Canonical realization of Chua's circuit family

Chua, Leon O.; Lin, G. N.

doi:10.1109/31.55064

Cited by 302 publications

(107 citation statements)

References 9 publications

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“…In 2010, for the first time, a hidden chaotic attractor was discovered Bragin et al, 2011;Kuznetsov et al, 2011a;Leonov et al, 2011cLeonov et al, , 2012a] in Chua's circuit, described by three-dimensional dynamical system. Note that Chua himself, in analyzing various cases of attractors' existence in Chua's circuit [Chua & Lin, 1990] did not admit the existence of hidden attractor (discovered later) in the circuit.…”

Section: Hidden Attractor In Chua's Circuitsmentioning

confidence: 99%

“…Classical attractors in well-known chaotic dynamical systems of Lorenz [1963], Chua [Chua & Lin, 1990], Chen [Chen & Ueta, 1999], and many others are selfexcited attractors and can be obtained numerically by the standard computational procedure. However, there are chaotic attractors of another type: hidden chaotic attractors, for which the possibility of such simple computational approach turn out to be highly limited.…”

Section: Hidden Attractor In Chua's Circuitsmentioning

confidence: 99%

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Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Леонов

Kuznetsov

2013

Int. J. Bifurcation Chaos

758

414

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From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such attractors.At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which were devoted to constructing nested limit cycles in quadratic systems, that showed the necessity of studying hidden oscillations for solving this problem. Later, the problem of analyzing hidden oscillations arose from engineering problems in automatic control. In the 50-60s of the last century, the investigations of widely known Markus-Yamabe's, Aizerman's, and Kalman's conjectures on absolute stability have led to the finding of hidden oscillations in automatic control systems with a unique stable stationary point. In 1961, Gubar revealed a gap in Kapranov's work on phase locked-loops (PLL) and showed the possibility of the existence of hidden oscillations in PLL. At the end of the last century, the difficulties in analyzing hidden oscillations arose in simulations of drilling systems and aircraft's control systems (anti-windup) which caused crashes.Further investigations on hidden oscillations were greatly encouraged by the present authors' discovery, in 2010 (for the first time), of chaotic hidden attractor in Chua's circuit.This survey is dedicated to efficient analytical-numerical methods for the study of hidden oscillations. Here, an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods. †

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Section: Hidden Attractor In Chua's Circuitsmentioning

confidence: 99%

Section: Hidden Attractor In Chua's Circuitsmentioning

confidence: 99%

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Леонов

Kuznetsov

2013

Int. J. Bifurcation Chaos

758

414

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“…To demonstrate and verify the performance of the proposed method, the dynamics of Chua's circuit system [25] can be expressed as …”

Section: Illustrative Examplesmentioning

confidence: 99%

Chaos control of uncertain time‐delay chaotic systems with input dead‐zone nonlinearity

Pai

2014

Complexity

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This article presents an adaptive sliding mode control (SMC) scheme for the stabilization problem of uncertain time-delay chaotic systems with input dead-zone nonlinearity. The algorithm is based on SMC, adaptive control, and linear matrix inequality technique. Using Lyapunov stability theorem, the proposed control scheme guarantees the stability of overall closed-loop uncertain time-delay chaotic system with input dead-zone nonlinearity. It is shown that the state trajectories converge to zero asymptotically in the presence of input dead-zone nonlinearity, time-delays, nonlinear real-valued functions, parameter uncertainties, and external disturbances simultaneously. The selection of sliding surface and the design of control law are two important issues, which have been addressed. Moreover, the knowledge of upper bound of uncertainties is not required. The reaching phase and chattering phenomenon are eliminated. Simulation results demonstrate the effectiveness and robustness of the proposed scheme.

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“…The Chua circuit in the chaotic regime [35,36] has been used and we have added a measurement noise to the outcoming signal. The noise (white and Gaussian) has come from an electronic noise generator.…”

Section: Noise Level Estimation: Examplesmentioning

confidence: 99%

Noise-level estimation of time series using coarse-grained entropy

Urbanowicz

Hołyst

2003

Phys. Rev. E

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We present a method of noise level estimation that is valid even for high noise levels. The method makes use of the functional dependence of coarse grained correlation entropy K 2 (ε) on the threshold parameter ε. We show that the function K 2 (ε) depends in a characteristic way on the noise standard deviation σ. It follows that observing K 2 (ε) one can estimate the noise level σ.Although the theory has been developed for the gaussian noise added to the observed variable we have checked numerically that the method is also valid for the uniform noise distribution and for the case of Langevine equation corresponding to the dynamical noise. We have verified the validity of our method by applying it to estimate the noise level in several chaotic systems and in the Chua electronic circuit contaminated by noise.

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Canonical realization of Chua's circuit family

Cited by 302 publications

References 9 publications

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Hidden Attractors in Dynamical Systems. From Hidden Oscillations in Hilbert–kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits

Chaos control of uncertain time‐delay chaotic systems with input dead‐zone nonlinearity

Noise-level estimation of time series using coarse-grained entropy

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