Comment on “A hidden chaotic attractor in the classical Lorenz system”

2018 ◽  
Vol 113 ◽  
pp. 261-262 ◽  
Author(s):  
J.C. Sprott ◽  
Buncha Munmuangsaen
Keyword(s):  
Chaotic Attractor ◽  
Lorenz System ◽  
Hidden Chaotic Attractor

A hidden chaotic attractor in the classical Lorenz system

2018 ◽  
Vol 107 ◽  
pp. 61-66 ◽  
Author(s):  
Buncha Munmuangsaen ◽  
Banlue Srisuchinwong
Keyword(s):  
Chaotic Attractor ◽  
Lorenz System ◽  
Hidden Chaotic Attractor

scholarly journals A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 564 ◽  
Author(s):  
Jesus Munoz-Pacheco ◽  
Ernesto Zambrano-Serrano ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Jacques Kengne ◽  
...  
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Equilibrium Points ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Equilibrium System ◽  
Hidden Attractors ◽  
Hidden Chaotic Attractor

In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics.

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.

Complex Dynamics in the Unified Lorenz-Type System

2014 ◽  
Vol 24 (04) ◽  
pp. 1450055 ◽  
Author(s):  
Qigui Yang ◽  
Yuming Chen
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Dynamics ◽  
Type System ◽  
Global Dynamics ◽  
Two Systems ◽  
The Lorenz System ◽  
The Stability

This paper is devoted to the analysis of complex dynamics of the unified Lorenz-type system (ULTS) with six parameters, which contain common chaotic systems as its particular cases. First, some important local dynamics such as pitchfork bifurcation, Hopf bifurcation, and the stability of nondegenerate and double-zero equilibria are systematically investigated using the parameter-dependent center manifold theory combined with some bifurcation theories. Some adequate conditions for guaranteeing the occurrence of degenerate Hopf bifurcation (DHB) and the stability of the equilibria are given. Second, it is found that if DHB does not generate at the trivial equilibrium but generates at two symmetric nontrivial equilibria, then a small perturbation can lead that ULTS to exhibit a chaotic attractor. Interestingly, such a case can take place in the Chen and Lü systems (two common chaotic systems) but cannot take place in the Lorenz and Yang systems (the other two common chaotic systems), essentially distinguishing the Lorenz system from the Chen system. In addition, it is numerically verified that both of the latter two systems can exhibit the coexistence of both a chaotic attractor and multiple limit cycles but the former two systems seem not to have this property. If DHB takes place simultaneously at three equilibria of ULTS, then this system has an invariant algebraic surface, and rigorously prove the existence of some global dynamics such as periodic orbit, center, homoclinic/heteroclinic orbits. Third, it is shown that a singularly degenerate heteroclinic cycle can exist in the case of b = 0 (where b is a parameter of ULTS, like that in the Lorenz system), and a chaotic attractor can be generated by perturbing this cycle for small b > 0. These results altogether indicate that the ULTS can exhibit complex dynamics, and provide a more reasonable classification for chaos in the 3D autonomous chaotic ODE systems that were developed based on the Lorenz system, in contrast to the previous studies.

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.

EXPERIMENTAL VERIFICATION OF THE BUTTERFLY ATTRACTOR IN A MODIFIED LORENZ SYSTEM

2002 ◽  
Vol 12 (07) ◽  
pp. 1627-1632 ◽  
Author(s):  
S. ÖZOǦUZ ◽  
A. S. ELWAKIL ◽  
M. P. KENNEDY
Keyword(s):  
Experimental Verification ◽  
Chaotic Attractor ◽  
Electronic Circuit ◽  
Lorenz System ◽  
Essential Dynamics ◽  
Compound Structure ◽  
Circuit Realization

An electronic circuit realization of a modified Lorenz system, which is multiplier-free, is described. The well-known butterfly chaotic attractor is experimentally observed verifying that the proposed modified system does capture the essential dynamics of the original Lorenz system. Furthermore, we clarify that the butterfly attractor is a compound structure obtained by merging together two simple attractors after performing one mirror operation.

scholarly journals Complex Chaotic Attractor via Fractal Transformation

Entropy ◽  
2019 ◽  
Vol 21 (11) ◽  
pp. 1115 ◽  
Author(s):  
Shengqiu Dai ◽  
Kehui Sun ◽  
Shaobo He ◽  
Wei Ai
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Digital Circuit ◽  
Chaotic Attractors ◽  
Chaotic Sequences ◽  
Three Dimensional Space

Based on simplified Lorenz multiwing and Chua multiscroll chaotic systems, a rotation compound chaotic system is presented via transformation. Based on a binary fractal algorithm, a new ternary fractal algorithm is proposed. In the ternary fractal algorithm, the number of input sequences is extended from 2 to 3, which means the chaotic attractor with fractal transformation can be presented in the three-dimensional space. Taking Lorenz system, rotation Lorenz system and compound chaotic system as the seed chaotic systems, the dynamics of the complex chaotic attractors with fractal transformation are analyzed by means of bifurcation diagram, complexity and power spectrum, and the results show that the chaotic sequences with fractal transformation have higher complexity. As the experimental verification, one kind of complex chaotic attractors is implemented by DSP, and the result is consistent with that of the simulation, which verifies the feasibility of digital circuit implement.

scholarly journals Templates of Two Foliated Attractors — Lorenz and Chen Systems

2016 ◽  
Vol 26 (02) ◽  
pp. 1650037
Author(s):  
Martin Rosalie
Keyword(s):  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Lorenz System ◽  
Chen System ◽  
Attractor Solution ◽  
The Lorenz System ◽  
Foliated Structure

A chaotic attractor solution of the Lorenz system [Lorenz, 1963] with foliated structure is topologically characterized. Its template permits to both summarize the organization of its periodic orbits and detail the topology of the solution as a branched manifold. A template of an attractor solution of the Chen system [Chen & Ueta, 1999] with a similar foliated structure is also established.

A GALLERY OF LORENZ-LIKE AND CHEN-LIKE ATTRACTORS

2013 ◽  
Vol 23 (04) ◽  
pp. 1330011 ◽  
Author(s):  
XIONG WANG ◽  
GUANRONG CHEN
Keyword(s):  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Lorenz Attractor ◽  
Chen System ◽  
The Lorenz System ◽  
Algebraic Forms ◽  
General Dynamic

In this article, three-dimensional autonomous chaotic systems with two quadratic terms, similar to the Lorenz system in their algebraic forms, are studied. An attractor with two clearly distinguishable scrolls similar to the Lorenz attractor is referred to as a Lorenz-like attractor, while an attractor with more intertwine between the two scrolls similar to the Chen attractor is referred to as a Chen-like attractor. A gallery of Lorenz-like attractors and Chen-like attractors are presented. For several different families of such systems, through tuning only one real parameter gradually, each of them can generate a spectrum of chaotic attractors continuously changing from a Lorenz-like attractor to a Chen-like attractor. Some intrinsic relationships between the Lorenz system and the Chen system are revealed and discussed. Some common patterns of the Lorenz-like and Chen-like attractors are found and analyzed, which suggest that the instability of the two saddle-foci of such a system somehow determines the shape of its chaotic attractor. These interesting observations on the general dynamic patterns hopefully could shed some light for a better understanding of the intrinsic relationships between the algebraic structures and the geometric attractors of these kinds of chaotic systems.

Sign in / Sign up

Export Citation Format

Share Document