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

Deterministic Chaos

System of three non-autonomous oscillators with hyperbolic chaos. The model with DA-attractor

We consider a system of three coupled non-autonomous van der Pol oscillators, in which the behavior of the phases over a characteridtic period is described approximately by the Fibonacci map with modification of the «Smale surgery», which leads to the appearance of DA-attractor («Derived from Anosov»). According to the numerical results, the attractor of the stroboscopic map is placed approximately on a two-dimensional torus embedded in the six-dimensional phase space and has transverse Cantor-like structure typical for this kind of attractrors.

Investigation of regular and chaotic dynamics of one financial system

Based on complex numerical investigation for the nonlinear financial system introduced by Chen a map of dynamic regimes has been built, depending on the bifurcation parameters. All the major scenarios of transition to deterministic chaos have been found. Theorems of the existence of the globally exponentially attractive set and positive invariant, of periodic solutions, of Poincare–Andronov–Hopf bifurcation existence and theorems in the field of control of attractors are proved. 

The effect of weak nonlinear dissipation on the stochastic web

The effect of a weak nonlinear dissipation on the structure of the system’s phase space with stochastic web is invstigated. The bifurcation scenario of attractor transformations with the increase of dissipation is revealed.

Control of a chaotic microwave signal generation in nonautonomous ring system based on a ferromagnetic film

Experimental investigation results of the nonautonomous active ring system based on a ferromagnetic film at three-wave interactions were considered. The possibility of system dynamics controlling by the external harmonic signal was shown. A comparison of experimental data with simulation results was done.

Estimation of the main parameter values of nonlinear dynamic system with noise in experiment

We consider the method of parameter values estimation of dynamical system with noise in application to secure communication. We solve the problem of creating experimental radiophysical generator (Ressler generator) and comparison dynamics of numerical model with radiophysical experiment data. We analyse the influence of noise on the oscillator dynamics and parameter estimation error. We research the possibility of estimation of constant parameter and time-variable parameter, which can be modulated by different form signals.

Effect of filtering in dynamic system parameters estimation

A question on the distortion of the dynamic system parameters estimation using a time variation of the single component after exposing recursive filters with different order and with different cut-off frequency is analyzed. The Lorenz system is used as a test dynamic system for comparative evaluation of the correlation dimension and the dimension of the system state vector in the case of recursive filtering.

System of three nonautonomous oscillators with hyperbolic chaos. Part I The model with dynamics on attractor governed by Arnold’s cat map on torus

In this paper a system of three coupled nonautonomous self-oscillatory elements is studied, in which the behavior of oscillators phases on a period of the coefficients variation in the equations corresponds to the Anosov map demonstrating chaotic dynamics. Results of numerical studies allow us to conclude that the attractor of the Poincare map can be viewed as an object roughly represented by a two-dimensional torus embedded in the sixdimensional phase space of the Poincare map, on which the dynamics is the hyperbolic ´ chaos intrinsic to Anosov’s systems.

Robust chaos in autonomous time-delay system

We consider an autonomous system constructed as modification of the logistic differential equation with delay that generates successive trains of oscillations with phases evolving according to chaotic maps. The system contains two feedback loops characterized by two generally distinct  retarding time parameters. In the case of their equality, chaotic dynamics is associated with the  Smale–Williams attractor that corresponds to the double-expanding circle map for the phases of the carrier of the oscillatory trains.

Noise induced parametric instability and stochastic oscillations in the oscillator with nonlinear dissipation

The appearance of the instability of oscillator equilibrium state in a case of noisy modulation of the natural frequency is considered in the work. The threshold of instability and the properties of stochastic oscillations arising over the threshold are studied for the different noise characteristics.

Multiparametrical analysis based on Melnikov criterion and optimal chaos suppression in periodically driven dynamic systems

The results that illustrate the fruitfulness of the idea of optimal parametric correction for the analysis and optimization of the class of periodically driven chaotic systems are presented. Two problems that reveal the peculiarities of suppression of chaotic dynamics and present the method of regulation of the behavior of dissipative nonlinear oscillator were solved with the help of Melnikov criterion.