Global Bifurcation Analysis of the Double Scroll Circuit

Komuro, M.; Tokunaga, Ryuji; Matsumoto, Takashi; Chua, Leon O.; Hotta, Akitoshi

doi:10.1142/s0218127491000105

Cited by 56 publications

(34 citation statements)

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“…Shrimps should not be confused with their innermost main domain of periodicity." Results by Gaspard et al 45 in 1984, Rössler et al 46 in 1989, and Komuro et al 47 in 1991 already demonstrated the existence of self-similar periodic structures in a 2D-mapping of the Chua's system, the logistic map, and the mapping of the double scroll circuit, respectively. However, it was after the pioneering paper of Gallas 48 in 1993 studying the parameter space of the Hénon map that the parameter space attracted much attention, and since then, it has already been shown that such shrimp-shaped domains can be found in many theoretical models.…”

Section: Introductionmentioning

confidence: 95%

Shrimp-shape domains in a dissipative kicked rotator

Oliveira

Robnik

Leonel

2011

Chaos

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Some dynamical properties for a dissipative kicked rotator are studied. Our results show that when dissipation is taken into account a drastic change happens in the structure of the phase space in the sense that the mixed structure is modified and attracting fixed points and chaotic attractors are observed. A detailed numerical investigation in a two-dimensional parameter space based on the behavior of the Lyapunov exponent is considered. Our results show the existence of infinite self-similar shrimp-shaped structures corresponding to periodic attractors, embedded in a large region corresponding to the chaotic regime. A strongly dissipative kicked rotator is studied. The investigation is based mainly on the asymptotic behavior of the Lyapunov exponents. Our results show that by reducing the two-dimensional map to a one-dimensional map in the limit of infinite kicks, the Feigenbaum's d is recovered. An investigation in the parameter space (the dissipation parameter and the kicking parameter) reveals the existence of infinite families of self-similar structures of shrimp-shape embedded in a large region corresponding to the chaotic regime. The organization of stability shrimps reported here for the discrete-time kicked rotator agrees well with the parameter space organization reported recently in the literature for several flows (continuous-time systems).

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Section: Introductionmentioning

confidence: 95%

Shrimp-shape domains in a dissipative kicked rotator

Oliveira

Robnik

Leonel

2011

Chaos

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“…The double-scroll circuit [10,11] illustrated in Fig. 1(a) has two capacitors (C 1 and C 2 ), one inductor (L), a linear resistor represented by its admittance (g = 1/R), and a nonlinear resistor (R N ).…”

Section: The Double-scroll Circuitmentioning

confidence: 99%

“…We choose to work with the double-scroll circuit [10,11] because it is a piecewise system of the family for which the Shilnikov theorem applies. That is, a homoclinic orbit exists for a set of parameter ranges for which a chaotic attracting set also exists.…”

Section: Introductionmentioning

confidence: 99%

Homoclinic orbits in a piecewise system and their relation with invariant sets

Medrano-T

Baptista

Caldas

2003

Physica D: Nonlinear Phenomena

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“…The behavior of a single oscillator is widely described in the literature (see e.g. [Komuro et al, 1991]). It is characterized by period-doubling bifurcations cascade and bistability, when two symmetric attractors formed near two nontrivial equilibria P 1 and P 2 coexist in the phase space.…”

Section: The Evolution Of Traveling Waves With Parameters Changingmentioning

confidence: 99%

Developing Chaos on Base of Traveling Waves in a Chain of Coupled Oscillators With Period-Doubling: Synchronization and Hierarchy of Multistability Formation

Shabunin

Astakhov

Анищенко

2002

Int. J. Bifurcation Chaos

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The work is devoted to the analysis of dynamics of traveling waves in a chain of self-oscillators with period-doubling route to chaos. As a model we use a ring of Chua's circuits symmetrically coupled via a resistor. We consider how complicated are temporal regimes with parameters changing influences on spatial structures in the chain. We demonstrate that spatial periodicity exists until transition to chaos through period-doubling and tori birth bifurcations of regular regimes. Temporal quasi-periodicity does not induce spatial quasi-periodicity in the ring. After transition to chaos exact spatial periodicity is changed by the spatial periodicity in the average. The periodic spatial structures in the chain are connected with synchronization of oscillations. For quantity researching of the synchronization we propose a measure of chaotic synchronization based on the coherence function and investigate the dependence of the level of synchronization on the strength of coupling and on the chaos developing in the system. We demonstrate that the spatial periodic structure is completely destroyed as a consequence of loss of coherence of oscillations on base frequencies.

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Global Bifurcation Analysis of the Double Scroll Circuit

Cited by 56 publications

References 0 publications

Shrimp-shape domains in a dissipative kicked rotator

Shrimp-shape domains in a dissipative kicked rotator

Homoclinic orbits in a piecewise system and their relation with invariant sets

Developing Chaos on Base of Traveling Waves in a Chain of Coupled Oscillators With Period-Doubling: Synchronization and Hierarchy of Multistability Formation

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