PIECEWISE-LINEAR CHAOTIC SYSTEMS WITH A SINGLE EQUILIBRIUM POINT

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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A Novel Method for Constructing Grid Multi-Wing Butterfly Chaotic Attractors via Nonlinear Coupling Control

Journal of Electrical and Computer Engineering ◽

10.1155/2016/9143989 ◽

2016 ◽

Vol 2016 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Yun Huang

Keyword(s):

Piecewise Linear ◽

Equilibrium Points ◽

Three Dimensional ◽

Piecewise Linear Function ◽

Chaotic Attractors ◽

Nonlinear Coupling ◽

Linear Coupling ◽

First Order ◽

Novel Method ◽

Chaotic Characteristic

A new method is presented to construct grid multi-wing butterfly chaotic attractors. Based on the three-dimensional Lorenz system, two first-order differential equations are added along with one linear coupling controller, respectively. And a piecewise linear function, which is taken into the linear coupling controller, is designed to form a nonlinear coupling controller; thus a five-dimensional chaotic system is produced, which is able to generate gird multi-wing butterfly chaotic attractors. Through the analysis of the equilibrium points, Lyapunov exponent spectrums, bifurcation diagrams, and Poincaré mapping in this system, the chaotic characteristic of the system is verified. Apart from the research above, an electronic circuit is designed to implement the system. The circuit experimental results are in accordance with the results of numerical simulation, which verify the availability and feasibility of this method.

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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 GENERALIZED 3-D FOUR-WING CHAOTIC SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409025171 ◽

2009 ◽

Vol 19 (11) ◽

pp. 3841-3853 ◽

Cited By ~ 3

Author(s):

ZENGHUI WANG ◽

GUOYUAN QI ◽

YANXIA SUN ◽

MICHAËL ANTONIE VAN WYK ◽

BAREND JACOBUS VAN WYK

Keyword(s):

Phase Diagrams ◽

Lyapunov Exponents ◽

Chaotic System ◽

Chaotic Attractor ◽

Autonomous System ◽

Chaotic Systems ◽

Three Dimensional ◽

Bifurcation Diagrams ◽

Basic Properties ◽

New System

In this paper, several three-dimensional (3-D) four-wing smooth quadratic autonomous chaotic systems are analyzed. It is shown that these systems have similar features. A simpler and generalized 3-D continuous autonomous system is proposed based on these features which can be extended to existing 3-D four-wing chaotic systems by adding some linear and/or quadratic terms. The new system can generate a four-wing chaotic attractor with simple topological structures. Some basic properties of the new system is analyzed by means of Lyapunov exponents, bifurcation diagrams and Poincaré maps. Phase diagrams show that the equilibria are related to the existence of multiple wings.

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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 GALLERY OF ATTRACTORS FROM SMOOTH CHUA'S EQUATION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127405011990 ◽

2005 ◽

Vol 15 (01) ◽

pp. 1-49 ◽

Cited By ~ 73

Author(s):

AKIO TSUNEDA

Keyword(s):

Numerical Simulations ◽

Linear Function ◽

Piecewise Linear ◽

Piecewise Linear Function ◽

Bifurcation Diagrams ◽

Cubic Nonlinearity ◽

Linear Characteristic ◽

Cubic Polynomial

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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A Novel Cubic–Equilibrium Chaotic System with Coexisting Hidden Attractors: Analysis, and Circuit Implementation

Journal of Circuits System and Computers ◽

10.1142/s0218126618500664 ◽

2017 ◽

Vol 27 (04) ◽

pp. 1850066 ◽

Cited By ~ 36

Author(s):

Viet-Thanh Pham ◽

Christos Volos ◽

Sajad Jafari ◽

Tomasz Kapitaniak

Keyword(s):

Chaotic System ◽

Autonomous System ◽

Chaotic Systems ◽

Electronic Circuit ◽

Complex Dynamics ◽

Three Dimensional ◽

Chaotic Attractors ◽

Circuit Implementation ◽

Hidden Attractors ◽

Dynamical Properties

Chaotic systems with a curve of equilibria have attracted considerable interest in theoretical researches and engineering applications because they are categorized as systems with hidden attractors. In this paper, we introduce a new three-dimensional autonomous system with cubic equilibrium. Fundamental dynamical properties and complex dynamics of the system have been investigated. Of particular interest is the coexistence of chaotic attractors in the proposed system. Furthermore, we have designed and implemented an electronic circuit to verify the feasibility of such a system with cubic equilibrium.

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ON CONSTRUCTING COMPLEX GRID MULTIWING CHAOTIC SYSTEM BY SWITCHING CONTROL AND MIRROR SYMMETRY CONVERSION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413501150 ◽

2013 ◽

Vol 23 (07) ◽

pp. 1350115 ◽

Cited By ~ 2

Author(s):

CHAOXIA ZHANG ◽

SIMIN YU ◽

QIANHUA HE ◽

JINXIN RUAN

Keyword(s):

Chaotic System ◽

Mirror Symmetry ◽

Chaotic Systems ◽

Equilibrium Points ◽

Three Dimensional ◽

Quadratic Function ◽

Chaotic Attractors ◽

Novel Method ◽

Switching Controller ◽

Index 2

This paper further investigates a novel method, namely the mirror and double-mirror symmetry conversion both in z direction and in y direction, for generating complex grid multiwing chaotic attractors from a three-dimensional quadratic chaotic system. First, by designing a switching controller with even-symmetry multisegment quadratic function to extend the number of saddle-focus equilibrium points with index 2 in x direction, multiwing chaotic attractors are obtained. Based on this approach, then, different from the methods proposed in the previous literature, by the mirror and double-mirror symmetry conversion with respect to x-axis and y-axis respectively, various intended grid n × m-wing chaotic systems can be obtained. The principle and method for generating grid multiwing are also given. Numerical simulations and circuit realizations have demonstrated the feasibility and effectiveness of the proposed approaches.

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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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