# Методические заметки по нелинейной динамике

## About some concepts of oscillation theory of nonconservative systems with asymmetrical couplings

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.

## Simulation of field nonlinear phase shift dynamics in ring interferometer in case of two-frequency influence

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.

## Spatial deterministic chaos: the model and demonstration of phenomenon in computing experiment

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.

## How objects which "must not exist really" can be seen in experiment

Using nonautonomous nonlinear oscillator, we demonstrate the experimental approach, which permit to illustrate the role of unstability in complicated dynamics formation of nonlinear systems. The experimental system for observation of nonequilibrium processes is described, which permits to see on oscilloscop the unstable cycles in face space, to estimate the stability of states, to investigate the structure of chaotic attractors.

## Lorenz type attractor in electronic parametric generator and its transformation outside the accurate parametric resonance

The paper deals with a parametric oscillator composed of three LC-circuits and a quadratic nonlinear reactive element built on the basis of an operational amplifier and an analog multiplier; the equations for amplitudes of the interacting modes are derived. Motivation is a desire to implement the mechanism of parametric interaction of oscillatory modes giving rise to emergence of a strange attractor of Lorenz type without distortions introduced by nonlinearities of order three and higher.

## Хаотическая динамика простой электронной схемы

Лабораторная работа

## Maps with quasi-periodicity of different dimension and quasi-periodic bifurcations

The paper discusses the construction of convenient and informative three-dimensional mappings demonstrating the existence of 2-tori and 3-tori. The first mapping is obtained by discretizing the continuous time system – a generator of quasi-periodic oscillations. The second is obtained via discretization of the Lorentz-84 climate model. The third mapping was proposed in the theory of quasi-periodic bifurcations by Simo, Broer, Vitolo.