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

Smale–Williams solenoid

Autonomous system generating hyperbolic chaos: circuit simulation and experiment

We consider an electronic device, which represents an autonomous dynamical system with hyperbolic attractor of the Smale–Williams type in the Poincare map. Simulation of chaotic dynamics in the software environment Multisim has been undertaken. The generator of hyperbolic chaos is implemented as a laboratory model; its experimental investigation is carried out, and good compliance with the observed dynamics in the numerical and circuit simulation has been demonstrated.

Parametric generators with chaotic amplitude dynamics corresponding to attractors of smale–williams type

A new approach is considered to design of parametric generators of chaos with hyperbolic attractors on the basis of two alternately excited subsystems, each consisting of three oscillators, one of which plays the role of the pump source. In contrast to previously proposed schemes, the angular variable undergoing a multiple increase over each characteristic period is a quantity characterizing the amplitude ratio of two oscillators, rather then the phase of successive oscillation trains.

Electronic circuits manifesting hyperbolic chaos and simulation of their dynamics using software package multisim

We consider several electronic circuits, which are represented dynamical systems with hyperbolic chaotic attractors, such as Smale–Williams and Plykin attractors, and present results of their simulation using the software package NI Multisim 10. The approach developed is useful as an intermediate step of constructing real electronic devices with structurally stable hyperbolic chaos, which may be applicable in systems of secure communication, noise radar, for cryptographic systems, for random number generators.

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 Q-switched van der Pol oscillator

We present a way to realize hyperchaotic behavior for a system based on Q-switched van der Pol oscillator with non-linear 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. 

Hyperbolic strange attractors of physically realizable systems

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 time-delay systems.

Uniformly hyperbolic attractor in a system based on coupled oscillators with «figure-eight» separatrix

A new autonomous system with chaotic dynamics corresponding to Smale–Williams attractor in Poincare map is introduced. The system is constructed on the basis of the model with «figure-eight» separatrix on the phase plane discussed in former times by Y.I. Neimark. Our system is composed of two Neimark subsystems with generalized coordinates x and y. It is described by the equations with additional terms due to which the system becomes self-oscillating.