A novel 4D chaotic system with multistability: Dynamical analysis, circuit implementation, control design

2019 ◽  
Vol 33 (21) ◽  
pp. 1950240 ◽  
Author(s):  
Jian-Jun He ◽  
Bang-Cheng Lai
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Electronic Circuit ◽  
Period Doubling ◽  
Control Design ◽  
Critical Value ◽  
Dynamical Analysis ◽  
Chaotic Attractors ◽  
Hopf Bifurcations ◽  
The Novel

The purpose of this work is to introduce a novel 4D chaotic system and investigate its multistability. The novel system has an unstable origin and two stable symmetrical hyperbolic equilibria. When the parameter increases across a critical value, the equilibria lose their stability and double Hopf bifurcations occur with the appearance of limit cycles. A pair of point, periodic, chaotic attractors are observed in the system from different initial values for given parameters. The chaos of the system is yielded via period-doubling bifurcation. A double-scroll chaotic attractor is numerically observed as well. By using the electronic circuit, the chaotic attractor of the system is realized. The control problem of the system is reported. An effective controller is designed to stabilize the system.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

A NEW CHAOTIC SYSTEM AND BEYOND: THE GENERALIZED LORENZ-LIKE SYSTEM

2004 ◽  
Vol 14 (05) ◽  
pp. 1507-1537 ◽  
Author(s):  
JINHU LÜ ◽  
GUANRONG CHEN ◽  
DAIZHAN CHENG
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Compound Structure ◽  
Constant Input ◽  
New Class ◽  
Dynamical Behaviors ◽  
Generalized Lorenz System ◽  
New Chaotic System

This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display (i) two 1-scroll chaotic attractors simultaneously, with only three equilibria, and (ii) two 2-scroll chaotic attractors simultaneously, with five equilibria. Several issues such as some basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new chaotic system are then investigated, either analytically or numerically. Of particular interest is the fact that this chaotic system can generate a complex 4-scroll chaotic attractor or confine two attractors to a 2-scroll chaotic attractor under the control of a simple constant input. Furthermore, the concept of generalized Lorenz system is extended to a new class of generalized Lorenz-like systems in a canonical form. Finally, the important problems of classification and normal form of three-dimensional quadratic autonomous chaotic systems are formulated and discussed.

Dynamical Analysis on a Finance System with Nonconstant Elasticity of Demand

2020 ◽  
Vol 30 (10) ◽  
pp. 2050148
Author(s):  
Ting Yang
Keyword(s):  
Chaotic Attractor ◽  
Period Doubling ◽  
Approximate Expression ◽  
Algebraic Surfaces ◽  
Dynamical Analysis ◽  
Periodic Attractor ◽  
Normal Form Theory ◽  
Finance System ◽  
Elasticity Of Demand ◽  
Invariant Algebraic Surfaces

This paper investigates a finance system with nonconstant elasticity of demand. First, under some conditions, the system has invariant algebraic surfaces and the analytic expressions of the surfaces are given. Furthermore, when the two surfaces coincide and become one surface, the dynamics on the surface are analyzed and a globally stable equilibrium is found. Second, by using the normal form theory, the Hopf bifurcation is studied and the approximate expression and stability of the bifurcating periodic orbit are obtained. Third, the chaotic behaviors are investigated and the route to chaos is period-doubling bifurcations. Moreover, it is found that the system has coexisting attractors, including periodic attractor and periodic attractor, chaotic attractor and chaotic attractor. With the change of parameter, the two chaotic attractors coincide and then a symmetrical chaotic attractor arises.

scholarly journals Generation of Multi-Scroll Chaotic Attractors from a Jerk Circuit with a Special Form of a Sine Function

Electronics ◽  
2020 ◽  
Vol 9 (5) ◽  
pp. 842
Author(s):  
Pengfei Ding ◽  
Xiaoyi Feng
Keyword(s):  
Chaotic System ◽  
Special Form ◽  
Electronic Circuit ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Sine Function ◽  
Sign Function ◽  
System Parameters ◽  
Left And Right

A novel chaotic system for generating multi-scroll attractors based on a Jerk circuit using a special form of a sine function (SFSF) is proposed in this paper, and the SFSF is the product of a sine function and a sign function. Although there are infinite equilibrium points in this system, the scroll number of the generated chaotic attractors is certain under appropriate system parameters. The dynamical properties of the proposed chaotic system are studied through Lyapunov exponents, phase portraits, and bifurcation diagrams. It is found that the scroll number of the chaotic system in the left and right part of the x-y plane can be determined arbitrarily by adjusting the values of the parameters in the SFSF, and the size of attractors is dominated by the frequency of the SFSF. Finally, an electronic circuit of the proposed chaotic system is implemented on Pspice, and the simulation results of electronic circuit are in agreement with the numerical ones.

scholarly journals A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 564 ◽  
Author(s):  
Jesus Munoz-Pacheco ◽  
Ernesto Zambrano-Serrano ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Jacques Kengne ◽  
...  
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Equilibrium System ◽  
Hidden Attractors ◽  
Hidden Chaotic Attractor

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

scholarly journals Analysis, adaptive control and circuit simulation of a novel finance system with dissaving

2016 ◽  
Vol 26 (1) ◽  
pp. 95-115 ◽  
Author(s):  
Ourania I. Tacha ◽  
Christos K. Volos ◽  
Ioannis N. Stouboulos ◽  
Ioannis M. Kyprianidis
Keyword(s):  
Adaptive Control ◽  
Nonlinear Theory ◽  
Circuit Simulation ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Dynamical Analysis ◽  
Phase Portraits ◽  
The Novel ◽  
Finance System ◽  
Simulation Tools

In this paper a novel 3-D nonlinear finance chaotic system consisting of two nonlinearities with negative saving term, which is called ‘dissaving’ is presented. The dynamical analysis of the proposed system confirms its complex dynamic behavior, which is studied by using wellknown simulation tools of nonlinear theory, such as the bifurcation diagram, Lyapunov exponents and phase portraits. Also, some interesting phenomena related with nonlinear theory are observed, such as route to chaos through a period doubling sequence and crisis phenomena. In addition, an interesting scheme of adaptive control of finance system’s behavior is presented. Furthermore, the novel nonlinear finance system is emulated by an electronic circuit and its dynamical behavior is studied by using the electronic simulation package Cadence OrCAD in order to confirm the feasibility of the theoretical model.

GENERATION OF AN EIGHT-WING CHAOTIC ATTRACTOR FROM QI 3-D FOUR-WING CHAOTIC SYSTEM

2012 ◽  
Vol 22 (12) ◽  
pp. 1250287 ◽  
Author(s):  
GUOYUAN QI ◽  
ZHONGLIN WANG ◽  
YANLING GUO
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Poincaré Map ◽  
Electronic Circuit ◽  
Constant Parameter ◽  
Poincare Map ◽  
Compound Structure ◽  
Topological Structures ◽  
Dynamical Behaviors ◽  
Map Analysis

This paper presents an eight-wing chaotic attractor by replacing a constant parameter with a switch function in Qi four-wing 3-D chaotic system. The eight-wing chaotic attractor has more complicated topological structures and dynamics than the original one. Some basic dynamical behaviors and the compound structure of the proposed 3-D system are investigated. Poincaré-map analysis shows that the system has an extremely rich dynamics. The physical existence of the eight-wing chaotic attractor is verified by an electronic circuit FPGA.

DYNAMICAL ANALYSIS OF A CHAOTIC SYSTEM WITH TWO DOUBLE-SCROLL CHAOTIC ATTRACTORS

2004 ◽  
Vol 14 (03) ◽  
pp. 971-998 ◽  
Author(s):  
WENBO LIU ◽  
GUANRONG CHEN
Keyword(s):  
Chaotic System ◽  
Three Dimensional ◽  
Dynamical Analysis ◽  
Chaotic Attractors ◽  
Compound Structure ◽  
Numerical Examples ◽  
Routes To Chaos ◽  
Basic Properties ◽  
Dynamical Behaviors

Dynamical behaviors of a three-dimensional autonomous chaotic system with two double-scroll attractors are studied. Some basic properties such as bifurcation, routes to chaos, periodic windows and compound structure are demonstrated with various numerical examples. System equilibria and their stabilities are discussed, and chaotic features of the attractors are justified numerically.

A new three-dimensional chaotic system, its dynamical analysis and electronic circuit applications

Optik ◽  
2016 ◽  
Vol 127 (18) ◽  
pp. 7062-7071 ◽  
Author(s):  
Akif Akgul ◽  
Shafqat Hussain ◽  
Ihsan Pehlivan
Keyword(s):  
Chaotic System ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
Dynamical Analysis

scholarly journals Sequence of Routes to Chaos in a Lorenz-Type System

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Fangyan Yang ◽  
Yongming Cao ◽  
Lijuan Chen ◽  
Qingdu Li
Keyword(s):  
Limit Cycles ◽  
Chaotic Attractor ◽  
Type System ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Main Route ◽  
Route To Chaos ◽  
Period Doubling Bifurcations

This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.

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