scholarly journals Analysis of a Lorenz-Like Chaotic System by Lyapunov Functions

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-6 ◽  
Author(s):  
Fuchen Zhang
Keyword(s):  
Chaotic System ◽  
Chaos Synchronization ◽  
Lorenz System ◽  
Chen System ◽  
Lyapunov Function Method ◽  
Exponential Attractivity ◽  
Positively Invariant Set ◽  
Ultimate Bound ◽  
Special Case ◽  
Theoretical Results

In this paper, we investigate the ultimate bound set and positively invariant set of a 3D Lorenz-like chaotic system, which is different from the well-known Lorenz system, Rössler system, Chen system, Lü system, and even Lorenz system family. Furthermore, we investigate the global exponential attractive set of this system via the Lyapunov function method. The rate of the trajectories going from the exterior of the globally exponential attractive set to the interior of the globally exponential attractive set is also obtained for all the positive parameters values a,b,c. The innovation of this paper is that our approach to construct the ultimate bounded and globally exponential attractivity sets assumes that the corresponding sets depend on some artificial parameters (λ and m); that is, for the fixed parameters of the system, we have a series of sets depending on λ and m. The results contain the known result as a special case for the fixed λ and m. The efficiency of the scheme is shown numerically. The theoretical results may find wide applications in chaos control and chaos synchronization.

Chen's attractor exists if Lorenz repulsor exists: The Chen system is a special case of the Lorenz system

2013 ◽  
Vol 23 (3) ◽  
pp. 033108 ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Lorenz System ◽  
Chen System ◽  
The Lorenz System ◽  
Special Case

Generating the New Chaotic Attractors of the Lorenz System Family via Anti-Controlling Method

2012 ◽  
Vol 542-543 ◽  
pp. 1042-1046 ◽  
Author(s):  
Xin Deng
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Attractors ◽  
Chen System ◽  
Lü System ◽  
New Chaotic System ◽  
Dynamical Properties ◽  
The Lorenz System ◽  
Lu System

In this paper, the first new chaotic system is gained by anti-controlling Chen system,which belongs to the general Lorenz system; also, the second new chaotic system is gained by anti-controlling the first new chaotic system, which belongs to the general Lü system. Moreover,some basic dynamical properties of two new chaotic systems are studied, either numerically or analytically. The obtained results show clearly that Chen chaotic system and two new chaotic systems also can form another Lorenz system family and deserve further detailed investigation.

A CHAOTIC SYSTEM WITH ONE SADDLE AND TWO STABLE NODE-FOCI

2008 ◽  
Vol 18 (05) ◽  
pp. 1393-1414 ◽  
Author(s):  
QIGUI YANG ◽  
GUANRONG CHEN
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Autonomous System ◽  
Lorenz System ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Type Systems ◽  
Stable Node ◽  
Compound Structure ◽  
Chen System

This paper reports the finding of a chaotic system with one saddle and two stable node-foci in a simple three-dimensional (3D) autonomous system. The system connects the original Lorenz system and the original Chen system and represents a transition from one to the other. The algebraical form of the chaotic attractor is very similar to the Lorenz-type systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the chaotic system has a chaotic attractor, one saddle and two stable node-foci. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions, and the compound structure of the system are analyzed and demonstrated with careful numerical simulations.

scholarly journals Complex Dynamical Behaviors of Lorenz-Stenflo Equations

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 513 ◽  
Author(s):  
Fuchen Zhang ◽  
Min Xiao
Keyword(s):  
Gravity Waves ◽  
Chaos Synchronization ◽  
Chaos Control ◽  
Acoustic Gravity Waves ◽  
Dynamical Behaviors ◽  
Lyapunov Function Method ◽  
Attractive Sets ◽  
Globally Attractive ◽  
Generalized Lyapunov Function ◽  
Theoretical Results

A mathematical chaos model for the dynamical behaviors of atmospheric acoustic-gravity waves is considered in this paper. Boundedness and globally attractive sets of this chaos model are studied by means of the generalized Lyapunov function method. The innovation of this paper is that it not only proves this system is globally bounded but also provides a series of global attraction sets of this system. The rate of trajectories entering from the exterior of the trapping domain to its interior is also obtained. Finally, the detailed numerical simulations are carried out to justify theoretical results. The results in this study can be used to study chaos control and chaos synchronization of this chaos system.

ULTIMATE BOUND ESTIMATION OF A CLASS OF HIGH DIMENSIONAL QUADRATIC AUTONOMOUS DYNAMICAL SYSTEMS

2011 ◽  
Vol 21 (09) ◽  
pp. 2679-2694 ◽  
Author(s):  
PEI WANG ◽  
DAMEI LI ◽  
XIAOQUN WU ◽  
JINHU LÜ ◽  
XINGHUO YU
Keyword(s):  
Dynamical Systems ◽  
Chaotic System ◽  
Lorenz System ◽  
High Dimensional ◽  
Hyperchaotic System ◽  
Sufficient Condition ◽  
System A ◽  
Ultimate Bound ◽  
Autonomous Dynamical Systems ◽  
Bound Estimation

This paper aims to propose a unified approach for the ultimate bound estimation of a class of High Dimensional Quadratic Autonomous Dynamical Systems (HDQADS). Using the proposed method and the optimization idea, a sufficient condition is then given for estimating the ultimate bounds of a class of HDQADS. To validate the above sufficient condition, this paper further investigates the ultimate bound estimation of a hyperchaotic system, a 6D and a 9D chaotic system, separately. Moreover, the ultimate bounds for a general Lorenz system, a low-order atmospheric circulation model, and a new 3D chaotic system are also discussed in detail. In particular, it should be pointed out that a unified and accurate ultimate bound estimation is attained for the generalized Lorenz system and it includes several well-known results as its special cases. Some numerical simulations are also given to verify and visualize the corresponding theoretical results.

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