CAN A THREE-DIMENSIONAL SMOOTH AUTONOMOUS QUADRATIC CHAOTIC SYSTEM GENERATE A SINGLE FOUR-SCROLL ATTRACTOR?

2004 ◽  
Vol 14 (04) ◽  
pp. 1395-1403 ◽  
Author(s):  
WENBO LIU ◽  
GUANRONG CHEN
Keyword(s):  
Phase Space ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
System Parameters ◽  
New Chaotic System

Recently, we have investigated a new chaotic system of three-dimensional autonomous quadratic ordinary differential equations, and found that the system visually displays a four-scroll chaotic attractor confirmed by both numerical simulations and circuit implementation. In this paper, we further study the following question: Is it really true that this system can generate a four-scroll chaotic attractor, or is it only a numerical artifact? By a more careful theoretical analysis along with some further numerical simulations, we conclude that the four-scroll chaotic attractor of this system, which we observed on both computer and oscilloscope, cannot actually exist in theory. The fact is that this system has two co-existing two-scroll chaotic attractors that are arbitrarily close in the phase space for some system parameters, therefore extremely tiny numerical round-off errors or signal fluctuations will nudge the system orbit to switch from one attractor to another, thereby forming the seemingly single four-scroll chaotic attractor on screen display.

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.

A New Chaotic Attractor Around a Pre-Located Ring

2017 ◽  
Vol 27 (10) ◽  
pp. 1750152 ◽  
Author(s):  
Zhen Wang ◽  
Zhe Xu ◽  
Ezzedine Mliki ◽  
Akif Akgul ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Specific Property ◽  
Unique Property ◽  
Circuit Implementation ◽  
New Chaotic System ◽  
New Chaotic Attractor ◽  
New System

Designing chaotic systems with specific features is a very interesting topic in nonlinear dynamics. However most of the efforts in this area are about features in the structure of the equations, while there is less attention to features in the topology of strange attractors. In this paper, we introduce a new chaotic system with unique property. It has been designed in such a way that a specific property has been injected to it. This new system is analyzed carefully and its real circuit implementation is presented.

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

2017 ◽  
Vol 27 (04) ◽  
pp. 1850066 ◽  
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.

A New Chaotic System Family Dynamics Analysis and Circuit Implementation

2013 ◽  
Vol 392 ◽  
pp. 232-236
Author(s):  
Shu Min Duan ◽  
Guo Zeng Wu
Keyword(s):  
Chaotic System ◽  
Electronic Circuit ◽  
Physical Phenomenon ◽  
Three Dimensional ◽  
Dynamics Analysis ◽  
Circuit Implementation ◽  
Parallel Lines ◽  
New Chaotic System ◽  
Dynamical Properties ◽  
Electronic Circuit Implementation

A new three-dimensional chaotic system is presented in this paper. Some basic dynamical Properties of this chaotic system are investigated by means of Poincaré mapping, Lyapunov exponents and bifurcation diagram. The dynamical behaviours of this system are proved not only by performing numerical simulation and brief theoretical analysis but also by conducting an electronic circuit implementation. It is new physical phenomenon that the Poincaré mapping of this system is a group of parallel lines.

AN EXTENDED ŠIL'NIKOV HOMOCLINIC THEOREM AND ITS APPLICATIONS

2009 ◽  
Vol 19 (05) ◽  
pp. 1679-1693 ◽  
Author(s):  
BAOYING CHEN ◽  
TIANSHOU ZHOU ◽  
GUANRONG CHEN
Keyword(s):  
Theoretical Analysis ◽  
Equilibrium Point ◽  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
The Other ◽  
Complex Numbers ◽  
New Chaotic System ◽  
Focus Type

The classical Šil'nikov homoclinic theorem provides an analytic criterion for proving the existence of chaos in three-dimensional autonomous systems, but it can only be applied to systems with fixed points of the saddle-focus type. This paper extends this powerful theorem to a degenerate case where one of the eigenvalues of the Jacobian evaluated at an equilibrium point is zero and the other two are a pair of conjugate complex numbers, and consequently establishes a set of criteria for proving the existence of chaos in the sense of having Smale horseshoes. Based on this new extended Šil'nikov homoclinic theorem, a new chaotic system is constructed, whose corresponding bounded chaotic attractor is first verified numerically through phase trajectories, Lyapunov exponents, bifurcation routes and Poincaré mappings, followed by theoretical analysis on the existence of one homoclinic orbit, the key component of the extended Šil'nikov homoclinic theorem.

scholarly journals Generation of a New Three Dimension Autonomous Chaotic Attractor and its Four Wing Type

2013 ◽  
Vol 3 (1) ◽  
pp. 352-358
Author(s):  
F. Yu ◽  
C. Wang
Keyword(s):  
Fractal Dimension ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Topological Structure ◽  
Nonlinear Term ◽  
Chaotic Attractors ◽  
Three Dimension ◽  
Hyperbolic Sine ◽  
New Chaotic System ◽  
Dynamical Properties

In this paper, a new three-dimension (3D) autonomous chaotic system with a nonlinear term in the form of a hyperbolic sine (or cosine) function is reported. Some interesting and complex attractors are obtained. Basic dynamical properties of the new chaotic system are demonstrated in terms of Lyapunov exponents, Poincare mapping, fractal dimension and continuous spectrum. Meanwhile, for further enhancing the complexity of the topological structure of the new chaotic attractors, the attractors are changed from two-wing to four-wing through making axis doubly polarized, theoretically analyzed and numerically simulated. The obtained results clearly show that the chaotic system deserves further detailed investigation.

scholarly journals Complex Chaotic Attractor via Fractal Transformation

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1115 ◽  
Author(s):  
Shengqiu Dai ◽  
Kehui Sun ◽  
Shaobo He ◽  
Wei Ai
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Digital Circuit ◽  
Chaotic Attractors ◽  
Chaotic Sequences ◽  
Three Dimensional Space

Based on simplified Lorenz multiwing and Chua multiscroll chaotic systems, a rotation compound chaotic system is presented via transformation. Based on a binary fractal algorithm, a new ternary fractal algorithm is proposed. In the ternary fractal algorithm, the number of input sequences is extended from 2 to 3, which means the chaotic attractor with fractal transformation can be presented in the three-dimensional space. Taking Lorenz system, rotation Lorenz system and compound chaotic system as the seed chaotic systems, the dynamics of the complex chaotic attractors with fractal transformation are analyzed by means of bifurcation diagram, complexity and power spectrum, and the results show that the chaotic sequences with fractal transformation have higher complexity. As the experimental verification, one kind of complex chaotic attractors is implemented by DSP, and the result is consistent with that of the simulation, which verifies the feasibility of digital circuit implement.

A New 3D Cubic Chaotic System and its Circuitry Implementation

2011 ◽  
Vol 130-134 ◽  
pp. 3924-3927
Author(s):  
Wei Deng ◽  
Yan Feng Wang ◽  
Jie Fang
Keyword(s):  
Theoretical Analysis ◽  
Chaotic System ◽  
Electronic Circuit ◽  
Nonlinear Term ◽  
Dynamic Properties ◽  
Three Dimensional ◽  
Lyapunov Dimension ◽  
New Chaotic System ◽  
Lyapunov Exponent Spectrum ◽  
New System

A new three-dimensional cubic chaotic system is reported. This new system contains five system parameters and each equation contains nonlinear term. Moreover, two equations of nonlinear term is cubic. The basic properties of the new system are investigated via theoretical analysis, numerical simulation, Lyapunov exponent spectrum, bifurcation diagram, Lyapunov dimension and Poincare diagram. The different dynamic behaviors of the new system are analyzed when each system parameter is changed .An electronic circuit was designed to realize the new chaotic system. Experimental chaotic behaviors of the system were found to be identical to the dynamic properties predicted by theoretical analysis and numerical simulations.

A Chaotic System with Different Families of Hidden Attractors

2016 ◽  
Vol 26 (08) ◽  
pp. 1650139 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Sundarapandian Vaidyanathan ◽  
Tomasz Kapitaniak ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Chaotic System ◽  
Infinite Number ◽  
Mathematical Analysis ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Circuit Implementation ◽  
Hidden Attractors ◽  
Dynamical Behaviors

The presence of hidden attractors in dynamical systems has received considerable attention recently both in theory and applications. A novel three-dimensional autonomous chaotic system with hidden attractors is introduced in this paper. It is exciting that this chaotic system can exhibit two different families of hidden attractors: hidden attractors with an infinite number of equilibrium points and hidden attractors without equilibrium. Dynamical behaviors of such system are discovered through mathematical analysis, numerical simulations and circuit implementation.

HYPERCHAOS IN THE FRACTIONAL-ORDER NONAUTONOMOUS CHEN'S SYSTEM AND ITS SYNCHRONIZATION

2005 ◽  
Vol 16 (05) ◽  
pp. 815-826 ◽  
Author(s):  
HONGBIN ZHANG ◽  
CHUNGUANG LI ◽  
GUANRONG CHEN ◽  
XING GAO
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Chaotic System ◽  
Three Dimensional ◽  
Synchronization Problem ◽  
Driving Signal

Recently, a new hyperchaos generator, obtained by controlling a three-dimensional autonomous chaotic system — Chen's system — with a periodic driving signal, has been found. In this letter, we formulate and study the hyperchaotic behaviors in the corresponding fractional-order hyperchaotic Chen's system. Through numerical simulations, we found that hyperchaos exists in the fractional-order hyperchaotic Chen's system with order less than 4. The lowest order we found to have hyperchaos in this system is 3.4. Finally, we study the synchronization problem of two fractional-order hyperchaotic Chen's systems.

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