Infinitely Many Heteroclinic Orbits of a Complex Lorenz System

2017 ◽  
Vol 27 (07) ◽  
pp. 1750110 ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Lorenz System ◽  
Mathematical Problem ◽  
Heteroclinic Orbit ◽  
Heteroclinic Orbits ◽  
Essential Difference ◽  
Limit Set ◽  
Analytical Proof ◽  
Infinite Set ◽  
Hyperchaotic Systems ◽  
Complex Lorenz System

The existence of heteroclinic orbits of a chaotic system is a difficult yet interesting mathematical problem. Nowadays, a rigorous analytical proof for the existence of a heteroclinic orbit can be carried out only for some special chaotic and hyperchaotic systems, and few results are known for the complex systems. In this paper, by revisiting a complex Lorenz system, it is found that this system possesses an infinite set of heteroclinic orbits to the origin and its circle equilibria. However, it is impossible for the corresponding real Lorenz system to have infinitely many heteroclinic orbits. The theoretical tools for proving the main results are Lyapunov functions and the definitions of [Formula: see text]-limit set and [Formula: see text]-limit set. Numerical simulations show the effectiveness and correctness of the theoretical conclusions. The investigations not only enrich the related results for the complex Lorenz system, but also find the essential difference between the complex Lorenz system and its corresponding real version: the complex Lorenz system has infinitely many heteroclinic orbits whereas its corresponding real one does not.

Heteroclinic Orbit, Forced Lorenz System, and Chaos

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
Dibakar Ghosh ◽  
Anirban Ray ◽  
A. Roy Chowdhury
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lorenz System ◽  
Heteroclinic Orbit ◽  
Convergent Series ◽  
Heteroclinic Orbits ◽  
Onset Of Chaos ◽  
Uniformly Convergent ◽  
Analytic Expression

Forced Lorenz system, important in modeling of monsoonlike phenomena, is analyzed for the existence of heteroclinic orbit. This is done in the light of the suggested new mechanism for the onset of chaos by Magnitskii and Sidorov (2006, “Finding Homoclinic and Heteroclinic Contours of Singular Points of Nonlinear Systems of Ordinary Differential Equations,” Diff. Eq., 39, pp. 1593–1602), where heteroclinic orbits plays important and dominant roles. The analysis is performed based on the theory laid down by Shilnikov. An analytic expression in the form of uniformly convergent series is obtained. The same orbit is also obtained numerically by a technique enunciated by Magnitskii and Sidorov, reproducing the necessary important features.

HETEROCLINIC ORBITS FOR MONOTONE TWIST MAPS THROUGH A VARIATIONAL PRINCIPLE

2001 ◽  
Vol 11 (09) ◽  
pp. 2451-2461
Author(s):  
TIFEI QIAN
Keyword(s):  
Variational Principle ◽  
Variational Method ◽  
Fixed Points ◽  
Heteroclinic Orbit ◽  
Geometric Method ◽  
Heteroclinic Orbits ◽  
Constructive Approach ◽  
Connecting Orbits ◽  
Relative Ease ◽  
Twist Maps

The variational method has shown many advantages over the geometric method in proving the existence of connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained by a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits. In this paper a variational principle is used to construct a heteroclinic orbit between an adjacent minimal pair of fixed points for monotone twist maps on (ℝ/ℤ) × ℝ. Application of our results to a standard map is also given.

The characteristics and self-time-delay synchronization of two-time-delay complex Lorenz system

2019 ◽  
Vol 356 (1) ◽  
pp. 334-350 ◽  
Author(s):  
Baojiang Sun ◽  
Min Li ◽  
Fangfang Zhang ◽  
Hui Wang ◽  
Jian Liu
Keyword(s):  
Time Delay ◽  
Lorenz System ◽  
Complex Lorenz System

Analysis of a Nonlinear System with a Random Parameter

2014 ◽  
Vol 721 ◽  
pp. 366-369
Author(s):  
Hong Gang Dang ◽  
Xiao Ya Yang ◽  
Wan Sheng He
Keyword(s):  
Nonlinear System ◽  
Fractional Order ◽  
Approximation Method ◽  
Lorenz System ◽  
Laguerre Polynomial ◽  
Random Parameter ◽  
Order System ◽  
Order Complex ◽  
Ensemble Mean ◽  
Complex Lorenz System

In this paper, a nonlinear system with random parameter, which is called stochastic fractional-order complex Lorenz system, is investigated. The Laguerre polynomial approximation method is used to study the system. Then, the stochastic fractional-order system is reduced into the equivalent deterministic one with Laguerre approximation. The ensemble mean and sample responses of the stochastic system can be obtained.

scholarly journals Generalized Projective Synchronization for Different Hyperchaotic Dynamical Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
M. M. El-Dessoky ◽  
E. Saleh
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Lorenz System ◽  
Sufficient Conditions ◽  
Projective Synchronization ◽  
Hyperchaotic Lorenz System ◽  
Generalized Projective Synchronization ◽  
Hyperchaotic Systems

Projective synchronization and generalized projective synchronization have recently been observed in the coupled hyperchaotic systems. In this paper a generalized projective synchronization technique is applied in the hyperchaotic Lorenz system and the hyperchaotic Lü. The sufficient conditions for achieving projective synchronization of two different hyperchaotic systems are derived. Numerical simulations are used to verify the effectiveness of the proposed synchronization techniques.

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.

scholarly journals Inverse Projective Synchronization between Two Different Hyperchaotic Systems with Fractional Order

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Yi Chai ◽  
Liping Chen ◽  
Ranchao Wu
Keyword(s):  
Fractional Order ◽  
Lorenz System ◽  
Projective Synchronization ◽  
Generalized Synchronization ◽  
Fractional Order Systems ◽  
Chen System ◽  
Fractional Order Differential Equations ◽  
Hyperchaotic Lorenz System ◽  
The Stability ◽  
Hyperchaotic Systems

This paper mainly investigates a novel inverse projective synchronization between two different fractional-order hyperchaotic systems, that is, the fractional-order hyperchaotic Lorenz system and the fractional-order hyperchaotic Chen system. By using the stability theory of fractional-order differential equations and Lyapunov equations for fractional-order systems, two kinds of suitable controllers for achieving inverse projective synchronization are designed, in which the generalized synchronization, antisynchronization, and projective synchronization of fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system are also successfully achieved, respectively. Finally, simulations are presented to demonstrate the validity and feasibility of the proposed method.

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