Coexisting Chaotic Attractors in Chua's Circuit

Lozi, René; Ushiki, Shigehiro

doi:10.1142/s021812749100066x

Cited by 25 publications

(8 citation statements)

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“…6 The coexistence of two stable limit cycles of different amplitudes (and frequencies) separated by an unstable limit cycle is the signature of birhythmicity. Birhythmicity may also appear in chaotic oscillators, e.g., the coexistence of two chaotic attractors has been reported in the literature 7 and studied in detail in Ref. 8.…”

Section: Introductionmentioning

confidence: 95%

Control of birhythmicity: A self-feedback approach

Biswas

Banerjee

Kurths

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Birhythmicity occurs in many natural and artificial systems. In this paper, we propose a selffeedback scheme to control birhythmicity. To establish the efficacy and generality of the proposed control scheme, we apply it on three birhythmic oscillators from diverse fields of natural science, namely, an energy harvesting system, the p53-Mdm2 network for protein genesis (the OAK model), and a glycolysis model (modified Decroly-Goldbeter model). Using the harmonic decomposition technique and energy balance method, we derive the analytical conditions for the control of birhythmicity. A detailed numerical bifurcation analysis in the parameter space establishes that the control scheme is capable of eliminating birhythmicity and it can also induce transitions between different forms of bistability. As the proposed control scheme is quite general, it can be applied for control of several real systems, particularly in biochemical and engineering systems. Multistability appears in diverse forms, and their study is an exciting topic of research in science and engineering. A particular form of multistability is bistability: it shows many variants, such as the coexistence of two stable steady states (SSS), one stable steady state and one stable limit cycle (LC), two stable limit cycles, or two chaotic attractors. Birhythmicity is the phenomenon of coexistence of two stable limit cycles separated by an unstable limit cycle with different amplitudes and frequencies. In many physical systems, birhythmicity is undesirable as in energy harvesting systems, but in most biological systems, e.g., enzymatic oscillations, it is desirable. Therefore, control of birhythmicity is of utmost importance. Although the control of multistability is a well studied topic, the control of birhythmicity has not been explored to that extent. In this paper, we propose a control scheme that can effectively control and, whenever required, can eliminate birhythmicity. We theoretically explore and numerically establish the technique of control of birhythmicity and transitions to any desired attractor. A number of engineering and biological systems are investigated with the proposed control scheme to establish the efficacy and generality of the scheme. The main essence of this control scheme lies in the fact that it is easily realizable and offers an efficient mean to control birhythmicity.

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Section: Introductionmentioning

confidence: 95%

Control of birhythmicity: A self-feedback approach

Biswas

Banerjee

Kurths

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The resulting three-dimensional system was only producing limit cycles at the time. At an international congress on rhythmic functions held on September [8][9][10][11][12]1975 in Vienna, he met Art Winfree, a theoretical biologist who challenged him to find a biochemical reaction reproducing the Lorenz attractor. Rössler failed to find a chemical or biochemical reaction producing the Lorenz attractor but, he instead found a simpler type of chaos in a paper he wrote during the 1975 Christmas holidays.…”

Section: Rössler Attractor (1976)mentioning

confidence: 99%

“…62 The confinor theory, when applied to Chua's circuit allowed the discovery of coexisting chaotic regimes. 8 …”

Section: Chua Attractor (1983)mentioning

confidence: 99%

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Can we trust in numerical computations of chaotic solutions of dynamical systems ?

Lozi

2013

Topology and Dynamics of Chaos

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From Laser Dynamics to Topology of Chaos, (celebrating the 70th birthday of Prof Robert Gilmore, Rouen, June 28-30, 2011), Ed. Ch. Letellier.Since the famous paper of E. Lorenz in 1963 numerical computations using computers play a central role in order to display and analyze solutions of nonlinear dynamical systems. By these means new structures have been emphasized like hyperbolic and/or strange attractors. However theoretical proofs of their existence are very di¢ cult and limited to very special linear cases. Computer aided proofs are also complex and require special interval arithmetic analysis. Nevertheless, numerous researchers in several fields linked to chaotic dynamical systems are confident in the numerical solutions they found using popular software and publish without checking carefully the reliability of their results. In the simple case of discrete dynamical systems (e.g. Hénon map) there are concerns about the nature of what a computer find out : long unstable pseudo-orbits or strange attractors? The shadowing property and its generalizations which ensure that pseudo-orbits of a homeomorphism can be traceable by actual orbits even if rounding errors are not inevitable are not of great help in order to validate the numerical results. Continuous dynamical systems (e.g. Chua, Lorenz, Rössler) are even more difficult to handle in this scope and researchers have to be very cautious to back up theory with numerical computations. We present a survey of the topic based on these, only few, but well studied models

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“…Limit cycles, homoclinic orbits and chaos are observed in three-dimensional dynamical systems [34][35][36][37]. We stress that these trajectories were discovered in the modulation equations of different mechanical systems [38][39][40].…”

Section: Non-linear Beam Oscillationsmentioning

confidence: 99%

Non-Linear Beam Oscillations Excited by Lateral Force at Combination Resonance

Аврамов¹

2002

Journal of Sound and Vibration

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Coexisting Chaotic Attractors in Chua's Circuit

Cited by 25 publications

References 0 publications

Control of birhythmicity: A self-feedback approach

Control of birhythmicity: A self-feedback approach

Can we trust in numerical computations of chaotic solutions of dynamical systems ?

Non-Linear Beam Oscillations Excited by Lateral Force at Combination Resonance

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