MODIFIED GENERALIZED LORENZ SYSTEM AND FOLDED CHAOTIC ATTRACTORS

2009 ◽  
Vol 19 (08) ◽  
pp. 2573-2587 ◽  
Author(s):  
BOCHENG BAO ◽  
ZHONG LIU ◽  
JUEBANG YU
Keyword(s):  
Chaotic Systems ◽  
Normal Forms ◽  
Lorenz System ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors ◽  
Generalized Lorenz System ◽  
Parameter Values ◽  
Lyapunov Exponent Spectrum

A modified generalized Lorenz system in a canonical form extended from the generalized Lorenz system is proposed in this paper. This novel system has a folded factor and can display complex 2-scroll folded attractors and 1-scroll folded attractors at different parameter values. Three typical normal forms, called Lorenz-like, Chen-like and Lü-like chaotic system respectively, of three-dimensional quadratic autonomous chaotic systems are derived, and their dynamical behaviors are further investigated by employing Lyapunov exponent spectrum, bifurcation diagram, Poincaré mapping and phase portrait, etc. Of particular interest is the fact that the folded factor makes Chen-like and Lü-like chaotic systems exhibit complicated nonlinear dynamical phenomena.

A Simple Method for Generating Mirror Symmetry Composite Multiscroll Chaotic Attractors

2020 ◽  
Vol 30 (11) ◽  
pp. 2050220
Author(s):  
Xuenan Peng ◽  
Yicheng Zeng
Keyword(s):  
Lyapunov Exponent ◽  
Mirror Symmetry ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Attractors ◽  
Simple Method ◽  
Dynamical Behaviors ◽  
The Lorenz System ◽  
Lyapunov Exponent Spectrum ◽  
New System

For further increasing the complexity of chaotic attractors, a new method for generating Mirror Symmetry Composite Multiscroll Chaotic Attractors (MSCMCA) is proposed. We take the Lorenz system as an example to explain the mechanism of the method. Firstly, by varying the signs and magnitudes of the nonlinear terms, the Lorenz system generates symmetrical attractors and different-magnitude attractors, respectively. Secondly, a modified Lorenz system is constructed by imposing several unified multilevel-logic pulse signals to the Lorenz system. The new system generates a novel chaotic attractor consisting of two pairs of different-magnitude symmetrical attractors. By adjusting the parameters of the pulse signals, the modified Lorenz system can also be controlled to generate novel grid multiscroll chaotic attractors, namely MSCMCA. Several dynamical behaviors of the new system are shown by equilibria analysis and Lyapunov exponent spectrum. Moreover, the method can be applied to other chaotic systems. Finally, a circuit of the modified Lorenz system is designed by Multisim software, and the simulation result proves the effectiveness of the method.

A NEW CHAOTIC SYSTEM AND BEYOND: THE GENERALIZED LORENZ-LIKE SYSTEM

2004 ◽  
Vol 14 (05) ◽  
pp. 1507-1537 ◽  
Author(s):  
JINHU LÜ ◽  
GUANRONG CHEN ◽  
DAIZHAN CHENG
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Compound Structure ◽  
Constant Input ◽  
New Class ◽  
Dynamical Behaviors ◽  
Generalized Lorenz System ◽  
New Chaotic System

This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display (i) two 1-scroll chaotic attractors simultaneously, with only three equilibria, and (ii) two 2-scroll chaotic attractors simultaneously, with five equilibria. Several issues such as some basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new chaotic system are then investigated, either analytically or numerically. Of particular interest is the fact that this chaotic system can generate a complex 4-scroll chaotic attractor or confine two attractors to a 2-scroll chaotic attractor under the control of a simple constant input. Furthermore, the concept of generalized Lorenz system is extended to a new class of generalized Lorenz-like systems in a canonical form. Finally, the important problems of classification and normal form of three-dimensional quadratic autonomous chaotic systems are formulated and discussed.

DESIGN AND IMPLEMENTATION OF MULTI-WING BUTTERFLY CHAOTIC ATTRACTORS VIA LORENZ-TYPE SYSTEMS

2010 ◽  
Vol 20 (01) ◽  
pp. 29-41 ◽  
Author(s):  
SIMIN YU ◽  
WALLACE K. S. TANG ◽  
JINHU LÜ ◽  
GUANRONG CHEN
Keyword(s):  
Analog Circuits ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Type Systems ◽  
Circuit Diagram ◽  
Chaotic Attractors ◽  
Dynamical Behaviors ◽  
Novel Approach ◽  
Index 2 ◽  
First Time

Lorenz system, as the first classical chaotic system, has been intensively investigated over the last four decades. Based on the sawtooth wave function, this paper initiates a novel approach for generating multi-wing butterfly chaotic attractors from the generalized first and second kinds of Lorenz-type systems. Compared with the traditional ring-shaped multi-scroll Lorenz chaotic attractors, the proposed multi-wing butterfly chaotic attractors are much easier to be designed and implemented by analog circuits. The dynamical behaviors of these multi-wing butterfly chaotic systems are further studied. Theoretical analysis shows that every index-2 saddle-focus equilibrium corresponds to a unique wing in the butterfly attractors. Finally, a module-based unified circuit diagram is constructed for realizing various multi-wing butterfly attractors. It should be especially pointed out that this is the first time in the literature that a maximal 10-wing butterfly chaotic attractor is experimentally verified by analog circuits.

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.

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.

Various Attractors, Coexisting Attractors and Antimonotonicity in a Simple Fourth-Order Memristive Twin-T Oscillator

2018 ◽  
Vol 28 (04) ◽  
pp. 1850050 ◽  
Author(s):  
Ling Zhou ◽  
Chunhua Wang ◽  
Xin Zhang ◽  
Wei Yao
Keyword(s):  
Fourth Order ◽  
Chaotic Systems ◽  
Basin Of Attraction ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Dynamical Behaviors ◽  
Random Bit Generator ◽  
In Series ◽  
Dynamical Features

By replacing the resistor in a Twin-T network with a generalized flux-controlled memristor, this paper proposes a simple fourth-order memristive Twin-T oscillator. Rich dynamical behaviors can be observed in the dynamical system. The most striking feature is that this system has various periodic orbits and various chaotic attractors generated by adjusting parameter [Formula: see text]. At the same time, coexisting attractors and antimonotonicity are also detected (especially, two full Feigenbaum remerging trees in series are observed in such autonomous chaotic systems). Their dynamical features are analyzed by phase portraits, Lyapunov exponents, bifurcation diagrams and basin of attraction. Moreover, hardware experiments on a breadboard are carried out. Experimental measurements are in accordance with the simulation results. Finally, a multi-channel random bit generator is designed for encryption applications. Numerical results illustrate the usefulness of the random bit generator.

DYNAMICAL ANALYSIS OF A CHAOTIC SYSTEM WITH TWO DOUBLE-SCROLL CHAOTIC ATTRACTORS

2004 ◽  
Vol 14 (03) ◽  
pp. 971-998 ◽  
Author(s):  
WENBO LIU ◽  
GUANRONG CHEN
Keyword(s):  
Chaotic System ◽  
Three Dimensional ◽  
Dynamical Analysis ◽  
Chaotic Attractors ◽  
Compound Structure ◽  
Numerical Examples ◽  
Routes To Chaos ◽  
Basic Properties ◽  
Dynamical Behaviors

Dynamical behaviors of a three-dimensional autonomous chaotic system with two double-scroll attractors are studied. Some basic properties such as bifurcation, routes to chaos, periodic windows and compound structure are demonstrated with various numerical examples. System equilibria and their stabilities are discussed, and chaotic features of the attractors are justified numerically.

scholarly journals Nonlinear dynamics induced anomalous Hall effect in topological insulators

2016 ◽  
Vol 2016 (DPC) ◽  
pp. 001543-001553
Author(s):  
Guanglei Wang ◽  
Hongya Xu ◽  
Ying-Cheng Lai
Keyword(s):  
Hall Effect ◽  
Three Dimensional ◽  
Anomalous Hall Effect ◽  
Transmission Probability ◽  
Spin Transfer Torque ◽  
Spin Transfer ◽  
Alternative Mechanism ◽  
Final States ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors

We uncover an alternative mechanism for anomalous Hall effect. In particular, we investigate the magnetisation dynamics of an insulating ferromagnet (FM) deposited on the surface of a three-dimensional topological insulator (TI), subject to an external voltage. The spin-polarised current on the TI surface induces a spin-transfer torque on the magnetisation of the top FM while its dynamics can change the transmission probability of the surface electrons through the exchange coupling and hence the current. We find a host of nonlinear dynamical behaviors including multistability, chaos, and phase synchronisation. Strikingly, a dynamics mediated Hall-like current can arise, which exhibits a nontrivial dependence on the channel conductance. We develop a physical understanding of the mechanism that leads to the anomalous Hall effect. The nonlinear dynamical origin of the effect stipulates that a rich variety of final states exist, implying that the associated Hall current can be controlled to yield desirable behaviors. The phenomenon can find applications in Dirac-material based spintronics.

scholarly journals Dynamical Behaviors of Coupled Memristor-Based Oscillators with Identical and Different Nonlinearities

2018 ◽  
Vol 2018 ◽  
pp. 1-20
Author(s):  
A. Elsonbaty ◽  
A. Abdelkhalek ◽  
A. Elsaid
Keyword(s):  
Mathematical Models ◽  
Coupled Oscillators ◽  
Perturbation Methods ◽  
Normal Forms ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Numerical Techniques ◽  
Ring Configuration ◽  
Dynamical Behaviors

In this work, the interesting dynamics of coupled nonlinear memristor-based oscillators in ring configuration are explored. The mathematical models are derived to describe the possible cases of employing identical or different nonlinearities. Analytical and numerical techniques, involving perturbation methods, normal forms, phase portraits, and Lyapunov exponents are used to investigate various types of dynamical behaviors along with their stability regions in parameters space. The effects of time-delayed coupling on the proposed system are numerically studied. It is demonstrated that the coupled oscillators show rich dynamics including periodic orbits, quasiperiodicity, two-dimensional, and three-dimensional tori.

ON A GENERALIZED LORENZ CANONICAL FORM OF CHAOTIC SYSTEMS

2002 ◽  
Vol 12 (08) ◽  
pp. 1789-1812 ◽  
Author(s):  
SERGEJ ČELIKOVSKÝ ◽  
GUANRONG CHEN
Keyword(s):  
Large Class ◽  
Canonical Form ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chen System ◽  
Scalar Parameter ◽  
Generalized Lorenz System ◽  
Special Cases ◽  
The One ◽  
Generalized Lorenz Canonical Form

This paper shows that a large class of systems, introduced in [Čelikovský & Vaněček, 1994; Vaněček & Čelikovský, 1996] as the so-called generalized Lorenz system, are state-equivalent to a special canonical form that covers a broader class of chaotic systems. This canonical form, called generalized Lorenz canonical form hereafter, generalizes the one introduced and analyzed in [Čelikovský & Vaněček, 1994; Vaněček & Čelikovský, 1996], and also covers the so-called Chen system, recently introduced in [Chen & Ueta, 1999; Ueta & Chen, 2000].Thus, this new generalized Lorenz canonical form contains as special cases the original Lorenz system, the generalized Lorenz system, and the Chen system, so that a comparison of the structures between two essential types of chaotic systems becomes possible. The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design. Some other closely related topics are also studied and discussed in the paper.

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