A GALLERY OF ATTRACTORS FROM SMOOTH CHUA'S EQUATION

In this tutorial paper, we present some interesting phenomena from Chua's equation with a cubic nonlinearity as well as that with a piecewise-linear characteristic, where a cubic polynomial approximates the original three-segment piecewise-linear function. A gallery of attractors and bifurcation diagrams obtained by numerical simulations are presented. We hope this will motivate researchers to study the smooth version of this extremely simple yet versatile equation with more than 20 attractors.

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DESIGN OF TIME DELAYED CHAOTIC CIRCUIT WITH THRESHOLD CONTROLLER

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127411028751 ◽

2011 ◽

Vol 21 (03) ◽

pp. 725-735 ◽

Cited By ~ 16

Author(s):

K. SRINIVASAN ◽

I. RAJA MOHAMED ◽

K. MURALI ◽

M. LAKSHMANAN ◽

SUDESHNA SINHA

Keyword(s):

Numerical Simulations ◽

Linear Function ◽

Piecewise Linear ◽

Piecewise Linear Function ◽

Operational Amplifiers ◽

Phase Portraits ◽

Chaotic Attractors ◽

Chaotic Oscillator ◽

Threshold Values ◽

Chaotic Circuit

A novel time delayed chaotic oscillator exhibiting mono- and double scroll complex chaotic attractors is designed. This circuit consists of only a few operational amplifiers and diodes and employs a threshold controller for flexibility. It efficiently implements a piecewise linear function. The control of piecewise linear function facilitates controlling the shape of the attractors. This is demonstrated by constructing the phase portraits of the attractors through numerical simulations and hardware experiments. Based on these studies, we find that this circuit can produce multi-scroll chaotic attractors by just introducing more number of threshold values.

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A HYPERCHAOTIC CHUA SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409025146 ◽

2009 ◽

Vol 19 (11) ◽

pp. 3823-3828 ◽

Cited By ~ 18

Author(s):

PAULO C. RECH ◽

HOLOKX A. ALBUQUERQUE

Keyword(s):

Linear Function ◽

Piecewise Linear ◽

Dynamical Behavior ◽

Piecewise Linear Function ◽

Hyperchaotic System ◽

Cubic Polynomial ◽

New Hyperchaotic System

In this paper, we report a new four-dimensional autonomous hyperchaotic system, constructed from a Chua system where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode has been replaced by a cubic polynomial. Analytical and numerical procedures are conducted to study the dynamical behavior of the proposed new hyperchaotic system.

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Nonlinear Analysis of a 2-DOF Piecewise Linear Aeroelastic System

Volume 1: 24th Conference on Mechanical Vibration and Noise, Parts A and B ◽

10.1115/detc2012-70038 ◽

2012 ◽

Author(s):

Tarek A. Elgohary ◽

Tamás Kalmár-Nagy

Keyword(s):

Linear Function ◽

Chaotic Behavior ◽

Initial Conditions ◽

Piecewise Linear ◽

Piecewise Linear Function ◽

Hopf Bifurcations ◽

System Behavior ◽

Bifurcation Diagrams ◽

Aerodynamic Forces ◽

Aeroelastic System

Aerodynamic forces for a 2-DOF aeroelastic system oscillating in pitch and plunge are modeled as a piecewise linear function. Equilibria of the piecewise linear model are obtained and their stability/bifurcations analyzed. Two of the main bifurcations are border collision and rapid/Hopf bifurcations. Continuation is used to generate the bifurcation diagrams of the system. Chaotic behavior following the intermittent route is also observed. To better understand the grazing phenomenon sets of initial conditions associated with the system behavior are defined and analyzed.

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A PARAMETER-SPACE OF A CHUA SYSTEM WITH A SMOOTH NONLINEARITY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409023676 ◽

2009 ◽

Vol 19 (04) ◽

pp. 1351-1355 ◽

Cited By ~ 12

Author(s):

HOLOKX A. ALBUQUERQUE ◽

PAULO C. RECH

Keyword(s):

Numerical Simulations ◽

Parameter Space ◽

Discrete Time ◽

Piecewise Linear ◽

Piecewise Linear Function ◽

Cubic Polynomial ◽

Self Similar ◽

Autonomous Differential Equations ◽

Chua Oscillator ◽

Discrete Time Models

In this paper we investigate, via numerical simulations, the parameter space of the set of autonomous differential equations of a Chua oscillator, where the piecewise-linear function usually taken to describe the nonlinearity of the Chua diode was replaced by a cubic polynomial. As far as we know, we are the first to report that this parameter-space presents islands of periodicity embedded in a sea of chaos, scenario typically observed only in discrete-time models until recently. We show that these islands are self-similar, and organize themselves in period-adding bifurcation cascades.

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PIECEWISE-LINEAR CHAOTIC SYSTEMS WITH A SINGLE EQUILIBRIUM POINT

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127400001286 ◽

2000 ◽

Vol 10 (09) ◽

pp. 2015-2060 ◽

Cited By ~ 12

Author(s):

TAO YANG ◽

LEON O. CHUA

Keyword(s):

Linear Function ◽

Autonomous System ◽

Chaotic Systems ◽

Random Search ◽

Piecewise Linear ◽

Equilibrium Points ◽

Three Dimensional ◽

Piecewise Linear Function ◽

Chaotic Attractors ◽

Bifurcation Diagrams

As a unique paradigm for chaos, the various versions of Chua's circuits and equations consists of a three-dimensional autonomous system with a three-segment piecewise-linear function which gives rise to three equilibrium points. This paper considers the possibility of simplifying the system configurations of piecewise-linear chaotic systems based on the structures of Chua's systems. We study a new class of piecewise-linear three-dimensional autonomous system with a three-segment piecewise-linear function. However, unlike Chua's systems, the systems we study in this paper have only single equilibrium points. To find chaotic attractors from this class of systems, we use a systematic random-search process to search the parameter space. The searching process consists of three stages. For the first stage, we simply count the number of points on a Poincaré section and find candidates for chaotic attractors. At the second stage, Lyapunov exponents are calculated for selecting chaotic attractors from the candidates. Finally, bifurcation diagrams constructed around the located chaotic attractors are used to find different types of chaotic attractors. Many qualitatively different chaotic attractors of this class of systems had been found and presented in this paper. Another method to simplify the configurations of a piecewise-linear chaotic system is to reduce the number of segments of the piecewise-linear function. We have developed some chaotic systems with a two-segment piecewise-linear function and which gives rise to two equilibrium points. Many color illustrations of chaotic attractors and bifurcation diagrams are presented.

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MINLP formulations for continuous piecewise linear function fitting

Computational Optimization and Applications ◽

10.1007/s10589-021-00268-5 ◽

2021 ◽

Author(s):

Noam Goldberg ◽

Steffen Rebennack ◽

Youngdae Kim ◽

Vitaliy Krasko ◽

Sven Leyffer

Keyword(s):

Linear Function ◽

Piecewise Linear ◽

Piecewise Linear Function ◽

Mixed Integer ◽

Function Fitting ◽

Computational Performance ◽

Integer Nonlinear Programming ◽

Minlp Model ◽

Necessary Constraints ◽

Continuous Piecewise Linear Function

AbstractWe consider a nonconvex mixed-integer nonlinear programming (MINLP) model proposed by Goldberg et al. (Comput Optim Appl 58:523–541, 2014. 10.1007/s10589-014-9647-y) for piecewise linear function fitting. We show that this MINLP model is incomplete and can result in a piecewise linear curve that is not the graph of a function, because it misses a set of necessary constraints. We provide two counterexamples to illustrate this effect, and propose three alternative models that correct this behavior. We investigate the theoretical relationship between these models and evaluate their computational performance.

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