Strange Nonchaotic Attractors of Chua's Circuit with Quasiperiodic Excitation

Zhu, Zhiwen; Liu, Zhong

doi:10.1142/s0218127497000169

Cited by 34 publications

(10 citation statements)

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“…( 1) above, on an analog simulator, and verified SNA dynamics by computing power spectra and Lyapunov experiments. Similar experiments have been implemented in the forced Ueda's circuit [Liu and Zhu, 1996], the forced Chua's circuit [Zhu and Liu, 1997] and the forced Murali-Lakshmanan-Chua circuit [Yang and Bilimgut, 1997].…”

Section: Experiments and Applicationsmentioning

confidence: 99%

Strange Nonchaotic Attractors

Prasad

Negi

Ramaswamy

2001

Int. J. Bifurcation Chaos

148

113

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Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic attractors (SNAs). Such attractors are generic in quasiperiodically driven nonlinear systems, and like strange attractors, are geometrically fractal. The largest Lyapunov exponent is zero or negative: trajectories do not show exponential sensitivity to initial conditions. In recent years, SNAs have been seen in a number of diverse experimental situations ranging from quasiperiodically driven mechanical or electronic systems to plasma discharges. An important connection is the equivalence between a quasiperiodically driven system and the Schrödinger equation for a particle in a related quasiperiodic potential, giving a correspondence between the localized states of the quantum problem with SNAs in the related dynamical system. In this review we discuss the main conceptual issues in the study of SNAs, including the different bifurcations or routes for the creation of such attractors, the methods of characterization, and the nature of dynamical transitions in quasiperiodically forced systems. The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, has emerged as an important means of studying fractal attractors, and this analysis finds useful application here. The ubiquity of such attractors, in conjunction with their several unusual properties, suggest novel applications.

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Section: Experiments and Applicationsmentioning

confidence: 99%

Strange Nonchaotic Attractors

Prasad

Negi

Ramaswamy

2001

Int. J. Bifurcation Chaos

148

113

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“…Its dynamics in the environment of a sinusoidal excitation has been extensively studied in a series of works by Murali and Lakshmanan [Murali & Lakshmanan, 1990;Lakshmanan & Murali, 1991;Murali & Lakshmanan, 1992a,c,b]. Due to its simplicity and the rich content of its nonlinear dynamical phenomena, the driven Chua's circuit continues to evoke renewed interest by researchers in the field of non-linear electronics [Anishchenko et al, 1993;Zhu & Liu, 1997;Elwakil, 2002;Srinivasan et al, 2009] The driven memristive Chua's circuit is obtained by replacing the Chua's diode in the original driven Chua's circuit with the active flux controlled active memristor introduced by the authors in 2011, as its nonlinearity. Further the inductances in parallel in the Driven Chua's circuit are replaced by a single equivalent inductance.…”

Section: Memristive Driven Chua Oscillatormentioning

confidence: 99%

Sliding Bifurcations in the Memristive Murali-Lakshmanan-Chua Circuit and the Memristive Driven Chua Oscillator

Ahamed¹,

Lakshmanan

2020

Preprint

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In this paper we report the occurrence of sliding bifurcations admitted by the memristive Murali-Lakshmanan-Chua circuit [Ishaq & Lakshmanan, 2013] and the memristive driven Chua oscillator [Ishaq et al., 2011]. Both of these circuits have a flux-controlled active memristor designed by the authors in 2011, as their non-linear element. The three segment piecewise-linear characteristic of this memristor bestows on the circuits two discontinuity boundaries, dividing their phase spaces into three sub-regions. For proper choice of parameters, these circuits take on a degree of smoothness equal to one at each of their two discontinuities, thereby causing them to behave as Filippov systems. Sliding bifurcations, which are characteristic of Filippov systems, arise when the periodic orbits in each of the sub-regions, interact with the discontinuity boundaries, giving rise to many interesting dynamical phenomena. The numerical simulations are carried out after incorporating proper zero time discontinuity mapping (ZDM) corrections. These are found to agree well with the experimental observations which we report here appropriately.

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“…2 Thus far, SNAs have been studied in various nonlinear models [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and have been observed in experiments using electronic circuits, [22][23][24][25] magnetoelastic ribbons, 26 and electrochemical cells 27 (see also Refs. 28-30, and references therein).…”

Section: Introductionmentioning

confidence: 99%

“…Experimental distinction between chaotic and strange nonchaotic attractors on the basis of consistency Seiji Uenohara, 1,a) Takahito Mitsui, 2,3 Yoshito Hirata, 3 Takashi Morie, 1 Yoshihiko Horio, 4 and Kazuyuki Aihara 3 We experimentally study strange nonchaotic attractors (SNAs) and chaotic attractors by using a nonlinear integrated circuit driven by a quasiperiodic input signal. An SNA is a geometrically strange attractor for which typical orbits have nonpositive Lyapunov exponents.…”

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confidence: 99%

Experimental distinction between chaotic and strange nonchaotic attractors on the basis of consistency

Uenohara

Mitsui

Hirata

et al. 2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We experimentally study strange nonchaotic attractors (SNAs) and chaotic attractors by using a nonlinear integrated circuit driven by a quasiperiodic input signal. An SNA is a geometrically strange attractor for which typical orbits have nonpositive Lyapunov exponents. It is a difficult problem to distinguish between SNAs and chaotic attractors experimentally. If a system has an SNA as a unique attractor, the system produces an identical response to a repeated quasiperiodic signal, regardless of the initial conditions, after a certain transient time. Such reproducibility of response outputs is called consistency. On the other hand, if the attractor is chaotic, the consistency is low owing to the sensitive dependence on initial conditions. In this paper, we analyze the experimental data for distinguishing between SNAs and chaotic attractors on the basis of the consistency. Strange nonchaotic attractors (SNAs) often appear in dynamical systems driven by a quasiperiodic signal. An SNA is a geometrically strange attractor for which typical orbits have nonpositive Lyapunov exponents. In fact, typical orbits are insensitive to initial conditions under a common quasiperiodic signal. Owing to their characteristics reminiscent of both quasiperiodic order and chaos, SNAs have attracted the attention of researchers from both theoretical and experimental perspectives. Conventionally, the information dimension is often employed to distinguish between SNAs and chaotic attractors in experiments. However, using the information dimension for the experimental distinction between these types of attractors is difficult as the estimation of the information dimension is highly sensitive to noise, and requires rather long time series. Recently, Ngamga et al. 1 proposed a method for distinguishing between SNAs and chaotic attractors, which utilizes the consistency property of SNAs. Consistency here means that a nonlinear system shows the same response to the same input signal after some transient time, regardless of the initial conditions. In the method proposed by Ngamga et al., the determinism for a cross-recurrence plot of two response time series to the same input signal is used as a consistency measure. However, to our knowledge, there is no research paper on using their cross-recurrence method to experimentally distinguish between SNAs and chaotic attractors. In this study, we evaluated the consistency of a system by using the zero-delay normalized cross-correlation and Ngamga et al.'s cross-recurrence methods. By combining spectrum analysis and evaluation of consistency, we showed that SNAs and chaotic attractors can be distinguished more precisely as compared to conventional methods.

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Strange Nonchaotic Attractors of Chua's Circuit with Quasiperiodic Excitation

Cited by 34 publications

References 0 publications

Strange Nonchaotic Attractors

Strange Nonchaotic Attractors

Sliding Bifurcations in the Memristive Murali-Lakshmanan-Chua Circuit and the Memristive Driven Chua Oscillator

Experimental distinction between chaotic and strange nonchaotic attractors on the basis of consistency

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