New Results to a Three-Dimensional Chaotic System With Two Different Kinds of Nonisolated Equilibria

2015 ◽  
Vol 10 (6) ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic System ◽  
Three Dimensional ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Heteroclinic Cycle ◽  
Homoclinic And Heteroclinic Orbits ◽  
Singular Orbits ◽  
The Stability ◽  
Singularly Degenerate Heteroclinic Cycle

In the paper by Liu et al. (2009, “A Novel Three-Dimensional Autonomous Chaos System,” Chaos Solitons Fractals, 39(4), pp. 1950–1958), the three-dimensional (3D) chaotic system x·=-ax-ey2,y·=by-kxz,z·=-cz+mxy is investigated, and some of its dynamics according to theoretical and numerical analyses only for the parameters (a, e, b, k, c, m) = (1, 1, 2.5, 4, 5, 4) are discussed. In 2013, the same chaotic system x·1=-ax1- fx2x3,x·2=cx2-dx1x3,x·3=-bx3+ex22 by Li et al. (2013, “Analysis of a Novel Three-Dimensional Chaotic System,” Optik, 124(13), pp. 1516–1522) was mainly discussed by numerical simulation. In this article, by some deeper investigations, combining some numerical simulations, we formulate some new results of the system. First, after some problems in the first paper are pointed out, we display that its parameters e, k, and m may be kicked out by some homothetic transformations. Second, some of its rich nonlinear dynamics hiding and not found previously, such as the stability and Hopf bifurcation of its isolated equilibria, the behavior of its nonisolated equilibria, the existence of singular orbits (including singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbits, etc.), and its dynamics at infinity, etc., are clearly formulated. What's more interesting, we find, this system has two different kinds of nonisolated equilibria Ex and Ez, and new chaotic attractors can be bifurcated out with the disappearance of Ex, but this system has no such properties at Ez. In the meantime, several problems about the existence of singular orbits deserving further investigations are presented. Our results better complement and improve the known ones.

scholarly journals SINGULAR ORBITS AND DYNAMICS AT INFINITY OF A CONJUGATE LORENZ-LIKE SYSTEM

2015 ◽  
Vol 20 (2) ◽  
pp. 148-167 ◽  
Author(s):  
Fengjie Geng ◽  
Xianyi Li
Keyword(s):  
Theoretical Analysis ◽  
Lyapunov Exponents ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Basic Properties ◽  
Homoclinic And Heteroclinic Orbits ◽  
Quadratic Nonlinearities ◽  
Singular Orbits

A conjugate Lorenz-like system which includes only two quadratic nonlinearities is proposed in this paper. Some basic properties of this system, such as the distribution of its equilibria and their stabilities, the Lyapunov exponents, the bifurcations are investigated by some numerical and theoretical analysis. The forming mechanisms of compound structures of its new chaotic attractors obtained by merging together two simple attractors after performing one mirror operation are also presented. Furthermore, some of its other complex dynamical behaviours, which include the existence of singularly degenerate heteroclinic cycles, the existence of homoclinic and heteroclinic orbits and the dynamics at infinity, etc, are formulated in detail. In the meantime, some problems deserving further investigations are presented.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

More Dynamical Properties Revealed from a 3D Lorenz-like System

2014 ◽  
Vol 24 (10) ◽  
pp. 1450133 ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Mathematical Work ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Properties ◽  
Theoretical Results

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

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.

scholarly journals Hopf Bifurcation and Dynamic Analysis of an Improved Financial System with Two Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
G. Kai ◽  
W. Zhang ◽  
Z. Jin ◽  
C. Z. Wang
Keyword(s):  
Nonlinear Dynamics ◽  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Chaotic System ◽  
Financial System ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

The complex chaotic dynamics and multistability of financial system are some important problems in micro- and macroeconomic fields. In this paper, we study the influence of two-delay feedback on the nonlinear dynamics behavior of financial system, considering the linear stability of equilibrium point under the condition of single delay and two delays. The system undergoes Hopf bifurcation near the equilibrium point. The stability and bifurcation directions of Hopf bifurcation are studied by using the normal form method and central manifold theory. The theoretical results are verified by numerical simulation. Furthermore, one feature of the proposed financial chaotic system is that its multistability depends extremely on the memristor initial condition and the system parameters. It is shown that the nonlinear dynamics of financial chaotic system can be significantly changed by changing the values of time delays.

A 3D Autonomous System with Infinitely Many Chaotic Attractors

2019 ◽  
Vol 29 (12) ◽  
pp. 1950166 ◽  
Author(s):  
Ting Yang ◽  
Qigui Yang
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Autonomous System ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Double Zero ◽  
Finite Dimensional ◽  
Periodic Attractors ◽  
The Stability ◽  
Autonomous Dynamical System

Intuitively, a finite-dimensional autonomous system of ordinary differential equations can only generate finitely many chaotic attractors. Amazingly, however, this paper finds a three-dimensional autonomous dynamical system that can generate infinitely many chaotic attractors. Specifically, this system can generate infinitely many coexisting chaotic attractors and infinitely many coexisting periodic attractors in the following three cases: (i) no equilibria, (ii) only infinitely many nonhyperbolic double-zero equilibria, and (iii) both infinitely many hyperbolic saddles and nonhyperbolic pure-imaginary equilibria. By analyzing the stability of double-zero and pure-imaginary equilibria, it is shown that the classic Shil’nikov criteria fail in verifying the existence of chaos in the above three cases.

HOMOCLINIC AND HETEROCLINIC ORBITS AND BIFURCATIONS OF A NEW LORENZ-TYPE SYSTEM

2011 ◽  
Vol 21 (09) ◽  
pp. 2695-2712 ◽  
Author(s):  
XIANYI LI ◽  
HAIJUN WANG
Keyword(s):  
Chaotic Attractor ◽  
Lorenz System ◽  
Type System ◽  
Pitchfork Bifurcation ◽  
Heteroclinic Orbits ◽  
Chen System ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
The Stability

In this paper, a new Lorenz-type system with chaotic attractor is formulated. The structure of the chaotic attractor in this new system is found to be completely different from that in the Lorenz system or the Chen system or the Lü system, etc., which motivates us to further study in detail its complicated dynamical behaviors, such as the number of its equilibrium, the stability of the hyperbolic and nonhyperbolic equilibrium, the degenerate pitchfork bifurcation, the Hopf bifurcation and the local manifold character, etc., when its parameters vary in their space. The existence or nonexistence of homoclinic and heteroclinic orbits of this system is also rigorously proved. Numerical simulation evidences are also presented to examine the corresponding theoretical analytical results.

Antimonotonicity, Chaos and Multiple Attractors in a Novel Autonomous Jerk Circuit

2017 ◽  
Vol 27 (07) ◽  
pp. 1750100 ◽  
Author(s):  
J. Kengne ◽  
A. Nguomkam Negou ◽  
Z. T. Njitacke
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Electrical Circuit ◽  
Semiconductor Diode ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Multiple Attractors ◽  
Systematic Analysis ◽  
The Stability

We perform a systematic analysis of a system consisting of a novel jerk circuit obtained by replacing the single semiconductor diode of the original jerk circuit described in [Sprott, 2011a] with a pair of semiconductor diodes connected in antiparallel. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a control system with nonlinear velocity feedback. The stability of the (unique) fixed point, the local bifurcations, and the discrete symmetries of the model equations are discussed. The complex behavior of the system is categorized in terms of its parameters by using bifurcation diagrams, Lyapunov exponents, time series, Poincaré sections, and basins of attraction. Antimonotonicity, period doubling bifurcation, symmetry restoring crises, chaos, and coexisting bifurcations are reported. More interestingly, one of the key contributions of this work is the finding of various regions in the parameters’ space in which the proposed (“elegant”) jerk circuit experiences the unusual phenomenon of multiple competing attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). The basins of attraction of various coexisting attractors display complexity (i.e. fractal basins boundaries), thus suggesting possible jumps between coexisting attractors in experiment. Results of theoretical analyses are perfectly traced by laboratory experimental measurements. To the best of the authors’ knowledge, the jerk circuit/system introduced in this work represents the simplest electrical circuit (only a quadruple op amplifier chip without any analog multiplier chip) reported to date capable of four disconnected periodic and chaotic attractors for the same parameters setting.

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.

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