Chaos in the Periodically Parametrically Excited Lorenz System

2021 ◽  
Vol 31 (08) ◽  
pp. 2130024
Author(s):  
Weisheng Huang ◽  
Xiao-Song Yang
Keyword(s):  
Chaotic Dynamics ◽  
Lorenz System ◽  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Time Varying ◽  
Asymptotically Stable ◽  
Parametrically Excited ◽  
The Lorenz System

We demonstrate in this paper a new chaotic behavior in the Lorenz system with periodically excited parameters. We focus on the parameters with which the Lorenz system has only two asymptotically stable equilibrium points, a saddle and no chaotic dynamics. A new mechanism of generating chaos in the periodically excited Lorenz system is demonstrated by showing that some trajectories can visit different attractor basins due to the periodic variations of the attractor basins of the time-varying stable equilibrium points when a parameter of the Lorenz system is varying periodically.

scholarly journals Chaotic dynamics of fractional Vallis system for El-Niño

2019 ◽  
Vol 22 (3) ◽  
pp. 825-842
Author(s):  
Amey Deshpande ◽  
Varsha Daftardar-Gejji
Keyword(s):  
El Niño ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
El Nino ◽  
Critical Value ◽  
Additional Parameter ◽  
Weather Phenomenon ◽  
The Lorenz System ◽  
Range Of Values

Abstract Vallis proposed a simple model for El-Niño weather phenomenon (referred as Vallis system) by adding an additional parameter p to the Lorenz system. He showed that the chaotic behavior of the Vallis system is related to the El-Niño effect. In the present article we study fractional version of Vallis system in detail. We investigate bifurcations and chaos present in the fractional Vallis system and the effect of variation of system parameter p. It is observed that the range of values of parameter p for which the Vallis system is chaotic, reduces with the reduction of the fractional order. Further we analyze the incommensurate fractional Vallis system and find the critical value below which the system loses chaos. We also synchronize Vallis system with Bhalekar-Gejji system.

scholarly journals Stability Analysis of the Chaotic Lorenz System with a State-Feedback Controller

2021 ◽  
Author(s):  
Dan Jones
Keyword(s):  
Strange Attractor ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Geometric Approach ◽  
Stable Equilibrium Point ◽  
The Lorenz System ◽  
The Stability ◽  
Parameter Values

The Lorenz model is considered a benchmark system in chaotic dynamics in that it displays extraordinary sensitivity to initial conditions and the strange attractor phenomenon. Even though the system tends to amplify perturbations, it is indeed possible to convert a strange attractor to a non-chaotic one using various control schemes. In this work it is shown that the chaotic behavior of the Lorenz system can be suppressed through the use of a feedback loop driven by a quotient controller. The stability of the controlled Lorenz system is evaluated near its equilibrium points using Routh-Hurwitz testing, and the global stability of the controlled system is established using a geometric approach. It is shown that the controlled Lorenz system has only one globally stable equilibrium point for the set of parameter values under consideration.

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.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

Jacobi analysis for an unusual 3D autonomous system

2020 ◽  
Vol 17 (04) ◽  
pp. 2050062 ◽  
Author(s):  
Chunsheng Feng ◽  
Qiujian Huang ◽  
Yongjian Liu
Keyword(s):  
Chaotic System ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Chen System ◽  
Parameter Region ◽  
Stable Equilibria ◽  
The Lorenz System ◽  
Deviation Vector

Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.

Jacobi Stability Analysis of the Chen System

2019 ◽  
Vol 29 (10) ◽  
pp. 1950139 ◽  
Author(s):  
Qiujian Huang ◽  
Aimin Liu ◽  
Yongjian Liu
Keyword(s):  
Periodic Orbit ◽  
Lorenz System ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Chen System ◽  
Nonlinear Connection ◽  
Berwald Connection ◽  
The Lorenz System ◽  
Nonzero Equilibrium ◽  
Deviation Vector

In this paper, the research of the Jacobi stability of the Chen system is performed by using the KCC-theory. By associating a nonlinear connection and a Berwald connection, five geometrical invariants of the Chen system are obtained. The Jacobi stability of the Chen system at equilibrium points and a periodic orbit is investigated in terms of the eigenvalues of the deviation curvature tensor. The obtained results show that the origin is always Jacobi unstable, while the Jacobi stability of the other two nonzero equilibrium points depends on the values of the parameters. And a periodic orbit of the Chen system is proved to be also Jacobi unstable. Furthermore, Jacobi stability regions of the Chen system and the Lorenz system are compared. Finally, the dynamical behavior of the components of the deviation vector near the equilibrium points is also discussed.

A New Approach for Estimating the Attraction Domain for Hopfield-Type Neural Networks

2009 ◽  
Vol 21 (1) ◽  
pp. 101-120 ◽  
Author(s):  
Dequan Jin ◽  
Jigen Peng
Keyword(s):  
Neural Networks ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Numerical Results ◽  
New Method ◽  
Attraction Domain ◽  
Network System ◽  
Asymptotically Stable ◽  
New Approach

In this letter, using methods proposed by E. Kaslik, St. Balint, and their colleagues, we develop a new method, expansion approach, for estimating the attraction domain of asymptotically stable equilibrium points of Hopfield-type neural networks. We prove theoretically and demonstrate numerically that the proposed approach is feasible and efficient. The numerical results that obtained in the application examples, including the network system considered by E. Kaslik, L. Brăescu, and St. Balint, indicate that the proposed approach is able to achieve better attraction domain estimation.

Tangent bundle viewpoint of the Lorenz system and its chaotic behavior

2010 ◽  
Vol 374 (11-12) ◽  
pp. 1315-1319 ◽  
Author(s):  
Takahiro Yajima ◽  
Hiroyuki Nagahama
Keyword(s):  
Tangent Bundle ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
The Lorenz System

scholarly journals Author’s reply to: Comments on “Asymptotically stable equilibrium points in new chaotic systems”

Nova Scientia ◽  
2017 ◽  
Vol 9 (19) ◽  
pp. 906-909
Author(s):  
K. Casas-García ◽  
L. A. Quezada-Téllez ◽  
S. Carrillo-Moreno ◽  
J. J. Flores-Godoy ◽  
Guillermo Fernández-Anaya
Keyword(s):  
Dynamic Systems ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Heteroclinic Orbits ◽  
Stable Equilibrium Point ◽  
Asymptotically Stable ◽  
Linear Quadratic ◽  
The Monte Carlo Method

Since theorem 1 of (Elhadj and Sprott, 2012) is incorrect, some of the systems found in the article (Casas-García et al. 2016) may have homoclinic or heteroclinic orbits and may seem chaos in the Shilnikov sense. However, the fundamental contribution of our paper was to find ten simple, three-dimensional dynamic systems with non-linear quadratic terms that have an asymptotically stable equilibrium point and are chaotic, which was achieved. These were obtained using the Monte Carlo method applied specifically for the search of these systems.

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.

Sign in / Sign up

Export Citation Format

Share Document