Methodical Papers on Nonlinear Dynamics

Topic and aim. It is known that the dual representation of problems (via the basic and conjugate functions in the Lagrange sense) allows us to formulate an effective version of the theory of small perturbations, but expanding its scope by including the next terms in the perturbation theory series sharply complicates the solution procedure. In this regard, a number of works have been undertaken to find alternative approaches.

The aim of the article is to demonstrate an algorithm for constructing measure-preserving three-dimensional chaotic maps defined in domains formed by rotation bodies. On the one hand, the class of multidimensional chaotic mappings is expended, and on the other hand, we obtain formulas for simulating pseudo-random quantities that are in demand in problems solving by the Monte Carlo method.

For sampling of time in a differential equation of movement of van der Pol oscillator (generator) it is offered to use a combination of the numerical method of finite differences and the asymptotic method of the slowl-changing amplitudes. The difference approximations of temporal derivatives are selected so that, first, to save conservatism and natural frequency of the linear circuit of self-oscillatory system in the discrete time.

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.

Лабораторная работа

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.

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.