Homoclinic bifurcations in Chua's circuit

Kahan, Sandra; Sicardi-Schifino, A.

doi:10.1016/s0378-4371(98)00389-6

Cited by 21 publications

(13 citation statements)

References 13 publications

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“…23 This parameter space is not completely new, because it was already partially drawn in Ref. 24, in which a few orbits can be observed. In the present work, though, we show its microscopic structure, by revealing the scaling at which subsidiary homoclinic orbits appear in extended regions of the parameter space.…”

Section: Introductionmentioning

confidence: 92%

Shilnikov homoclinic orbit bifurcations in the Chua’s circuit

Medrano-T

Baptista

Caldas

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We analytically describe the complex scenario of homoclinic bifurcations in the Chua's circuit. We obtain a general scaling law that gives the ratio between bifurcation parameters of different nearby homoclinic orbits. As an application of this theoretical approach, we estimate the number of higher order subsidiary homoclinic orbits that appear between two consecutive lower order subsidiary orbits. Our analytical finds might be valid for a large class of dynamical systems and are numerically confirmed in the parameter space of the Chua's circuit.

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Section: Introductionmentioning

confidence: 92%

Shilnikov homoclinic orbit bifurcations in the Chua’s circuit

Medrano-T

Baptista

Caldas

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Bifurcations separate qualitatively different dynamics in dynamical systems as one or more parameters are changed. Bifurcations have been mathematically characterized in elastic-plastic materials 1 , electronic circuits 2 , or in open quantum systems 3 . Also, bifurcations have been theoretically described in population dynamics 4 – 6 , in socioecological systems 7 , 8 , as well as in fixation of alleles in population genetics and computer virus propagation, to name a few examples 9 , 10 .…”

Section: Introductionmentioning

confidence: 99%

Finite-time scaling in local bifurcations

Corral

Sardanyés

Alsedà

2018

Sci Rep

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Finite-size scaling is a key tool in statistical physics, used to infer critical behavior in finite systems. Here we have made use of the analogous concept of finite-time scaling to describe the bifurcation diagram at finite times in discrete (deterministic) dynamical systems. We analytically derive finite-time scaling laws for two ubiquitous transitions given by the transcritical and the saddle-node bifurcation, obtaining exact expressions for the critical exponents and scaling functions. One of the scaling laws, corresponding to the distance of the dynamical variable to the attractor, turns out to be universal, in the sense that it holds for both bifurcations, yielding the same exponents and scaling function. Remarkably, the resulting scaling behavior in the transcritical bifurcation is precisely the same as the one in the (stochastic) Galton-Watson process. Our work establishes a new connection between thermodynamic phase transitions and bifurcations in low-dimensional dynamical systems, and opens new avenues to identify the nature of dynamical shifts in systems for which only short time series are available.

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“…The Chua's circuit has been the object of study of hundreds of papers, where its topological, numerical, physical, and dynamical characterizations are deeply investigated. See [12][13][14][15] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Inductorless Chua's Circuit: Experimental Time Series Analysis

Rubinger

Nascimento

Mello

et al. 2007

Mathematical Problems in Engineering

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We have implemented an operational amplifier inductorless realization of the Chua's circuit. We have registered time series from its dynamical variables with the resistorRas the control parameter and varying from 1300Ωto 2000Ω. Experimental time series at fixedRwere used to reconstruct attractors by the delay vector technique. The flow attractors and their Poincaré maps considering parameters such as the Lyapunov spectrum, its subproduct the Kaplan-Yorke dimension, and the information dimension are also analyzed here. The results for a typical double scroll attractor indicate a chaotic behavior characterized by a positive Lyapunov exponent and with a Kaplan-Yorke dimension of 2.14. The occurrence of chaos was also investigated through numerical simulations of the Chua's circuit set of differential equations.

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Homoclinic bifurcations in Chua's circuit

Cited by 21 publications

References 13 publications

Shilnikov homoclinic orbit bifurcations in the Chua’s circuit

Shilnikov homoclinic orbit bifurcations in the Chua’s circuit

Finite-time scaling in local bifurcations

Inductorless Chua's Circuit: Experimental Time Series Analysis

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