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.

CLASSIFICATION OF CHAOTIC REGIMES IN THE T SYSTEM BY USE OF COMPETITIVE MODES

2010 ◽  
Vol 20 (11) ◽  
pp. 3785-3793 ◽  
Author(s):  
ROBERT A. VAN GORDER ◽  
S. ROY CHOUDHURY
Keyword(s):  
Chaotic Behavior ◽  
Three Dimensional ◽  
Determine Parameter ◽  
System A ◽  
Great Utility ◽  
Stable Equilibria ◽  
Competitive Modes ◽  
The Lorenz System ◽  
The Many ◽  
T System

We study chaotic behavior of the T system, a three-dimensional autonomous nonlinear system introduced by G. Tigan [Analysis of a dynamical system derived from the Lorenz system, Sci. Bull. Politehnica Univ Timisoara50 (2005) 61–72] which has potential application in secure communications. The recently-developed technique of competitive modes analysis is applied to determine parameter regimes for which the system may exhibit chaotic behavior. We verify that the T system exhibits interesting behaviors in the many parameter regimes thus obtained, thereby demonstrating the great utility of the competitive modes approach in delineating chaotic regimes in multiparemeter systems, where their identification can otherwise involve tedious numerical searches. An additional, novel finding is that one may use competitive modes "at infinity" in order to identify parameter regimes admitting stable equilibria in dynamical models such as the T system.

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.

CONTROL OF THE DIFFERENTIALLY FLAT LORENZ SYSTEM

2001 ◽  
Vol 11 (07) ◽  
pp. 1989-1996 ◽  
Author(s):  
JIN MAN JOO ◽  
JIN BAE PARK
Keyword(s):  
Equilibrium State ◽  
Lorenz System ◽  
Degree Of Freedom ◽  
Design Approach ◽  
State Trajectory ◽  
System A ◽  
Full State ◽  
Tracking Controller ◽  
The Lorenz System ◽  
Two Degree Of Freedom

This paper presents an approach for the control of the Lorenz system. We first show that the controlled Lorenz system is differentially flat and then compute the flat output of the Lorenz system. A two degree of freedom design approach is proposed such that the generation of full state feasible trajectory incorporates with the design of a tracking controller via the flat output. The stabilization of an equilibrium state and the tracking of a feasible state trajectory are illustrated.

Chaos in the Periodically Parametrically Excited Lorenz System

2021 ◽  
Vol 31 (08) ◽  
pp. 2130024
Author(s):  
Weisheng Huang ◽  
Xiao-Song Yang
Keyword(s):  
Chaotic Dynamics ◽  
Lorenz System ◽  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Time Varying ◽  
Asymptotically Stable ◽  
Parametrically Excited ◽  
The Lorenz System

We demonstrate in this paper a new chaotic behavior in the Lorenz system with periodically excited parameters. We focus on the parameters with which the Lorenz system has only two asymptotically stable equilibrium points, a saddle and no chaotic dynamics. A new mechanism of generating chaos in the periodically excited Lorenz system is demonstrated by showing that some trajectories can visit different attractor basins due to the periodic variations of the attractor basins of the time-varying stable equilibrium points when a parameter of the Lorenz system is varying periodically.

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.

Tangent bundle viewpoint of the Lorenz system and its chaotic behavior

2010 ◽  
Vol 374 (11-12) ◽  
pp. 1315-1319 ◽  
Author(s):  
Takahiro Yajima ◽  
Hiroyuki Nagahama
Keyword(s):  
Tangent Bundle ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
The Lorenz System

Study on Chaotic Behavior of the Restricted Four-Body Problem with an Equilateral Triangle Configuration

2017 ◽  
Vol 27 (02) ◽  
pp. 1750026 ◽  
Author(s):  
Xuhua Cheng ◽  
Zhikun She
Keyword(s):  
Equilateral Triangle ◽  
Body Problem ◽  
Chaotic Behavior ◽  
Homoclinic Orbits ◽  
Simple Zero ◽  
Integral Function ◽  
Smale Horseshoe ◽  
Melnikov Integral ◽  
Transversal Homoclinic Orbits ◽  
Four Body Problem

In this paper, the chaotic behavior of a planar restricted four-body problem with an equilateral triangle configuration is analytically studied. Firstly, according to the perturbation method of Melnikov, the planar restricted four-body problem is regarded as a perturbation of the two-body model. Then, we show that the Melnikov integral function has a simple zero, arriving at the existence of transversal homoclinic orbits. Afterwards, since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle, we alternatively construct an invertible map [Formula: see text] and check that [Formula: see text] is a Smale horseshoe map, showing that our restricted four-body problem possesses chaotic behavior of the Smale horseshoe type.

Feedback Linearization of the Lorenz System: Stabilization and Tracking Control

2002 ◽  
Author(s):  
L. G. Crespo ◽  
J. Q. Sun
Keyword(s):  
Feedback Linearization ◽  
Lorenz System ◽  
The State ◽  
Differential Flatness ◽  
State Transformation ◽  
System A ◽  
Excellent Control ◽  
High Control ◽  
The Lorenz System ◽  
Control Study

This paper presents a control study of the Lorenz system by using feedback linearization. The effects of the state transformation on the dynamics of the Lorenz system are studied first. Then, composite controllers are developed for both stabilization and tracking of the system. The controls are designed to overcome the barrier in controllability imposed by the state transformation. The transition through the manifold defined by such a singularity is achieved by inducing the chaotic response within a boundary layer that contains the singularity. Outside this region, a conventional feedback nonlinear control is applied. In this fashion, the authority of the control is enlarged to the whole state space and the need for high control efforts is substantially mitigated. Tracking problems that involve single and cooperative objectives are studied by using the differential flatness property of the system. A good understanding of the system dynamics proves to be invaluable in the design of better controls. In all numerical examples, the proposed approach led to excellent control performances.

COLOR MAP OF LYAPUNOV EXPONENTS OF INVARIANT SETS

1999 ◽  
Vol 09 (07) ◽  
pp. 1459-1463 ◽  
Author(s):  
MARCO MONTI ◽  
WILLIAM B. PARDO ◽  
JONATHAN A. WALKENSTEIN ◽  
EPAMINONDAS ROSA ◽  
CELSO GREBOGI
Keyword(s):  
Lyapunov Exponent ◽  
Parameter Space ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Invariant Sets ◽  
Largest Lyapunov Exponent ◽  
Specific Behavior ◽  
Practical Applications ◽  
The Lorenz System ◽  
Different Levels

The largest Lyapunov exponent of the Lorenz system is used as a measure of chaotic behavior to construct parameter space color maps. Each color in these maps corresponds to different values of the Lyapunov exponent and indicates, in parameter space, the locations of different levels of chaos for the Lorenz system. Practical applications of these maps include moving in parameter space from place to place without leaving a region of specific behavior of the system.

Sign in / Sign up

Export Citation Format

Share Document