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.

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.

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.

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.

scholarly journals Generating novel multi-scroll chaotic attractors via fractal transformation

2021 ◽  
Author(s):  
Dengwei Yan ◽  
Musha Ji’e ◽  
Lidan Wang ◽  
Shukai Duan ◽  
Xinyu Du
Keyword(s):  
Efficient Method ◽  
Theoretical Basis ◽  
Chaotic Systems ◽  
Chaotic Attractors ◽  
Hardware Platform ◽  
New Class ◽  
Chaotic Sequences ◽  
Offset Boosting ◽  
Before And After ◽  
First Time

Abstract The fractal and chaos are bound tightly, and their relevant researches are well-established. Few of them, however, concentrates on the research of the possibility of combining the fractal and the chaotic systems to generate multi-scroll chaotic attractors. This paper presents a novel non-equilibrium point chaotic system, exhibiting extremely rich and complex hidden behaviors including chaos, hyper-chaos, multi-scroll attractors, extreme multi-stability and initial offset-boosting. The proposed system is combined with fractal transformation respectively, and a new class of multi-scroll attractors, such as multi-ring attractors and separated-scroll attractors, is observed. Particularly, swallow-shaped attractors for the first time is found. Moreover, another efficient method to generate a different class of chaotic attractors uses parabola transformation and triangle transformation. Additionally, the spectrum entropy ( SE ) complexity is employed to discuss the complexity of the proposed system before and after fractal, resulting in a chaotic sequences with fractal transformation that has higher complexity. Finally, we develop a hardware platform to implement the presented attractors before and after fractal in a way to confirm the accuracy of the numerical simulations, providing a theoretical basis for the next application in image encryption.

Constructing a New 3D Chaotic System with Any Number of Equilibria

2019 ◽  
Vol 29 (05) ◽  
pp. 1950060 ◽  
Author(s):  
Qigui Yang ◽  
Xinmei Qiao
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Complex Dynamics ◽  
Type Systems ◽  
Dynamical Behaviors ◽  
Stable Equilibria ◽  
Relationship Of ◽  
Number Of Equilibria

In the chaotic polynomial Lorenz-type systems (including Lorenz, Chen, Lü and Yang systems) and Rössler system, their equilibria are unstable and the number of the hyperbolic equilibria are no more than three. This paper shows how to construct a simple analytic (nonpolynomial) chaotic system that can have any preassigned number of equilibria. A special 3D chaotic system with no equilibrium is first presented and discussed. Using a methodology of adding a constant controller to the third equation of such a chaotic system, it is shown that a chaotic system with any preassigned number of equilibria can be generated. Two complete mathematical characterizations for the number and stability of their equilibria are further rigorously derived and studied. This system is very interesting in the sense that some complex dynamics are found, revealing many amazing properties: (i) a hidden chaotic attractor exists with no equilibria or only one stable equilibrium; (ii) the chaotic attractor coexists with unstable equilibria, including two/five unstable equilibria; (iii) the chaotic attractor coexists with stable equilibria and unstable equilibria, including one stable and two unstable equilibria/94 stable and 93 unstable equilibria; (iv) the chaotic attractor coexists with infinitely many nonhyperbolic isolated equilibria. These results reveal an intrinsic relationship of the global dynamical behaviors with the number and stability of the equilibria of some unusual 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 UNIFIED LORENZ-TYPE SYSTEM AND ITS CANONICAL FORM

2006 ◽  
Vol 16 (10) ◽  
pp. 2855-2871 ◽  
Author(s):  
QIGUI YANG ◽  
GUANGRONG CHEN ◽  
TIANSHOU ZHOU
Keyword(s):  
Canonical Form ◽  
Special Form ◽  
Lorenz System ◽  
Type System ◽  
Chaotic Attractors ◽  
Lorenz Attractor ◽  
System A ◽  
Generalized Lorenz System ◽  
Two Parameters ◽  
First Time

Based on the generalized Lorenz system, a conjugate Lorenz-type system is introduced, and a new unified Lorenz-type system containing these two classes of systems is naturally constructed in the paper. Such a unified system is state-equivalent to a simple special form, which is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, three new chaotic attractors, called conjugate attractors, are found for the first time, which are conjugate to the Lorenz attractor, the Chen attractor, and the Lü attractor, respectively.

3D Grid Multi-Wing Chaotic Attractors

2018 ◽  
Vol 28 (04) ◽  
pp. 1850045 ◽  
Author(s):  
Nan Yu ◽  
Yan-Wu Wang ◽  
Xiao-Kang Liu ◽  
Jiang-Wen Xiao
Keyword(s):  
Numerical Simulations ◽  
Bifurcation Diagram ◽  
Mirror Symmetry ◽  
Electrical Circuit ◽  
Circuit Diagram ◽  
Chaotic Attractors ◽  
Switching Functions ◽  
Rotation Transformation ◽  
Index 2 ◽  
Dynamical Properties

As reported in the existing literature, wing attractors are confined to 1D [Formula: see text]-wing attractors, 2D [Formula: see text]-grid wing attractors. In this paper, we break this limitation and generate 3D [Formula: see text]-grid multi-wing chaotic attractors (GMWCAs). The 3D GMWCAs are produced via the following three steps: (1) applying rotation transformation to a double-wing Lorenz-like system to ensure that its saddle-focus equilibria with index 2 are located on the plane [Formula: see text]; (2) extending the wing attractors of the transformed Lorenz-like system along the [Formula: see text]-axis to have mirror symmetry; (3) introducing stair switching functions to increase the number of saddle-focus equilibria with index 2 along the [Formula: see text]-axis and [Formula: see text]-axis. Furthermore, some basic dynamical properties of the 3D chaotic system, including equilibria, symmetry, dissipativity, Lyapunov exponents and bifurcation diagram, are investigated and a module-based unified circuit diagram is designed. The effectiveness of this approach is confirmed by both numerical simulations and electrical circuit experiment.

DESIGN AND REALIZATION OF MULTI-WING CHAOTIC ATTRACTORS VIA SWITCHING CONTROL

2011 ◽  
Vol 25 (16) ◽  
pp. 2183-2194 ◽  
Author(s):  
CHAO-XIA ZHANG ◽  
SI-MIN YU ◽  
YUN ZHANG
Keyword(s):  
Numerical Simulation ◽  
Equilibrium Points ◽  
Switching Control ◽  
Linear Functions ◽  
Experimental Result ◽  
Chaotic Attractors ◽  
Dynamical Behaviors ◽  
Novel Approach ◽  
Switching Controller ◽  
Maximum Lyapunov Exponents

In this paper, a novel approach for generating multi-wing chaotic attractors via switching control is proposed. By using a switching controller, multi-wing chaotic attractors can be generated from a double-wing system. The presented method is different from that of classical multi-scroll chaotic attractors generated by odd-symmetric multi-segment linear functions from Chua system. Furthermore, the basic dynamical behaviors, including equilibrium points, maximum Lyapunov exponents and bifurcations, are further investigated. An improved module-based unified circuit is designed for realizing 4, 6, 8 and 10-wing chaotic attractors, and the experimental result is also demonstrated, which is consistent with the numerical simulation.

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.

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