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


hyperbolic chaos

Experimental studies of chaotic dynamics near the Theorist

The purpose of this work is to review of works in which experimental studies of the regularities of chaotic dynamics revealed theoretically in works of S.P. Kuznetsov were carried out. Methods. The research methods used are primarily based on the construction of experimental schemes; they correspond most closely to the mathematical models proposed and theoretically and numerically investigated by S.P. Kuznetsov. These are systems of radio engineering oscillators with various types of communication and impact, autogenerators with various types of feedback. Results.

Theoretical models of physical systems with rough chaos

The purpose of this review is to present in a unified manner the latest results on mathematical modeling of rough hyperbolic chaos in systems of various physical nature. Main research Methods are the numerical solution of systems of differential equations and partial differential equations, numerical extraction of the phase of oscillatory processes or spatial patterns, calculating of Lyapunov exponents, and studying the mutual arrangement of the stable and unstable manifolds of chaotic trajectories, the calculation of Gaussian curvature of surfaces.

Synchronization of oscillators with hyperbolic chaotic phases

Synchronization in a population
of oscillators with hyperbolic chaotic phases is studied for two
models. One is based on the Kuramoto dynamics of the phase oscillators and
on the Bernoulli map applied to these phases. This system
possesses an Ott-Antonsen invariant manifold, allowing for a derivation of a
map for the evolution of the complex order parameter. Beyond a critical coupling strength,
this model demonstrates bistability synchrony-disorder. Another model

The study of the unidirectionally coupled generators of robust chaos and wide band communication scheme based on its synchronization

A numerical simulation of a wide band or secure communication scheme, based on nonlinear admixture of an information signal to the chaotic one, and on synchronization of the transmitter and receiver generators, manifesting hyperbolic chaos. Synchronization of the transmitter and receiver is provided by a strong unidirectional coupling between them. The study of the possibility of synchronization between subsystems and functionality of the communication scheme are presented.

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.

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.

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

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.

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.

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.

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