Hidden Attractor in Chua’s Circuits

Kuznetsov, Nikolay; Леонов, Г. А.; Vagaytsev, Vladimir

doi:10.5220/0003530702790283

Cited by 6 publications

(6 citation statements)

References 17 publications

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“…In 2009, as a farther development of effective analytical-numerical methods for the study of oscillations [7], the idea of constructing a hidden chaotic Chua attractor was first proposed by Nikolay Kuznetsov [6,[8][9][10] and, in 2011, the first hidden chaotic attractor in the classical Chua circuit [11][12][13] was discovered. This hidden attractor has a very "thin" basin of attraction, which is not connected with equilibria, and coexists with a stable zero equilibrium, thus being "hidden" for a while for standard physics experiments and mathematical modeling of the circuit with random initial data.…”

Section: Introductionmentioning

confidence: 99%

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

et al. 2022

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After the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real existence of chaos in the study of a physical system developed in two directions. Within the first direction, effective analytic-numerical methods were invented providing the so-called computer-assisted proof of the existence of a chaotic attractor. In the framework of the second direction, attempts were made to detect chaotic behavior directly in a physical experiment, by designing a proper experimental setup. The first remarkable result in this direction is the experiment of L. Chua, in which he designed a simple RLC circuit (Chua circuit) containing a nonlinear element (Chua diode), and managed to demonstrate the real evidence of chaotic behavior in this circuit on the screen of oscilloscope. The mathematical model of the Chua circuit (further, Chua system) is also known to be the first example of a system in which the existence of a chaotic hidden attractor was discovered and the bifurcation scenario of its birth was described. Despite the nontriviality of this discovery and cogency of the procedure for hidden attractor localization, the question of detecting this type of attractor in a physical experiment remained open. This article aims to give an exhaustive answer to this question, demonstrating both a detailed formulation of a radiophysical experiment on the localization of a hidden attractor in the Chua circuit, as well as a thorough description of the relationship between a physical experiment, mathematical modeling, and computer simulation.

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Section: Introductionmentioning

confidence: 99%

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

et al. 2022

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“…After the idea of a \hidden attractor" was introduced and the rst hidden Chua attractor was discovered [69,73,76,80,103,111,112], hidden attractors have received much attention. Results on the study of hidden attractors were presented in a number of invited survey and plenary lectures at various international conferences 2 .…”

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confidence: 99%

Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

Леонов

Kuznetsov

Mokaev

2015

Eur. Phys. J. Spec. Top.

364

191

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Abstract. In this paper, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known GlukhovskyDolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.

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“…Consider the following example of discontinuous system -modified Chua system with discontinues characteristic [40,41,50] [38,43,54,[57][58][59]. So as to model system (9.27) with the help of both Filippov and Gelig-Leonov-Yakubovich definitions, the special event-driven numerical method, described in [71], was used.…”

Section: Numerical Modeling Of Chua Systemmentioning

confidence: 99%

“…Consider the following example of discontinuous system -modified Chua system with discontinues characteristic [40,41,50]…”

Section: Numerical Modeling Of Chua Systemmentioning

confidence: 99%

Theory of Differential Inclusions and Its Application in Mechanics

Kiseleva

Kuznetsov

Леонов

2017

New Perspectives and Applications of Modern Control Theory

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The following chapter deals with systems of differential equations with discontinuous right-hand sides. The key question is how to define the solutions of such systems. The most adequate approach is to treat discontinuous systems as systems with multivalued right-hand sides (differential inclusions). In this work three well-known definitions of solution of discontinuous system are considered. We will demonstrate the difference between these definitions and their application to different mechanical problems. Mathematical models of drilling systems with discontinuous friction torque characteristics are considered. Here, opposite to classical Coulomb symmetric friction law, the friction torque characteristic is asymmetrical. Problem of sudden load chance is studied. Analytical methods of investigation of systems with such asymmetrical friction based on the use of Lyapunov functions are demonstrated. The Watt governor and Chua system are considered to show different aspects of computer modeling of discontinuous systems.

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Hidden Attractor in Chua’s Circuits

Abstract: Notion of hidden attractor (basin does not contain neighborhoods of equilibria) is discussed. Effective analytical-numerical procedure for hidden attractors localization is considered. Existence of hidden attractor in Chua's circuits is demonstrated.

Cited by 6 publications

References 17 publications

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

Theory of Differential Inclusions and Its Application in Mechanics

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