Generalization of the simplest autonomous chaotic system

Munmuangsaen, Buncha; Srisuchinwong, Banlue; Sprott, Julien Clinton

doi:10.1016/j.physleta.2011.02.028

Cited by 70 publications

(20 citation statements)

References 14 publications

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“…Other oscillators described by systems of three ODEs with only quadratic nonlinearities yield chaotic strange attractors with fractal dimensions D KY that span practically the whole interval from 2 to 3. Those with values slightly greater than 2 come from simple flows with just one or two quadratic nonlinear terms in all three equations [18], while those systems with more quadratic terms yield higher values of D KY , often between 2.5 and 3 [3]. The Rössler and Lorenz systems have their fractal dimensions of 2.01 and 2.06, respectively, since they contain only one (Rössler) and two (Lorenz) quadratic terms.…”

Section: Comparison Of Fractal Dimensionsmentioning

confidence: 99%

Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

Marszałek

2012

Circuits Syst Signal Process

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We present two dual oscillating circuits having a wide spectrum of dynamical properties but relatively simple topologies. Each circuit has five bifurcating parameters, one nonlinear element of cubic current-voltage characteristics, one controlled element, LCR components and a constant biasing source. The circuits can be considered as two coupled oscillators (linear and nonlinear) that form dual jerk circuits. Bifurcation diagrams of the circuits show a rather surprising result that the bifurcation patterns are of the Farey sequence structure and the circuits' dynamics is of a fractal type. The circuits' fractal dimensions of the box counting (capacity) algorithm, Kaplan-Yorke (Lyapunov) type and its modified (improved) version are all estimated to be between 2.26 and 2.52. Our analysis is based on numerical calculations which confirm a close relationship of the circuits' bifurcation patterns with those of the Ford circles and Stern-Brocot trees.

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Section: Comparison Of Fractal Dimensionsmentioning

confidence: 99%

Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

Marszałek

2012

Circuits Syst Signal Process

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“…This latter expression shows that our model belongs to the larger class of "elegant" jerk dynamical systems described in [25]. More interestingly, our model (4) represents one of the simplest autonomous 3D systems reported to date, capable of exhibiting asymmetric double strange attractors (see Sections 4 and 5) depending uniquely on the choice of initial conditions [3,4,17].…”

Section: State Equationmentioning

confidence: 60%

Asymmetric Double Strange Attractors in a Simple Autonomous Jerk Circuit

et al. 2018

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The dynamics of a simple autonomous jerk circuit previously introduced by Sprott in 2011 are investigated. In this paper, the model is described by a three-time continuous dimensional autonomous system with an exponential nonlinearity. Using standard nonlinear techniques such as time series, bifurcation diagrams, Lyapunov exponent plots, and Poincaré sections, the dynamics of the system are characterized with respect to its parameters. Period-doubling bifurcations, periodic windows, and coexisting bifurcations are reported. As a major result of this work, it is found that the system experiences the unusual phenomenon of asymmetric bistability marked by the presence of two different attractors (e.g., screw-like Shilnikov attractor with a spiralling-like Feigenbaum attractor) for the same parameters setting, depending solely on the choice of initial states. Among few cases of lower dimensional systems capable of such type of behavior reported to date (e.g., Colpitts oscillator, Newton-Leipnik system, and hyperchaotic oscillator with gyrators), the jerk circuit/system considered in this work represents the simplest prototype. Results of theoretical analysis are perfectly reproduced by laboratory experimental measurements.

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“…Symmetry always plays an important role in physical system. This property is found in a variety of system including nonlinear and chaotic systems [30][31][32][33][34][35][36][37][38][39][40]. Symmetric chaotic systems provide the possibility to observe coexisting attractors.…”

Section: Introductionmentioning

confidence: 99%

Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization

Tsafack¹,

Kengne²

2019

WJAP

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Background: Since the invention of Chua's circuit, numerous generalizations based on substitution of the nonlinear function have been reported. One of the generalizations is obtained by replacing the piecewise-linear with the cubic and/or quadratic polynomial. These nonlinearities are used to be implement using analog multipliers which are relatively expensive. In this realization we propose a different approach to synthetize both cubic and quadratic nonlinearities of empirical Chua's circuit. Methods: The idea is to use diodes, Opamps and resistors to derive a PWL approximation of the cubic and quadratic functions. To demonstrate some complex phenomena observed in the system using the fourth order Runge-Kutta numerical integration method with a very small integration step. The bifurcation diagram which is the plot of local maxima of the temporal trace of a system's coordinate as a function of the control parameter also constitutes an excellent tool for the study of dynamic systems. Results: The above mentioned standard nonlinear analysis tools have been exploited and it is found that the system with adjustable symmetry experiences a plethora of symmetric and asymmetric coexisting attractors. A particular feature of the system is related to the simplicity of the corresponding electronic analog circuit (no analog multiplier chip used to implement the cubic and quadratic nonlinearities). Conclusions: It is observed that the proposed Chua's circuit system is more flexible (both symmetric and asymmetric) and displays complex dynamics behaviors of symmetric and asymmetric coexisting attractors. Note that this striking dynamic can be exploited in encryption algorithms.

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Generalization of the simplest autonomous chaotic system

Cited by 70 publications

References 14 publications

Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

Circuits with Oscillatory Hierarchical Farey Sequences and Fractal Properties

Asymmetric Double Strange Attractors in a Simple Autonomous Jerk Circuit

Complex Dynamics of the Chua’s Circuit System with Adjustable Symmetry and Nonlinearity: Multistability and Simple Circuit Realization

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