arXiv subjects

Sergey P. Kuznetsov

Publications and source records attributed to Sergey P. Kuznetsov.

At least 19 recordsLinked to original sources

arXiv2020

Route to hyperbolic hyperchaos in a nonautonomous time-delay system

We consider a self-oscillator whose excitation parameter is varied. Frequency of the variation is much smaller then the natural frequency of the oscillator so that oscillations in the system are periodically excited and decay. Also a time delay as added such that when the oscillations start to grow at a new excitation stage they are influenced via the delay line by the oscillations at the penultimate excitation stage. Due to a nonlinearity the seeding from the past arrives with a doubled phase so that oscillation phase changes from stage to stage according to chaotic Bernoulli-type map. As a result, the system operates as two coupled hyperbolic chaotic subsystems. Varying the relation between the delay time and the excitation period we affect a coupling strength between these subsystems as well as intensity of the phase doubling mechanism responsible for the hyperbolicity. Due to this, a transition from non-hyperbolic to hyperbolic hyperchaos occurs. The following steps of the transition scenario are revealed and analyzed: (a) an intermittency as alternation of long staying near a fixed point at the origin and short chaotic bursts; (b) chaotic oscillations with often visits to the fixed point; (c) plain hyperchaos without hyperbolicity after termination visiting the fixed point; (d) transformation of hyperchaos to hyperbolic form.

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arXiv2019

Parameter space arrangement of a model system nearby domain of existence of Plykin type attractor

For a model system defined as combination of sequentially applied continuous transformations of a sphere, the question of arrangement of the parameter space around the domain of existence of the Plykin-type attractor is considered. Results of numerical calculations are presented, including charts of dynamical regimes and Lyapunov exponents on the parameter plane, as well as portraits of attractors in characteristic regions of chaotic and regular dynamics. The Plykin attractor region is determined and depicted using the computational procedure for checking hyperbolicity, which consists in analyzing angles between expanding and contracting tangent subspaces of typical trajectories on the attractor. The Plykin attractor takes place in a bounded continuous region of the parameter plane that corresponds to roughness (structural stability) of the hyperbolic dynamics. Outside that region, various types of dynamics are observed including non-hyperbolic chaos, periodic and quasiperiodic behaviors.

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arXiv2019

Complex dynamics in generalizations of the Chaplygin sleigh

The article considers Chaplygin sleigh on a plane in potential well, assuming that an external potential force is supplied at the mass center. Two particular cases are studied in some detail, namely, a one-dimensional potential valley and a potential with rotational symmetry; in both cases the models reduce to four-dimensional differential equations conserving mechanical energy. Assuming the potential functions quadratic, various behaviors are observed numerically depending of the energy, from those characteristic to conservative dynamics (regularity islands and chaotic sea) to strange attractors. This is another example of nonholonomic system manifesting these phenomena (similar to those for Celtic stone or Chaplygin top), which reflects a fundamental nature of these systems occupying intermediate position between conservative and dissipative dynamics.

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arXiv2019

Mechanical systems with hyperbolic chaotic attractors based on Froude pendulums

We discuss two mechanical systems with hyperbolic chaotic attractors of Smale - Williams type. Both models are based on Froude pendulums. The first system is composed of two coupled Froude pendulums with alternating periodic braking. The second system is Froude pendulum with time-delayed feedback and periodic braking. We demonstrate by means of numerical simulations that proposed models have chaotic attractors of Smale - Williams type. We specify regions of parameter values at which the dynamics corresponds to Smale - Williams solenoid. We check numerically hyperbolicity of the attractors.

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arXiv2019

Some lattice models with hyperbolic chaotic attractors

Examples of one-dimensional lattice systems are considered, in which patterns of different spatial scales arise alternately, so that the spatial phase over a full cycle undergo transformation according to expanding circle map that implies occurrence of Smale-Williams attractors in the multidimensional state space. These models can serve as a basis for design electronic generators of robust chaos within a paradigm of coupled cellular networks. One of the examples is a mechanical pendulum system interesting and demonstrative for research and educational experimental studies.

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arXiv2019

Dynamics of phases and chaos in models of locally coupled conservative or dissipative oscillators

We discuss Hamiltonian model of oscillator lattice with local coupling. Model describes spatial modes of nonlinear Schrödinger equation with periodic tilted potential. The Hamiltonian system manifests reversibility of Topaj - Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with dynamics exactly equivalent to the Topaj - Pikovsky model. We demonstrate the complexity of dynamics with results of numerical simulations. We also propose two dissipative models close to Topaj - Pikovsky system.

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arXiv2018

Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: Examples and numerical study

A system of $N$ rotators is investigated with a constraint given by a condition of vanishing sum of the cosines of the rotation angles. Equations of the dynamics are formulated and results of numerical simulation for the cases $N$=3, 4, and 5 are presented relating to the geodesic flows on a two-dimensional, three-dimensional, and four-dimensional manifold, respectively, in a compact region (due to the periodicity of the configuration space in angular variables). It is shown that a system of three rotators demonstrates chaos, characterized by one positive Lyapunov exponent, and for systems of four and five elements there are, respectively, two and three positive exponents (`hyperchaos'). An algorithm has been implemented that allows calculating the sectional curvature of a manifold in the course of numerical simulation of the dynamics at points of a trajectory. In the case of $N$=3, curvature of the two-dimensional manifold is negative (except for a finite number of points where it is zero), and Anosov's geodesic flow is realized. For $N$=4 and 5, the computations show that the condition of negative sectional curvature is not fulfilled. Also the methodology is explained and applied for testing hyperbolicity based on numerical analysis of the angles between the subspaces of small perturbation vectors; in the case of $N$=3, the hyperbolicity is confirmed, and for $N$=4 and 5 the hyperbolicity does not take place.

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arXiv2018

Lyapunov analysis of strange pseudohyperbolic attractors: angles between tangent subspaces, local volume expansion and contraction

In this paper we analyze local structure of several chaotic attractors recently suggested in literature as pseudohyperbolic. The absence of tangencies and thus the presence of the pseudohyperbolicity is verified using the method of angles that includes computation of distributions of the angles between the corresponding tangent subspaces. Also we analyze how volume expansion in the first subspace and the contraction in the second one occurs locally. For this purpose we introduce a family of instant Lyapunov exponents. Unlike the well known finite time ones, the instant Lyapunov exponents show expansion or contraction on infinitesimal time intervals. Two types of instant Lyapunov exponents are defined. One is related to ordinary finite time Lyapunov exponents computed in the course of standard algorithm for Lyapunov exponents. Their sums reveal instant volume expanding properties. The second type of instant Lyapunov exponents shows how covariant Lyapunov vectors grow or decay on infinitesimal time. Using both instant and finite time Lyapunov exponents we demonstrate that specific to the pseudohyperbolicity average expanding and contracting properties are typically violated on infinitesimal time. Instantly volumes from the first subspace can sometimes be contacted, directions in the second subspace can sometimes be expanded, and the instant contraction in the first subspace can sometimes be stronger than the contraction in the second subspace.

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arXiv2018

The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass

For a Chaplygin sleigh moving in the presence of weak friction, we present and investigate two mechanisms of arising acceleration due to oscillations of an internal mass. In certain parameter regions, the mechanism induced by small oscillations determines acceleration which is on average one-directional. The role of friction is that the velocity reached in the process of the acceleration is stabilized at a certain level. The second mechanism is due to the effect of parametric excitation of oscillations, when the internal moving particle is comparable in mass with the main platform, and, as occurs, a necessary condition for the acceleration is presence of friction. The parametric instability and the resulting acceleration of the sleigh turn out to be bounded if the line of oscillations of the moving mass is displaced from the center of mass. The steady-state regime of motion is in many cases associated with a chaotic attractor; accordingly, the motion of the sleigh turns out to be similar to the process of random walk on a plane.

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arXiv2017

Belykh attractor in Zaslavsky map and its transformation under smoothing

If we allow non-smooth or discontinuous functions in definition of an evolution operator for dynamical systems, then situations of quasi-hyperbolic chaotic dynamics often occur like, for example, on attractors in model Lozi map and in Belykh map. The present article deals with the quasihyperbolic attractor of Belykh in a map describing a rotator with dissipation driven by periodic kicks, the intensity of which depends on the instantaneous angular coordinate of the rotator as a sawtooth-like function, and also the transformation of the attractor under smoothing of that function is considered. Reduction of the equations to the standard form of the Belykh map is provided. Results of computations illustrating the dynamics of the system with continuous time on the Belykh attractor are presented. Also, results for the model with the smoothed sawtooth function are considered depending on the parameter characterizing the smoothing scale. On graphs of Lyapunov exponents versus a parameter, the smoothing of the sawtooth implies appearance of periodicity windows, which indicates violation of the quasi-hyperbolic nature of the attractor. Charts of dynamic regimes on the parameter plane of the system are also plotted, where regions of periodic motions ("Arnold's tongues") are present, which decrease in size with the decrease in the characteristic scale of the smoothing, and disappear in the limit case of the sawtooth function with a break. Since the Belykh attractor was originally introduced in the radiophysical context (phase-locked loops), the analysis undertaken here is of interest from the point of view of possible exploiting the chaotic dynamics on this attractor in electronic devices.

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arXiv2017

Chaos in three coupled rotators: From Anosov dynamics to hyperbolic attractors

Starting from Anosov chaotic dynamics of geodesic flow on a surface of negative curvature, we develop and consider a number of self-oscillatory systems including those with hinged mechanical coupling of three rotators and a system of rotators interacting through a potential function. These results are used to design an electronic circuit for generation of rough (structurally stable) chaos. Results of numerical integration of the model equations of different degree of accuracy are presented and discussed. Also, circuit simulation of the electronic generator is provided using the NI Multisim environment. Portraits of attractors, waveforms of generated oscillations, Lyapunov exponents, and spectra are considered and found to be in good correspondence for the dynamics on the attractive sets of the self-oscillatory systems and for the original Anosov geodesic flow. The hyperbolic nature of the dynamics is tested numerically using a criterion based on statistics of angles of intersection of stable and unstable subspaces of the perturbation vectors at a reference phase trajectory on the attractor.

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arXiv2017

Numerical test for hyperbolicity in chaotic systems with multiple time delays

We develop an extension of the fast method of angles for hyperbolicity verification in chaotic systems with an arbitrary number of time-delay feedback loops. The adopted method is based on the theory of covariant Lyapunov vectors and provides an efficient algorithm applicable for systems with high-dimensional phase space. Three particular examples of time-delay systems are analyzed and in all cases the expected hyperbolicity is confirmed.

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arXiv2017

Chaotic Communication with Robust Hyperbolic Transmitter and Receiver

New chaos-based communication schemes for transmission of analog and digital information are suggested. The carrier signal is produced by chaotic generator having well-defined oscillation phases at least on short time intervals. For data extraction, a special procedure of phase detection for chaotic signals is developed. An efficiency of the scheme is demonstrated by examples of coupled generators of structurally stable (robust) chaos with hyperbolic attractors of Smale-Williams and Arnold's "cat map" types. Robustness of the transmitter and the receiver makes it possible to transmit information in presence of parameter mismatch or additive noise in the communication channel.

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arXiv2016

Strange Nonchaotic Self-Oscillator

An example of strange nonchaotic attractor (SNA) is discussed in a dissipative system of mechanical nature driven by constant torque applied to one of the elements of the construction. So the external force is not oscillatory, and the system is autonomous. Components of the motion with incommensurable frequencies emerge due to the irrational ratio of sizes of the involved rotating elements. We regard the phenomenon as strange nonchaotic self-oscillations, and its existence sheds new light on the question of feasibility of SNA in autonomous systems.

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arXiv2016

Numerical test for hyperbolicity of chaotic dynamics in time-delay systems

We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.

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arXiv2016

From geodesic flow on a surface of negative curvature to electronic generator of robust chaos

Departing from the geodesic flow on a surface of negative curvature as a classic example of the hyperbolic chaotic dynamics, we propose an electronic circuit operating as a generator of rough chaos. Circuit simulation in NI Multisim software package and numerical integration of the model equations are provided. Results of computations (phase trajectories, time dependences of variables, Lyapunov exponents and Fourier spectra) show good correspondence between the chaotic dynamics on the attractor of the proposed system and of the Anosov dynamics for the original geodesic flow.

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arXiv2015

Four-dimensional system with torus attractor birth via saddle-node bifurcation of limit cycles in content of family of blue sky catastrophes

A new four-dimensional model with quasi-periodic dynamics is suggested. The torus attractor originates via the saddle-node bifurcation, which may be regarded as a member of a bifurcation family embracing different types of blue sky catastrophes. Also the torus birth trough the Neimark-Sacker bifurcation occurs in some other region of the parameter space.

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arXiv2015

Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories

Dynamical equations are formulated and a numerical study is provided for self-oscillatory model systems based on the triple linkage hinge mechanism of Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic mechanical constraint of three rotators as well as systems, where three rotators interact by potential forces. We present and discuss some quantitative characteristics of the chaotic regimes (Lyapunov exponents, power spectrum). Chaotic dynamics of the models we consider are associated with hyperbolic attractors, at least, at relatively small supercriticality of the self-oscillating modes; that follows from numerical analysis of the distribution for angles of intersection of stable and unstable manifolds of phase trajectories on the attractors. In systems based on rotators with interacting potential the hyperbolicity is violated starting from a certain level of excitation.

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