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.

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.

scholarly journals Existence of Chaos in the Chen System with Linear Time-Delay Feedback

2019 ◽  
Vol 29 (09) ◽  
pp. 1950114 ◽  
Author(s):  
Kun Tian ◽  
Hai-Peng Ren ◽  
Celso Grebogi
Keyword(s):  
Time Delay ◽  
Projection Method ◽  
Homoclinic Orbit ◽  
Linear Time ◽  
Autonomous Systems ◽  
Technical Problem ◽  
Three Dimensional ◽  
Heteroclinic Orbit ◽  
Real Eigenvalue ◽  
Chen System

It is mathematically challenging to analytically show that complex dynamical phenomena observed in simulations and experiments are truly chaotic. The Shil’nikov lemma provides a useful theoretical tool to prove the existence of chaos in three-dimensional smooth autonomous systems. It requires, however, the proof of existence of a homoclinic or heteroclinic orbit, which remains a very difficult technical problem if contigent on data. In this paper, for the Chen system with linear time-delay feedback, we demonstrate a homoclinic orbit by using a modified undetermined coefficient method and we propose a spiral involute projection method. In such a way, we identify experimentally the asymmetrical homoclinic orbit in order to apply the Shil’nikov-type lemma and to show that chaos is indeed generated in the Chen circuit with linear time-delay feedback. We also identify the presence of a single-scroll attractor in the Chen system with linear time-delay feedback in our experiments. We confirm that the Chen single-scroll attractor is hyperchaotic by numerically estimating the finite-time local Lyapunov exponent spectrum. By means of a linear scaling in the coordinates and the time, such a method can also be applied to the generalized Lorenz-like systems. The contribution of this work lies in: first, we treat the trajectories corresponding to the real eigenvalue and the image eigenvalues in different ways, which is compatible with the characteristics of the trajectory geometry; second, we propose a spiral involute projection method to exhibit the trajectory corresponding to the image eigenvalues; third, we verify the homoclinic orbit by experimental data.

A NEW CHAOTIC ATTRACTOR COINED

2002 ◽  
Vol 12 (03) ◽  
pp. 659-661 ◽  
Author(s):  
JINHU LÜ ◽  
GUANRONG CHEN
Keyword(s):  
Chaotic Attractor ◽  
Autonomous System ◽  
Three Dimensional ◽  
The Other ◽  
Lorenz Attractor ◽  
New Chaotic Attractor

This letter reports the finding of a new chaotic attractor in a simple three-dimensional autonomous system, which connects the Lorenz attractor and Chen's attractor and represents the transition from one to the other.

scholarly journals Stability and Bifurcation of Two Kinds of Three-Dimensional Fractional Lotka-Volterra Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jinglei Tian ◽  
Yongguang Yu ◽  
Hu Wang
Keyword(s):  
Hopf Bifurcation ◽  
Asymptotic Stability ◽  
Theoretical Analysis ◽  
Fractional Order ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Critical Value ◽  
The Other ◽  
Bifurcation Parameter ◽  
Volterra Systems

Two kinds of three-dimensional fractional Lotka-Volterra systems are discussed. For one system, the asymptotic stability of the equilibria is analyzed by providing some sufficient conditions. And bifurcation property is investigated by choosing the fractional order as the bifurcation parameter for the other system. In particular, the critical value of the fractional order is identified at which the Hopf bifurcation may occur. Furthermore, the numerical results are presented to verify the theoretical analysis.

ŠIL'NIKOV HOMOCLINIC ORBITS IN TWO CLASSES OF 3D AUTONOMOUS NONLINEAR SYSTEMS

2011 ◽  
Vol 25 (20) ◽  
pp. 2697-2712 ◽  
Author(s):  
BAOYING CHEN ◽  
TIANSHOU ZHOU
Keyword(s):  
Computer Simulation ◽  
Nonlinear Systems ◽  
Geometric Structure ◽  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Normal Forms ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Numerical Examples ◽  
Undetermined Coefficient

The Šil'nikov homoclinic theorem provides one analytic criterion for proving the existence of chaos in three-dimensional autonomous nonlinear systems. In applications of the theorem, however, the existence of a homoclinic orbit that usually determines the geometric structure of the chaotic attractor is not easily verified mainly because there are no available efficient methods. In this paper, based on the undetermined coefficient approach we present a framework of how to find homoclinic orbits in two classes of three-dimensional autonomous nonlinear systems of normal forms, including how to set a reasonable form of expanding series of the homoclinic orbit, how to determine all coefficients in the expansion, and how to find a numerical homoclinic orbit. Numerical examples show that the proposed framework in combination with computer simulation is very efficient.

CLASSIFICATION OF CHAOS IN 3-D AUTONOMOUS QUADRATIC SYSTEMS-I: BASIC FRAMEWORK AND METHODS

2006 ◽  
Vol 16 (09) ◽  
pp. 2459-2479 ◽  
Author(s):  
TIANSHOU ZHOU ◽  
GUANGRONG CHEN
Keyword(s):  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Heteroclinic Orbit ◽  
Homoclinic Orbits ◽  
Classification Problem ◽  
Heteroclinic Orbits ◽  
Orbit Type ◽  
Quadratic Systems ◽  
Hybrid Type

This paper is part I of a series of contributions on the classification problem of chaos in three-dimensional autonomous quadratic systems. We try to classify chaos, based on the Ši'lnikov criteria, in such a large class of systems into the following four types: (1) chaos of the Ši'lnikov homoclinic orbit type; (2) chaos of the Ši'lnikov heteroclinic orbit type; (3) chaos of the hybrid type; i.e. those with both Ši'lnikov homoclinic and homoclinic orbits; (4) chaos of other types. We are especially interested in finding out all the simplest possible forms of chaotic systems for each type of chaos. Our main contributions are to develop some effective classification methods and to provide a basic classification framework under which each of the four types of chaos can be justified by some examples that are useful for describing the feasibility and procedure of the classification. In particular, we show several novel chaotic attractors, e.g. one hybrid-type chaotic attractor with three equilibria, one heteroclinic orbit and one homoclinic orbit, and one 4-scroll chaotic attractor with five equilibria and two heteroclinic orbits.

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.

scholarly journals A New Four-Scroll Chaotic Attractor Consisted of Two-Scroll Transient Chaotic and Two-Scroll Ultimate Chaotic

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yuhua Xu ◽  
Bing Li ◽  
Yuling Wang ◽  
Wuneng Zhou ◽  
Jian-an Fang
Keyword(s):  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
The Other ◽  
Phase Portraits ◽  
Fractional Dimension ◽  
The Novel ◽  
New Chaotic System ◽  
New System

A new four-scroll chaotic attractor is found by feedback controlling method in this paper. The novel chaotic system can generate four scrolls two of which are transient chaotic and the other two of which are ultimate chaotic. Of particular interest is that this novel chaotic system can generate one-scroll, two 2-scroll and four-scroll chaotic attractor with variation of a single parameter. We analyze the new system by means of phase portraits, Lyapunov exponents, fractional dimension, bifurcation diagram, and Poincaré map, respectively. The analysis results show clearly that this is a new chaotic system which deserves further detailed investigation.

STRANGE ATTRACTORS OF THE SHIL'NIKOV TYPE IN CHUA'S CIRCUIT

1993 ◽  
Vol 03 (05) ◽  
pp. 1293-1298
Author(s):  
M. BLÁZQUEZ ◽  
E. TUMA
Keyword(s):  
Equilibrium Point ◽  
Homoclinic Orbit ◽  
Chaotic Behavior ◽  
Strange Attractors ◽  
Heteroclinic Orbits ◽  
Chua’S Circuit ◽  
Closed Contour ◽  
Chua's Circuit ◽  
Focus Type

The chaotic behavior of the solutions of Chua's circuit is studied in the neighborhood of a homoclinic orbit to an equilibrium point of the saddle-focus type and in a neighborhood of two heteroclinic orbits to saddle-focus points which form a closed contour.

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