The birth of the global stability theory and the theory of hidden oscillations

Kuznetsov, Nikolay; Lobachev, Mikhail Y.; Yuldashev, M. V.; Yuldashev, R. V.; Kudryashova, Elena V.; Rosenwasser, Efim N.; S.M, Abramovich

doi:10.23919/ecc51009.2020.9143726

Cited by 22 publications

(14 citation statements)

References 49 publications

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“…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. In recent years, the discovery of hidden Chua attractors led to the emergence of the theory of hidden oscillations [6,14,15], which represents the genesis of the modern era of Andronov's theory of oscillations and has attracted attention from the world's scientific community (see, e.g., [16] and references within). 1 In this work, using the Chua circuit as an example, we demonstrate the features of circuit simulation and the possibility of observing hidden attractors in a radiophysical experiment, and also compare the results obtained with mathematical modeling.…”

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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“…To analyse the pull-in range of system (2) with piecewise-linear PD characteristic, we apply the direct Lyapunov method and the corresponding theorem on global stability for the cylindrical phase space (see, e.g. [Leonov & Kuznetsov, 2014;Kuznetsov et al, 2020b]). If there is a continuous function V (x, θ e ) : R n → R such that (i) V (x, θ e + 2π) = V (x, θ e ) ∀x ∈ R n−1 , ∀θ e ∈ R;…”

Section: Global Stability Analysismentioning

confidence: 99%

The Gardner problem and cycle slipping bifurcation for type 2 phase-locked loops

Kuznetsov,

Arseniev,

Blagov

et al. 2021

Preprint

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“…The absence of hidden oscillations can be obtained by Barbashin-Krasovskii theorem (see e.g. [6,9]) and the Lyapunov function, if |r + 1| < 2, from Theorem 3 for ϑ := 0 and γ := σ we get the following Lyapunov function:…”

Section: Inner Estimation: the Global Stability And Trivial Attractorsmentioning

confidence: 99%

“…In some cases, the global stability property can be established analytically (e.g. by Lyapunov-type, either frequency-domain methods, see [6][7][8][9]), and the global stability boundary can be constructed in the parameter space. Crossing this analytical boundary can lead to the following scenarios.…”

Section: Introductionmentioning

confidence: 99%

Analytical and numerical study of the hidden boundary of practical stability: complex versus real Lorenz systems

Kuznetsov¹,

Mokaev²,

Shoreh³

et al. 2021

Preprint

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This work presents the continuation of the recent article "The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension" [1], published in the Nonlinear Dynamics journal. In this work, in comparison with the results for classical real-valued Lorenz system (henceforward -Lorenz system), the problem of analytical and numerical identification of the boundary of global stability for the complex-valued Lorenz system (henceforward -complex Lorenz system) is studied. As in the case of the Lorenz system, to estimate the inner boundary of global stability the possibility of using the mathematical apparatus of Lyapunov functions (namely, the Barbashin-Krasovskii and LaSalle theorems) is demonstrated. For additional analysis of homoclinic bifurcations in complex Lorenz system a special analytical approach by Vladimirov is utilized. To outline the outer boundary of global stability and identify the so-called hidden boundary of global stability, possible birth of hidden attractors and transient chaotic sets is analyzed.

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The birth of the global stability theory and the theory of hidden oscillations

Cited by 22 publications

References 49 publications

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

The Gardner problem and cycle slipping bifurcation for type 2 phase-locked loops

Analytical and numerical study of the hidden boundary of practical stability: complex versus real Lorenz systems

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