scholarly journals Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system

2006 ◽  
Vol 323 (2) ◽  
pp. 844-853 ◽  
Author(s):  
Damei Li ◽  
Jun-an Lu ◽  
Xiaoqun Wu ◽  
Guanrong Chen
Keyword(s):  
Chaotic System ◽  
Lorenz System ◽  
Invariant Set ◽  
Unified Chaotic System ◽  
The Lorenz System ◽  
Positively Invariant Set ◽  
Ultimate Bound

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.

BRIDGE THE GAP BETWEEN THE LORENZ SYSTEM AND THE CHEN SYSTEM

2002 ◽  
Vol 12 (12) ◽  
pp. 2917-2926 ◽  
Author(s):  
JINHU LÜ ◽  
GUANRONG CHEN ◽  
DAIZHAN CHENG ◽  
SERGEJ CELIKOVSKY
Keyword(s):  
Chaotic System ◽  
System Parameter ◽  
Lorenz System ◽  
Dual Systems ◽  
Chen System ◽  
Entire Spectrum ◽  
Unified Chaotic System ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
New System

This paper introduces a unified chaotic system that contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum. The new system represents the continued transition from the Lorenz to the Chen system and is chaotic over the entire spectrum of the key system parameter. Dynamical behaviors of the unified system are investigated in somewhat detail.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

An ultimate bound on the trajectories of the Lorenz system and its applications

Nonlinearity ◽  
2003 ◽  
Vol 16 (5) ◽  
pp. 1597-1605 ◽  
Author(s):  
A Yu Pogromsky ◽  
G Santoboni ◽  
H Nijmeijer
Keyword(s):  
Lorenz System ◽  
The Lorenz System ◽  
Ultimate Bound

GLOBALLY ATTRACTIVE AND POSITIVE INVARIANT SET OF THE LORENZ SYSTEM

2006 ◽  
Vol 16 (03) ◽  
pp. 757-764 ◽  
Author(s):  
PEI YU ◽  
XIAOXIN LIAO
Keyword(s):  
Lyapunov Function ◽  
Chaos Synchronization ◽  
Chaos Control ◽  
Lorenz System ◽  
Equilibrium Points ◽  
Invariant Set ◽  
The Lorenz System ◽  
Globally Attractive ◽  
Generalized Lyapunov Function ◽  
Positive Invariant Set

In this paper, based on a generalized Lyapunov function, a simple proof is given to improve the estimation of globally attractive and positive invariant set of the Lorenz system. In particular, a new estimation is derived for the variable x. On the globally attractive set, the Lorenz system satisfies Lipschitz condition, which is very useful in the study of chaos control and chaos synchronization. Applications are presented for globally, exponentially tracking periodic solutions, stabilizing equilibrium points and synchronizing two Lorenz systems.

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