When Two Dual Chaotic Systems Shake Hands

2014 ◽  
Vol 24 (06) ◽  
pp. 1450086 ◽  
Author(s):  
J. C. Sprott ◽  
Xiong Wang ◽  
Guanrong Chen
Keyword(s):  
Parameter Space ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Time Variable ◽  
Chen System ◽  
Common Parameter ◽  
The Lorenz System ◽  
The Relationship

This letter reports an interesting finding that the parametric Lorenz system and the parametric Chen system "shake hands" at a particular point of their common parameter space, as the time variable t → +∞ in the Lorenz system while t → -∞ in the Chen system. This helps better clarify and understand the relationship between these two closely related but topologically nonequivalent chaotic systems.

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 COMMON PHENOMENON IN CHAOTIC SYSTEMS LINKED BY TIME DELAY

2006 ◽  
Vol 16 (12) ◽  
pp. 3727-3736 ◽  
Author(s):  
PEI YU ◽  
FEI XU
Keyword(s):  
Time Delay ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Common Phenomenon ◽  
State Variables ◽  
Chen System ◽  
The Family ◽  
The Lorenz System ◽  
Parameter Values ◽  
Liu System

In this paper, we report a common phenomenon observed in chaotic systems linked by time delay. Recently, the Lorenz chaotic system has been extended to the family of Lorenz systems which includes the Chen and Lü systems. These three chaotic systems, corresponding to different sets of system parameter values, are topologically different. With the aid of numerical simulations, we have surprisingly found that a simple time delay, directly applied to one or more state variables, transforms the Lorenz system to the generalized Chen system or the generalized Lü system without any parameter changes. The existence of this phenomenon has also been found in other known chaotic systems: the Rössler system, the Chua's circuit and the 4-Liu system. This finding has shown a common characteristic of chaotic systems: a new chaotic "branch" can be created from a chaotic attractor by simply adding a time delay.

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.

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.

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

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.

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.

Multistability in the Lorenz System: A Broken Butterfly

2014 ◽  
Vol 24 (10) ◽  
pp. 1450131 ◽  
Author(s):  
Chunbiao Li ◽  
Julien Clinton Sprott
Keyword(s):  
Parameter Space ◽  
Limit Cycles ◽  
Lorenz System ◽  
Dynamical Behavior ◽  
Symmetric Pair ◽  
Physical Systems ◽  
System A ◽  
Plausible Model ◽  
Fluid Convection ◽  
The Lorenz System

In this paper, the dynamical behavior of the Lorenz system is examined in a previously unexplored region of parameter space, in particular, where r is zero and b is negative. For certain values of the parameters, the classic butterfly attractor is broken into a symmetric pair of strange attractors, or it shrinks into a small attractor basin intermingled with the basins of a symmetric pair of limit cycles, which means that the system is bistable or tristable under certain conditions. Although the resulting system is no longer a plausible model of fluid convection, it may have application to other physical systems.

GENERALIZED SYNCHRONIZATION OF NONIDENTICAL FRACTIONAL-ORDER CHAOTIC SYSTEMS

2013 ◽  
Vol 27 (30) ◽  
pp. 1350195 ◽  
Author(s):  
XING-YUAN WANG ◽  
ZUN-WEN HU ◽  
CHAO LUO
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Pole Placement ◽  
Generalized Synchronization ◽  
Chen System ◽  
Hyperchaotic Lorenz System ◽  
The Stability ◽  
Synchronization Scheme ◽  
Placement Technique

In this paper, a chaotic synchronization scheme is proposed to achieve the generalized synchronization between two different fractional-order chaotic systems. Based on the stability theory of fractional-order systems and the pole placement technique, a controller is designed and theoretical proof is given. Two groups of examples are shown to verify the effectiveness of the proposed scheme, the first one is to realize the generalized synchronization between the fractional-order Chen system and the fractional-order Rössler system, the second one is between the fractional-order Lü system and the fractional-order hyperchaotic Lorenz system. The corresponding numerical simulations verify the effectiveness of the proposed scheme.

scholarly journals Transient Dynamics of the Lorenz System with a Parameter Drift

2021 ◽  
Vol 31 (02) ◽  
pp. 2150029
Author(s):  
Julia Cantisán ◽  
Jesús M. Seoane ◽  
Miguel A. F. Sanjuán
Keyword(s):  
Lorenz System ◽  
Scaling Law ◽  
Transient State ◽  
Rate Of Change ◽  
Transient Dynamics ◽  
Change Rate ◽  
The Lorenz System ◽  
Sudden Transition ◽  
The Relationship

Nonautonomous dynamical systems help us to understand the implications of real systems which are in contact with their environment as it actually occurs in nature. Here, we focus on systems where a parameter changes with time at small but non-negligible rates before settling at a stable value, by using the Lorenz system for illustration. This kind of systems commonly show a long-term transient dynamics previous to a sudden transition to a steady state. This can be explained by the crossing of a bifurcation in the associated frozen-in system. We surprisingly uncover a scaling law relating the duration of the transient to the rate of change of the parameter for a case where a chaotic attractor is involved. Additionally, we analyze the viability of recovering the transient dynamics by reversing the parameter to its original value, as an alternative to the control theory for systems with parameter drifts. We obtain the relationship between the paramater change rate and the number of trajectories that tip back to the initial attractor corresponding to the transient state.

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