# Гиперболический хаос

## 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 selfoscillatory 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 twodimensional torus embedded in the sixdimensional phase space of the Poincare map, on which the dynamics is the hyperbolic chaos intrinsic to Anosov’s systems.

## On scenarios of hyperbolic chaos destruction in model maps on torus with dissipative perturbation

In this paper we investigate modi?ed «Arnold cat» map with dissipative terms, in which a hyperbolic chaos exists for small perturbation magnitudes, and in a certain range a hyperbolic chaotic attractor with Cantor transversal structure takes place, collapsing with a further perturbation amplitude increase.

## Attractor of smale–williams type in a ring system with periodic frequency modulation

A scheme of circular nonautonomous system is introduced, which is supposed to generate hyperbolic chaos. Its operation is based on doubling of phase on complete cycle of the signal transmission. This is a criterion for the Smale–Williams attractor to exist. The performance is realized due to smooth periodic variation of natural frequency in one of the two oscillatory subsystems, which compose the ring, from reference value to the doubled one.

## 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.

## Chaos in the phase dynamics of qswitched van der pol oscillator with additional delayed feedback loop

We present chaos generator based on a van der Pol oscillator with two additional delayed feedback loops. Oscillator alternately enters active and silence stages due to periodic variation of the parameter responsible for the Andronov–Hopf bifurcation. Excitation of the oscillations on each new activity stage is forced by signal resulting from mixing of the first and the second harmonics of signals from previous activity stages, transported through the feedback loops.

## Hyperchaos in a system with delayed feedback loop based on qswitched van der pol oscillator

We present a way to realize hyperchaotic behavior for a system based on Qswitched van der Pol oscillator with nonlinear signal transformation in the delayed feedback loop. The results of numerical studies are discussed: time dependences of variables, attractor portraits, Lyapunov exponents, and power spectrum.

## A new information transfer scheme based on phase modulation of a carrier chaotic signal

A new information transfer scheme based on dynamical chaos is suggested. An analog carrier signal is generated by selfexciting chaotic generator in a phasecoherent oscillatory regime. This carrier undergoes a modified procedure of phase modulation by information signal, which simultaneously affects upon the transmitting generator via the feedback loop. After the communication channel is passed, the signal modulated by information acts upon a receiving generator, so that a synchronous chaotic response arises in it.

## Circular nonautonomous generator of hyperbolic chaos

A scheme of circular system is introduced, which is supposed to generate hyperbolic chaos. Its operation is based on doubling of phase on each complete cycle of the signal transmission through the feedback ring. That is a criterion for the attractor of Smale–Williams type to exist. Mathematically, the model is described by the fourth order nonautonomous system of ordinary differential equations. The equations for slowly varying complex amplitudes are derived, and the Poincar ? e return map is obtained. Numerical simulation data are presented.