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.

A MODIFIED GENERALIZED LORENZ-TYPE SYSTEM AND ITS CANONICAL FORM

2009 ◽  
Vol 19 (06) ◽  
pp. 1931-1949 ◽  
Author(s):  
QIGUI YANG ◽  
KANGMING ZHANG ◽  
GUANRONG CHEN
Keyword(s):  
Special Form ◽  
Topological Structure ◽  
Complex Dynamics ◽  
Type System ◽  
Chaotic Attractors ◽  
Hopf Bifurcations ◽  
Compound Structure ◽  
Two Parameters ◽  
First Time ◽  
New System

In this paper, a modified generalized Lorenz-type system is introduced, which is state-equivalent to a simple and special form, and is parameterized by two parameters useful for chaos turning and system classification. More importantly, based on the parameterized form, two classes of new chaotic attractors are found for the first time in the literature, which are similar but nonequivalent in topological structure. To further understand the complex dynamics of the new system, some basic properties such as Lyapunov exponents, Hopf bifurcations and compound structure of the attractors are analyzed and demonstrated with careful numerical simulations.

CHAOTIC ATTRACTORS OF THE CONJUGATE LORENZ-TYPE SYSTEM

2007 ◽  
Vol 17 (11) ◽  
pp. 3929-3949 ◽  
Author(s):  
QIGUI YANG ◽  
GUANRONG CHEN ◽  
KUIFEI HUANG
Keyword(s):  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Type System ◽  
Chaotic Attractors ◽  
Compound Structure ◽  
Chen System ◽  
Numerical Examples ◽  
Special Cases ◽  
Dynamical Properties

A new conjugate Lorenz-type system is introduced in this paper. The system contains as special cases the conjugate Lorenz system, conjugate Chen system and conjugate Lü system. Chaotic dynamics of the system in the parametric space is numerically and thoroughly investigated. Meanwhile, a set of conditions for possible existence of chaos are derived, which provide some useful guidelines for searching chaos in numerical simulations. Furthermore, some basic dynamical 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 demonstrated with various numerical examples.

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.

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 Symmetries in Hidden Bifurcation Routes to Multiscroll Chaotic Attractors Generated by Saturated Function Series

2019 ◽  
Vol 3 (4) ◽  
pp. 511 ◽  
Author(s):  
Faiza Zaamoune ◽  
Tidjani Menacer ◽  
René Lozi ◽  
Guanrong Chen
Keyword(s):  
Open Access ◽  
Original Work ◽  
Range Extension ◽  
Chaotic Attractors ◽  
Function Series ◽  
Creative Commons ◽  
Basic Cell ◽  
Two Parameters ◽  
First Time ◽  
Open Access Article

In this paper, hidden bifurcation routes to multiscroll chaotic attractors generated by saturated function series are explored. The method to nd such hidden bifurcation routes (HBR) depending upon two parameters is similar to the method introduced by Menacer, et al. (2016) for Chua multiscroll attractors. These HBR are characterized by the maximal range extension (MARE) of their attractors and coding the appearance order of the scrolls under the control of the two parameters. Moreover, these HDR have interesting symmetries with respect to the two parameters. The novelty that this article introduces, is firstly the paradigm of MARE and the formula giving their approximate value depending upon parameters p and q, which is linked to the size of the scrolls; secondly the coding of the HBR which is dened for the first time including the basic cell; and thirdly unearthing the symmetries of these routes, allowing to obtain their coding without any numerical computation.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

scholarly journals Stochastic sensitivity of quasiperiodic and chaotic attractors of the discrete Lotka-Volterra model

2020 ◽  
Vol 55 ◽  
pp. 19-32
Author(s):  
A.V. Belyaev ◽  
T.V. Perevalova
Keyword(s):  
Period Doubling ◽  
Basins Of Attraction ◽  
Invariant Curve ◽  
Volterra Model ◽  
Chaotic Attractors ◽  
Deterministic System ◽  
System A ◽  
Stochastic Sensitivity ◽  
Two Parameters ◽  
Random States

The aim of the study presented in this article is to analyze the possible dynamic modes of the deterministic and stochastic Lotka-Volterra model. Depending on the two parameters of the system, a map of regimes is constructed. Parametric areas of existence of stable equilibria, cycles, closed invariant curves, and also chaotic attractors are studied. The bifurcations such as the period doubling, Neimark-Sacker and the crisis are described. The complex shape of the basins of attraction of irregular attractors (closed invariant curve and chaos) is demonstrated. In addition to the deterministic system, the stochastic system, which describes the influence of external random influence, is discussed. Here, the key is to find the sensitivity of such complex attractors as a closed invariant curve and chaos. In the case of chaos, an algorithm to find critical lines giving the boundary of a chaotic attractor, is described. Based on the found function of stochastic sensitivity, confidence domains are constructed that allow us to describe the form of random states around a deterministic attractor.

scholarly journals A Sinusoidally Driven Lorenz System and Circuit Implementation

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Chunyan Han ◽  
Simin Yu ◽  
Guangyi Wang
Keyword(s):  
Periodic Orbits ◽  
Analog Circuit ◽  
Lorenz System ◽  
Digital Circuit ◽  
Chaotic Attractors ◽  
Sine Function ◽  
Circuit Implementation ◽  
System A ◽  
Systematic Methodology ◽  
Sinusoidal Function

Another approach is developed for generating two-wing hyperchaotic attractor, four-wing chaotic attractor, and high periodic orbits such as period-14 from a sinusoidally driven based canonical Lorenz system. A sinusoidal function controller is introduced into a 3D autonomous Lorenz system, so that the abovementioned various hyperchaotic attractors, chaotic attractors, and high periodic orbits can be obtained, respectively, by adjusting the frequency of the sine function. In addition, an analog circuit and a digital circuit are also designed and implemented, with experimental results demonstrated. Both numerical simulations and circuit implementation together show the effectiveness of the proposed systematic methodology.

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 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.

scholarly journals A New 4D Piecewise Linear Multiscroll Chaotic System with Multistability and Its FPGA-Based Implementation

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Faqiang Wang ◽  
Hongbo Cao ◽  
Dingding Zhai
Keyword(s):  
Chaotic System ◽  
Piecewise Linear ◽  
Type System ◽  
Chaotic Attractors ◽  
Secure Communications ◽  
Attraction Basin ◽  
System A ◽  
New Chaotic System ◽  
C0 Complexity ◽  
Lyapunov Exponent Spectrum

Due to the complex behavior of a multiscroll chaotic system, it is a good candidate for the secure communications. In this paper, by adding an additional variable to the modified Lorenz-type system, a new chaotic system that includes only linear and piecewise items but can generate 4n + 4 scroll chaotic attractors via choosing the various values of natural number n is proposed. Its dynamics including bifurcation, multistability, and symmetric coexisting attractors, as well as various chaotic and periodic behaviors, are analyzed by means of attraction basin, bifurcation diagram, dynamic map, phase portrait, Lyapunov exponent spectrum, and C0 complexity in detail. The mechanism of the occurrence for generating multiscroll chaotic attractors is presented. Finally, this multiscroll chaotic system is implemented by using the Altera Cyclone IV EP4CE10F17C8 FPGA. It is found that this FPGA-based design has an advantage of requiring less resources for 0% of the embedded multipliers and 0% of the PLLs of this FPGA are occupied.

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