hyperbolic chaos

A review of studies aimed on revealing or constructing physical systems with hyperbolic strange attractors, like Plykin attractor and Smale–Williams solenoid, is presented. Examples of iterated maps, differential equations, and simple electronic devices with chaotic dynamics associated with such attractors are presented and discussed. A general principle is considered and illustrated basing on manipulation of phases in alternately excited oscillators and time­delay systems. Alternative approaches are reviewed outlined in literature, as well as the prospects of further researches.

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.

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.

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.

In this paper we investigate modified «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.

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.

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.