Generating Multiple Chaotic Attractors from Sprott B System

2016 ◽  
Vol 26 (11) ◽  
pp. 1650177 ◽  
Author(s):  
Qiang Lai ◽  
Shiming Chen
Keyword(s):  
Lyapunov Exponent ◽  
Function Method ◽  
Polynomial Function ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Phenomenon ◽  
Bifurcation Diagrams ◽  
Initial Values ◽  
Index 2 ◽  
Lyapunov Exponent Spectrum

Multiple chaotic attractors, implying several independent chaotic attractors generated simultaneously in a system from different initial values, are a very interesting and important nonlinear phenomenon, but there are few studies that have previously addressed it to our best knowledge. In this paper, we propose a polynomial function method for generating multiple chaotic attractors from the Sprott B system. The polynomial function extends the number of index-2 saddle foci, which determines the emergence of multiple chaotic attractors in the system. The analysis of the equilibria is presented. Two coexisting chaotic attractors, three coexisting chaotic attractors and four coexisting chaotic attractors are investigated for verifying the effectiveness of the method. The chaotic characteristics of the attractors are shown by bifurcation diagrams, Lyapunov exponent spectrum and phase portraits.

THE GENERATION AND ANALYSIS OF A NEW FOUR-DIMENSIONAL HYPERCHAOTIC SYSTEM

2007 ◽  
Vol 18 (06) ◽  
pp. 1013-1024 ◽  
Author(s):  
JIEZHI WANG ◽  
ZENGQIANG CHEN ◽  
ZHUZHI YUAN
Keyword(s):  
Lyapunov Exponent ◽  
Cross Product ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Bifurcation Diagrams ◽  
Dynamic Behaviors ◽  
System Parameters ◽  
Product Term ◽  
Two Parameters ◽  
Cross Product Term

A new four-dimensional continuous autonomous hyperchaotic system is considered. It possesses two parameters, and each equation of it has one quadratic cross product term. Some basic properties of it are studied. The dynamic behaviors of it are analyzed by the Lyapunov exponent (LE) spectrum, bifurcation diagrams, phase portraits, and Poincaré sections. The system has larger hyperchaotic region. When it is hyperchaotic, the two positive LE are both large and they are both larger than 1 if the system parameters are taken appropriately.

Generation and Circuit Implementation of Multi-Scroll Chaotic Attractors with Controllable Direction

2014 ◽  
Vol 513-517 ◽  
pp. 4559-4562
Author(s):  
Xiao Wen Luo ◽  
Chun Hua Wang
Keyword(s):  
Lyapunov Exponent ◽  
Control Function ◽  
Equilibrium Points ◽  
Nonlinear Function ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
Relative Coefficient ◽  
Jerk System ◽  
Lyapunov Exponent Spectrum

An approach for generating multi-scroll chaotic attractors with controllable direction in one plane is proposed. Firstly, an appropriate nonlinear function is selected to control the number and direction of multi-scroll chaotic attractors in the three-order Jerk system. Then, we add new control function to Jerk system and observe Lyapunov exponent spectrum of relative coefficient and the change of equilibrium points. Different multi-scroll chaotic attractors with controllable direction are generated by adjusting the coefficient of the control function in a plane. The implementation of circuit verifies the feasibility of this method.

CHAOTIC DYNAMICS OF A FIVE-DIMENSIONAL NONLINEAR NETWORK

2007 ◽  
Vol 18 (03) ◽  
pp. 335-342
Author(s):  
XUEWEI JIANG ◽  
DI YUAN ◽  
YI XIAO
Keyword(s):  
Lyapunov Exponent ◽  
Power Spectrum ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Chinese Traditional Medicine ◽  
Bifurcation Diagrams ◽  
Nonlinear Network ◽  
Dynamical Behaviors ◽  
Lyapunov Exponent Spectrum ◽  
Period Doubling Bifurcations

The dynamics of a five-dimensional nonlinear network based on the theory of Chinese traditional medicine is studied by the Lyapunov exponent spectrum, Poincaré, power spectrum and bifurcation diagrams. The result shows that this system has complex dynamical behaviors, such as chaotic ones. It also shows that the system evolves into chaos through a series of period-doubling bifurcations.

scholarly journals Symmetric Coexisting Attractors in a Novel Memristors-Based Chuas Chaotic System

2021 ◽  
Author(s):  
Shaohui Yan ◽  
Zhenlong Song ◽  
Wanlin Shi
Keyword(s):  
Lyapunov Exponent ◽  
Chaotic System ◽  
Analog Circuits ◽  
Complex Dynamics ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Initial Value ◽  
Field Programmable ◽  
A Charge ◽  
Lyapunov Exponent Spectrum

This paper introduces a charge-controlled memristor based on the classical Chuas circuit. It also designs a novel four-dimensional chaotic system and investigates its complex dynamics, including phase portrait, Lyapunov exponent spectrum, bifurcation diagram, equilibrium point, dissipation and stability. The system appears as single-wing, double-wings chaotic attractors and the Lyapunov exponent spectrum of the system is symmetric with respect to the initial value. In addition, symmetric and asymmetric coexisting attractors are generated by changing the initial value and parameters. The findings indicate that the circuit system is equipped with excellent multi-stability. Finally, the circuit is implemented in Field Programmable Gate Array (FPGA) and analog circuits.

scholarly journals A Meminductor-based Chaotic System

2020 ◽  
Vol 49 (2) ◽  
pp. 317-332
Author(s):  
Aixue Qi ◽  
Lei Ding ◽  
Wenbo Liu
Keyword(s):  
Theoretical Analysis ◽  
Chaotic System ◽  
Phase Trajectory ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Circuit Parameter ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Simulation Results

We propose a meminductor-based chaotic system. Theoretical analysis and numerical simulations reveal complex dynamical behaviors of the proposed meminductor-based chaotic system with five unstable equilibrium points and three different states of chaotic attractors in its phase trajectory with only a single change in circuit parameter. Lyapunov exponents, bifurcation diagrams, and phase portraits are used to investigate its complex chaotic and multi-stability behaviors, including its coexisting chaotic, periodic and point attractors. The proposed meminductor-based chaotic system was implemented using analog integrators, inverters, summers, and multipliers. PSPICE simulation results verified different chaotic characteristics of the proposed circuit with a single change in a resistor value.

A Simple Method for Generating Mirror Symmetry Composite Multiscroll Chaotic Attractors

2020 ◽  
Vol 30 (11) ◽  
pp. 2050220
Author(s):  
Xuenan Peng ◽  
Yicheng Zeng
Keyword(s):  
Lyapunov Exponent ◽  
Mirror Symmetry ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Attractors ◽  
Simple Method ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
Lyapunov Exponent Spectrum ◽  
New System

For further increasing the complexity of chaotic attractors, a new method for generating Mirror Symmetry Composite Multiscroll Chaotic Attractors (MSCMCA) is proposed. We take the Lorenz system as an example to explain the mechanism of the method. Firstly, by varying the signs and magnitudes of the nonlinear terms, the Lorenz system generates symmetrical attractors and different-magnitude attractors, respectively. Secondly, a modified Lorenz system is constructed by imposing several unified multilevel-logic pulse signals to the Lorenz system. The new system generates a novel chaotic attractor consisting of two pairs of different-magnitude symmetrical attractors. By adjusting the parameters of the pulse signals, the modified Lorenz system can also be controlled to generate novel grid multiscroll chaotic attractors, namely MSCMCA. Several dynamical behaviors of the new system are shown by equilibria analysis and Lyapunov exponent spectrum. Moreover, the method can be applied to other chaotic systems. Finally, a circuit of the modified Lorenz system is designed by Multisim software, and the simulation result proves the effectiveness of the method.

scholarly journals An Integer-Order Memristive System with Two- to Four-Scroll Chaotic Attractors and Its Fractional-Order Version with a Coexisting Chaotic Attractor

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Ping Zhou ◽  
Meihua Ke
Keyword(s):  
Lyapunov Exponent ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Chaotic Attractors ◽  
Largest Lyapunov Exponent ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Integer Order ◽  
Memristive System ◽  
Lyapunov Exponent Spectrum

First, based on a linear passive capacitor C, a linear passive inductor L, an active-charge-controlled memristor, and a fourth-degree polynomial function determined by charge, an integer-order memristive system is suggested. The proposed integer-order memristive system can generate two-scroll, three-scroll, and four-scroll chaotic attractors. The complex dynamics behaviors are investigated numerically. The Lyapunov exponent spectrum with respect to linear passive inductor L and the two-scroll, three-scroll, and four-scroll chaotic attractors are yielded by numerical calculation. Second, based on the integer-order memristive chaotic system with a four-scroll attractor, a fractional-order version memristive system is suggested. The complex dynamics behaviors of its fractional-order version are studied numerically. The largest Lyapunov exponent spectrum with respect to fractional-order p is yielded. The coexisting two kinds of three-scroll chaotic attractors and the coexisting three-scroll and four-scroll chaotic attractors can be found in its fractional-order version.

Control of Multistability in a Self-Excited Memristive Hyperchaotic Oscillator

2019 ◽  
Vol 29 (09) ◽  
pp. 1950119 ◽  
Author(s):  
T. Fonzin Fozin ◽  
R. Kengne ◽  
J. Kengne ◽  
K. Srinivasan ◽  
M. Souffo Tagueu ◽  
...  
Keyword(s):  
Coupling Parameter ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Weak Values ◽  
Controlled System ◽  
Nonlinear Dynamical ◽  
System Parameters ◽  
Control Scheme ◽  
Lyapunov Exponent Spectrum

This paper investigates the control of multistability in a self-excited memristive hyperchaotic oscillator using linear augmentation method. Such a method is advantageous in the case of system parameters that are inaccessible. The effectiveness of the applied control scheme is revealed numerically through the nonlinear dynamical tools including bifurcation diagrams, Lyapunov exponent spectrum, phase portraits, basins of attraction and relative basin sizes. Results of such numerical methods reveal that the asymmetric pair of chaotic attractors which were coexisting with the symmetric periodic one in the system, are progressively annihilated as the coupling parameter is increasing. The main transitions observed in the control system are the coexistence of three distinct attractors for weak values of the coupling strength. Above a certain critical value of the coupling parameter, only two attractors are now coexisting within the system. Finally, for higher values of the control strength, the controlled system becomes regular and monostable.

Hopf Bifurcation and Chaos Synchronization for a Class of Mechanical Centrifugal Flywheel Governor System

2007 ◽  
Vol 348-349 ◽  
pp. 349-352
Author(s):  
Yan Dong Chu ◽  
Jian Gang Zhang ◽  
Xian Feng Li ◽  
Ying Xiang Chang
Keyword(s):  
Hopf Bifurcation ◽  
Lyapunov Exponent ◽  
Chaos Synchronization ◽  
Chaotic Systems ◽  
Dynamical Equation ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Control Laws ◽  
Synchronization Scheme

Chaos and chaos synchronization of the centrifugal flywheel governor system are studied in this paper. By mechanics analyzing, the dynamical equation of the centrifugal flywheel governor system are established. Because of the non-linear terms of the system, the system exhibit both regular and chaotic motions. The evolution from Hopf bifurcation to chaos is shown by the bifurcation diagrams and a series of Poincaré sections under different sets of system parameters, and the bifurcation diagrams are verified by the related Lyapunov exponent spectra. This paper addresses control for the chaos synchronization of feedback control laws in two coupled non-autonomous chaotic systems with different coupling terms, which is demonstrated and verified by Lyapunov exponent spectra and phase portraits. Finally, numerical simulations are presented to show the effectiveness of the proposed chaos synchronization scheme.

scholarly journals Coexisting infinitely many attractors in a new chaotic system with a curve of equilibria: Its extreme multi-stability and Kolmogorov–Sinai entropy computation

2019 ◽  
Vol 11 (11) ◽  
pp. 168781401988804
Author(s):  
Atefeh Ahmadi ◽  
Xiong Wang ◽  
Fahimeh Nazarimehr ◽  
Fawaz E Alsaadi ◽  
Fuad E Alsaadi ◽  
...  
Keyword(s):  
Dynamical System ◽  
Mathematical Model ◽  
Lyapunov Exponent ◽  
Chaotic System ◽  
Phase Portraits ◽  
Initial Condition ◽  
Bifurcation Diagrams ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
The Mathematical Model

A new five-dimensional chaotic system with extreme multi-stability is introduced in this article. The mathematical model is established, and numerical simulations are done. This dynamical system complicates incident of extreme multi-stability. Most significantly, relied on the mathematical model, the recently proposed system has a curve of equilibria that ends in the occurrence of hidden attractors. We examine the initial-condition-dependent dynamics of this system. We inspect that there is an unrestricted number of coexistent attractors, which signifies the occurrence of extreme multi-stability strictly. In addition, the extreme multi-stability according to initial condition is investigated consuming the Lyapunov exponent spectra and bifurcation diagrams. The existence of coexisting infinitely many attractors is displayed with phase portraits. In the end, we calculate and debate Kolmogorov–Sinai entropy in the chaotic system. We direct trying the Kolmogorov–Sinai technique of entropic inspection for the dynamics of the system.

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