map.

The description of base concepts concerning to self-oscillations of nonconservative systems with asymmetrical couplings is given, in particular, such as the complex proper forms and frequencies, and complex normal coordinates.

Different methods for investigation of dissipative, nearly conservative and conservative systems have been demonstrated on the example of Ikeda map. The method for two-parameter analysis of dynamics of conservative systems has been proposed. Significant changes in the structure of the parameter and phase space of Ikeda map when dissipation decreases have been revealed. Tasks for seminars and computer practices have been proposed.

This paper is the book review. Its main purpose is to show that practically all kinds of canonical models of nonlinear dynamics are used in the present mathematical economics. 

The unusual technique of interpretation of numerical experiment results as sound waves is offered. Recommendations for practical implementation of the offered technique and for its application in various areas of research, designer and educational activity are given.

On the base of the cascade-bunching theory the calculation technique of starting current of multicavity klystron autogenerators is presented in this paper.

We present the electronic scheme of autonomous generator of two-frequency quasiperiodic motions and the experimental research results of effect of quasi-periodic motions synchronization under the external two-frequency force.

It was shown that addition of modulation of driving parameter with using delay can be considered as physically reasoned method of construction two-dimensional maps with nonfixed Jacobian. The examples of such two-parameter and three-parameter maps were presented. The conditions of Neumark–Sacker’s bifurcation, period doubling and resonance 1:2 were obtained. The structure of parameter space was studied by dynamical regimes maps method and the regions of quasiperiodic regimes and different synchronous regimes were revealed.

We study numerically chaotic behavior associated with the presence of a hyperbolic strange attractor of Plykin type in the model of Hunt, that is an artificially constructed dynamical system with continuous time. There are presented portraits of the attractor, plots of realizations for chaotic signal generated by the system, illustrations of the sensitive dependence on initial conditions for the trajectories on the attractor. Quantitative characteristics of the attractor are estimated, including the Lyapunov exponents and the attractor dimension.