Hidden attractors localization in Chua circuit via the describing function method

Kuznetsov, Nikolay; Кузнецова, О. А.; Леонов, Г. А.; Mokaev, Timur N.; Stankevich, Nataliya

doi:10.1016/j.ifacol.2017.08.470

Cited by 21 publications

(19 citation statements)

References 31 publications

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“…Note that the existence of hidden attractors in the Chua system can be effectively predicted by the describing function method (DFM) Rocha & Medrano-T, 2015;Kuznetsov et al, 2017a]. The classical DFM (see, e.g.…”

Section: Separated Twin-attractorsmentioning

confidence: 99%

“…Theorem Leonov & Kuznetsov, 2013;Kuznetsov et al, 2017a] Consider the describing function Φ(a) = 2π/ω 0 0 ψ(a cos(ω 0 t)) cos(ω 0 t)dt. If there exists a positive a 0 such that Φ(a 0 ) = 0, b 1 Φ (a 0 ) < 0, then system (9) has a stable 5 periodic solution with the initial data w 0 = a 0 + O(ε), 0, O(ε) and period T = 2π ω 0 + O(ε).…”

Section: Separated Twin-attractorsmentioning

confidence: 99%

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Scenario of the Birth of Hidden Attractors in the Chua Circuit

Stankevich

Kuznetsov

Леонов

et al. 2017

Int. J. Bifurcation Chaos

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Recently it was shown that in the dynamical model of Chua circuit both the classical selfexcited and hidden chaotic attractors can be found. In this paper the dynamics of the Chua circuit is revisited. The scenario of the chaotic dynamics development and the birth of selfexcited and hidden attractors is studied. It is shown a pitchfork bifurcation in which a pair of symmetric attractors coexists and merges into one symmetric attractor through an attractormerging bifurcation and a splitting of a single attractor into two attractors. The scenario relating the subcritical Hopf bifurcation near equilibrium points and the birth of hidden attractors is discussed.Comment: 20 pages, 11 figure

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Section: Separated Twin-attractorsmentioning

confidence: 99%

Section: Separated Twin-attractorsmentioning

confidence: 99%

Scenario of the Birth of Hidden Attractors in the Chua Circuit

Stankevich

Kuznetsov

Леонов

et al. 2017

Int. J. Bifurcation Chaos

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“…Also, global stability in the presence of locally unstable equilibria is a typical case for systems describing pendulums, PLLs [11,21,46,69], and electric machines [64]. 4 See, e.g., Andronov-Hopf bifurcation [2,5] and Bautin's "safe" and "dangerous" boundaries of stability [9], and corresponding birth of hidden Chua attractors [43,101]. model has a global bounded convex absorbing set, then over time, the state of the system, observed experimentally, will be attracted to the local attractor contained in the absorbing set.…”

Section: Introductionmentioning

confidence: 99%

The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

et al. 2020

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On the example of the famous Lorenz system, the difficulties and opportunities of reliable numerical analysis of chaotic dynamical systems are discussed in this article. For the Lorenz system, the boundaries of global stability are estimated and the difficulties of numerically studying the birth of selfexcited and hidden attractors, caused by the loss of global stability, are discussed. The problem of reliable numerical computation of the finite-time Lyapunov dimension along the trajectories over large time intervals is discussed. Estimating the Lyapunov dimension of attractors via the Pyragas time-delayed feedback control technique and the Leonov method is demonstrated. Taking into account the problems of reliable numerical experiments in the context of the shadowing and hyperbolicity theories, experiments are carried out on small time intervals and for trajectories on a grid of initial points in the attractor's basin of attraction.

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“…Notice that the chaotic systems mentioned above present quadratic nonlinearities. However, there are a few systems in the literature that generate hidden attractors from switching linear systems [37,38]. In these researches, the main methodology is to find hidden degeneracies in piecewise smooth dynamical systems through the sliding mode methods, and they use of efficient analytical-numerical methods for the study of hidden attractors.…”

Section: Introductionmentioning

confidence: 99%

Hidden attractors from the switching linear systems

et al. 2020

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Recently, chaotic behavior has been studied in dynamical systems that generates hidden attractors. Most of these systems have quadratic nonlinearities. This paper introduces a new methodology to develop a family of three-dimensional hidden attractors from the switching of linear systems. This methodology allows to obtain strange attractors with only one stable equilibrium, attractors with an infinite number of equilibria or attractors without equilibrium. The main matrix and the augmented matrix of every linear system are considered in Rouché-Frobenius theorem to analyze the equilibrium of the switching systems. Also, a systematic search assisted by a computer is used to find the chaotic behavior. Basic chaotic properties of the attractors are verified by the Lyapunov exponents.

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Hidden attractors localization in Chua circuit via the describing function method

Cited by 21 publications

References 31 publications

Scenario of the Birth of Hidden Attractors in the Chua Circuit

Scenario of the Birth of Hidden Attractors in the Chua Circuit

The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

Hidden attractors from the switching linear systems

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