CHAOS OR TURBULENCE?

1992 ◽  
Vol 02 (04) ◽  
pp. 1005-1009 ◽  
Author(s):  
RAY BROWN ◽  
LEON O. CHUA
Keyword(s):  
Dynamical System ◽  
Lorenz System ◽  
Three Dimensional ◽  
New Technique ◽  
Chaotic Dynamical System ◽  
The Lorenz System

We present a new three-dimensional autonomous chaotic dynamical system that appears to have a closer relationship to turbulence than the Lorenz system. We have developed this system using the new technique of dynamical synthesis.

Distinguishing Lorenz and Chen Systems Based Upon Hamiltonian Energy Theory

2017 ◽  
Vol 27 (02) ◽  
pp. 1750024 ◽  
Author(s):  
Shijian Cang ◽  
Aiguo Wu ◽  
Zenghui Wang ◽  
Zengqiang Chen
Keyword(s):  
Energy Function ◽  
Lorenz System ◽  
Three Dimensional ◽  
Type System ◽  
Chen System ◽  
First Order ◽  
Dimensional Phase Space ◽  
Ellipsoidal Surface ◽  
Generalized Hamiltonian Realization ◽  
The Lorenz System

Solving the linear first-order Partial Differential Equations (PDEs) derived from the unified Lorenz system, it is found that there is a unified Hamiltonian (energy function) for the Lorenz and Chen systems, and the unified energy function shows a hyperboloid of one sheet for the Lorenz system and an ellipsoidal surface for the Chen system in three-dimensional phase space, which can be used to explain that the Lorenz system is not equivalent to the Chen system. Using the unified energy function, we obtain two generalized Hamiltonian realizations of these two chaotic systems, respectively. Moreover, the energy function and generalized Hamiltonian realization of the Lü system and a four-dimensional hyperchaotic Lorenz-type system are also discussed.

scholarly journals On the topology of the Lorenz system

2017 ◽  
Vol 473 (2205) ◽  
pp. 20170374 ◽  
Author(s):  
Tali Pinsky
Keyword(s):  
Invariant Manifolds ◽  
Lorenz System ◽  
Three Dimensional ◽  
Lorenz Equations ◽  
New Paradigm ◽  
Anosov Flow ◽  
Trefoil Knot ◽  
The Lorenz System ◽  
Parameter Values ◽  
Special Parameter

We present a new paradigm for three-dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension 3 is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (i) for certain parameters, the Lorenz system has an invariant one-dimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (ii) The flow is topologically equivalent to an Anosov flow on the complement of this curve, and moreover to a geodesic flow. (iii) When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement. Different knots appear for different parameter values and each knot controls the dynamics at nearby parameters.

PERIODIC ORBITS OF THE LORENZ SYSTEM THROUGH PERTURBATION THEORY

2001 ◽  
Vol 11 (10) ◽  
pp. 2559-2566 ◽  
Author(s):  
J. PALACIÁN ◽  
P. YANGUAS
Keyword(s):  
Periodic Orbits ◽  
Lorenz System ◽  
Linear Part ◽  
Three Dimensional ◽  
Invariant Sets ◽  
Dimensional System ◽  
The Matrix ◽  
Lie Transformations ◽  
The Lorenz System ◽  
The Stability

Different transformations are applied to the Lorenz system with the aim of reducing the initial three-dimensional system into others of dimension two. The symmetries of the linear part of the system are determined by calculating the matrices which commute with the matrix associated to the linear part. These symmetries are extended to the whole system up to an adequate order by using Lie transformations. After the reduction, we formulate the resulting systems using the invariants associated to each reduction. At this step, we calculate for each reduced system the equilibria and their stability. They are in correspondence with the periodic orbits and invariant sets of the initial system, the stability being the same.

BIFURCATION ANALYSIS OF CHEN'S EQUATION

2000 ◽  
Vol 10 (08) ◽  
pp. 1917-1931 ◽  
Author(s):  
TETSUSHI UETA ◽  
GUANRONG CHEN
Keyword(s):  
Bifurcation Analysis ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Three Dimensional ◽  
Dynamical Properties ◽  
The Lorenz System

Anticontrol of chaos by making a nonchaotic system chaotic has led to the discovery of some new chaotic systems, particularly the continuous-time three-dimensional autonomous Chen's equation with only two quadratic terms. This paper further investigates some basic dynamical properties and various bifurcations of Chen's equation, thereby revealing its different features from some other chaotic models such as its origin, the Lorenz system.

scholarly journals Data-Rate Constrained Observers of Nonlinear Systems

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 282 ◽  
Author(s):  
Quentin Voortman ◽  
Alexander Pogromsky ◽  
Alexey Matveev ◽  
Henk Nijmeijer
Keyword(s):  
Dynamical System ◽  
Communication Channel ◽  
Lorenz System ◽  
Upper Bounds ◽  
Data Rate ◽  
The State ◽  
Box Dimension ◽  
Lyapunov Dimension ◽  
Remote Location ◽  
The Lorenz System

In this paper, the design of a data-rate constrained observer for a dynamical system is presented. This observer is designed to function both in discrete time and continuous time. The system is connected to a remote location via a communication channel which can transmit limited amounts of data per unit of time. The objective of the observer is to provide estimates of the state at the remote location through messages that are sent via the channel. The observer is designed such that it is robust toward losses in the communication channel. Upper bounds on the required communication rate to implement the observer are provided in terms of the upper box dimension of the state space and an upper bound on the largest singular value of the system’s Jacobian. Results that provide an analytical bound on the required minimum communication rate are then presented. These bounds are obtained by using the Lyapunov dimension of the dynamical system rather than the upper box dimension in the rate. The observer is tested through simulations for the Lozi map and the Lorenz system. For the Lozi map, the Lyapunov dimension is computed. For both systems, the theoretical bounds on the communication rate are compared to the simulated rates.

UNMASKING A MODULATED CHAOTIC COMMUNICATIONS SCHEME

1996 ◽  
Vol 06 (02) ◽  
pp. 367-375 ◽  
Author(s):  
KEVIN M. SHORT
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Lorenz System ◽  
Sinusoidal Signal ◽  
Chaotic Communication ◽  
Communication Scheme ◽  
Transmitted Signal ◽  
Chaotic Communications ◽  
The Lorenz System ◽  
Perfect Synchrony

This paper will address the problem of unmasking a new chaotic communication scheme using synchronizing circuits, where the Lorenz system is modulated by the message and the x-coordinate of the modulated system is added to the message and transmitted to the receiver. The receiver is driven into perfect synchrony with the transmitter even in the presence of the message, and since the message becomes part of the dynamics it provides very little distortion to the phase space of the dynamical system. However, this paper will demonstrate that it is still possible to extract a sinusoidal message from the transmitted signal. It will also be shown that it is possible to extract the sinusoidal signal solely from the x-coordinate, without secondarily adding back the message sinusoid before transmission. The message extraction is also shown to work for simple frequency-modulated and phase-modulated message signals. The modulated communication scheme does effectively nullify a multi-step unmasking technique which had been somewhat successful when applied to chaotic communication schemes which employed additive message signals.

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.

Shil’nikov Analysis of Homoclinic and Heteroclinic Orbits of the T System

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Robert A. Van Gorder ◽  
S. Roy Choudhury
Keyword(s):  
Lorenz System ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Secure Communications ◽  
Smale Horseshoe ◽  
System A ◽  
The Lorenz System ◽  
T System

We study the chaotic behavior of the T system, a three dimensional autonomous nonlinear system introduced by Tigan (2005, “Analysis of a Dynamical System Derived From the Lorenz System,” Scientific Bulletin Politehnica University of Timisoara, Tomul, 50, pp. 61–72), which has potential application in secure communications. Here, we first recount the heteroclinic orbits of Tigan and Dumitru (2008, “Analysis of a 3D Chaotic System,” Chaos, Solitons Fractals, 36, pp. 1315–1319), and then we analytically construct homoclinic orbits describing the observed Smale horseshoe chaos. In the parameter regimes identified by this rigorous Shil’nikov analysis, the occurrence of interesting behaviors thus predicted in the T system is verified by the use of numerical diagnostics.

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.

Sign in / Sign up

Export Citation Format

Share Document