Complex periodic structures in bi-dimensional bifurcation diagrams of a RLC circuit model with a nonlinear NDC device

Cardoso, Julio C.D.; Albuquerque, Holokx A.; Rubinger, R. M.

doi:10.1016/j.physleta.2009.04.024

Cited by 29 publications

(20 citation statements)

References 32 publications

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“…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. [49][50][51][52][53][54][55][56][57][58] Very recently, they were also observed experimentally in a circuit of the Nishio-Inaba family. 59 A recent result from one of the pioneers of chaos theory, 60 was also devoted to this intriguing and rich parameter space structures.…”

Section: Introductionmentioning

confidence: 79%

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: 79%

Shrimp-shape domains in a dissipative kicked rotator

Oliveira

Robnik

Leonel

2011

Chaos

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“…Numerical investigations in parameter‐spaces of continuous systems described by a set of autonomous first‐order differential equations have grown substantially in last years 1–5. The equations of motion that theoretically describe the dynamical behavior of the Chua's circuit 6, 7 are an example of that set of autonomous equations, and are given by where …”

Section: Introductionmentioning

confidence: 99%

Spiral periodic structure inside chaotic region in parameter‐space of a Chua circuit

Albuquerque

Rech

2012

Circuit Theory & Apps

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SUMMARYIn this letter we investigate, via numerical simulations, the parameter-space of the set of autonomous firstorder differential equations of a Chua circuit. We show that this parameter-space presents self-organized periodic structures immersed in a chaotic region, forming a single spiral structure that coils up around a focal point. Additionally, bifurcation diagrams are used to show that those periodic structures also organize themselves in period-adding cascades, along specific directions that point towards this same focal point.

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“…[1][2][3][4][5][6] Chaotic behaviors in continuous-time dynamical system models are known for a long time, but with the nowadays increased computational processing capacity, intricate phenomena became assessable and just recently have been reported in continuous-time models. As already known for discrete time parameter spaces, the abundant presence of self-similar periodic structures embedded in large chaotic regions has also been reported for continuous time models and such structures organize themselves in most varied forms.…”

Section: Introductionmentioning

confidence: 99%

High-resolution parameter space of an experimental chaotic circuit

Viana

Rubinger

Albuquerque

et al. 2010

Chaos

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A high-resolution codimension-two parameter space showing the abundance of complex periodic structures of an experimental chaotic circuit is reported. Such resolution was propitiated by the use of a 0.5 mV step dc voltage source as one of the control parameters. Those complex periodic structures organize themselves in a period-adding bifurcation cascade that accumulates in a chaotic region. Numerical investigations on the dynamical model were also carried out to corroborate several new features observed in the experimental high-resolution parameter space.

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Complex periodic structures in bi-dimensional bifurcation diagrams of a RLC circuit model with a nonlinear NDC device

Cited by 29 publications

References 32 publications

Shrimp-shape domains in a dissipative kicked rotator

Shrimp-shape domains in a dissipative kicked rotator

Spiral periodic structure inside chaotic region in parameter‐space of a Chua circuit

High-resolution parameter space of an experimental chaotic circuit

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