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

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.

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.

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.

Probabilistic Analysis of Bifurcations in Stochastic Nonlinear Dynamical Systems

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Ehsan Mirzakhalili ◽  
Bogdan I. Epureanu
Keyword(s):  
Dynamical Systems ◽  
Probability Density Function ◽  
Probability Density ◽  
Lorenz System ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical ◽  
Pitchfork Bifurcations ◽  
The Lorenz System

Bifurcation diagrams are limited most often to deterministic dynamical systems. However, stochastic dynamics can substantially affect the interpretation of such diagrams because the deterministic diagram often is not simply the mean of the probabilistic diagram. We present an approach based on the Fokker-Planck equation (FPE) to obtain probabilistic bifurcation diagrams for stochastic nonlinear dynamical systems. We propose a systematic approach to expand the analysis of nonlinear and linear dynamical systems from deterministic to stochastic when the states or the parameters of the system are noisy. We find stationary solutions of the FPE numerically. Then, marginal probability density function (MPDF) is used to track changes in the shape of probability distributions as well as determining the probability of finding the system at each point on the bifurcation diagram. Using MPDFs is necessary for multidimensional dynamical systems and allows direct visual comparison of deterministic bifurcation diagrams with the proposed probabilistic bifurcation diagrams. Hence, we explore how the deterministic bifurcation diagrams of different dynamical systems of different dimensions are affected by noise. For example, we show that additive noise can lead to an earlier bifurcation in one-dimensional (1D) subcritical pitchfork bifurcation. We further show that multiplicative noise can have dramatic changes such as changing 1D subcritical pitchfork bifurcations into supercritical pitchfork bifurcations or annihilating the bifurcation altogether. We demonstrate how the joint probability density function (PDF) can show the presence of limit cycles in the FitzHugh–Nagumo (FHN) neuron model or chaotic behavior in the Lorenz system. Moreover, we reveal that the Lorenz system has chaotic behavior earlier in the presence of noise. We study coupled Brusselators to show how our approach can be used to construct bifurcation diagrams for higher dimensional systems.

scholarly journals Chaotic Convection of Viscoelastic Fluid in Porous Medium Under G-Jitter

2019 ◽  
Vol 24 (1) ◽  
pp. 37-51 ◽  
Author(s):  
B.S. Bhadauria ◽  
A. Singh ◽  
M.K. Singh
Keyword(s):  
Viscoelastic Fluid ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Expansion Method ◽  
Suitable Parameter ◽  
Chaotic Convection ◽  
Chaotic Nature ◽  
The Lorenz System ◽  
Parameter Values ◽  
Retardation Parameter

Abstract The present article aims at investigating the effect of gravity modulation on chaotic convection of a viscoelastic fluid in porous media. For this, the problem is reduced into Lorenz system (non-autonomous) by employing the truncated Galerkin expansion method. The system shows transitions from periodic to chaotic behavior on increasing the scaled Rayleigh number R. The amplitude of modulation advances the chaotic nature in the system while the frequency of modulation has a tendency to delay the chaotic behavior which is in good agreement with the results due to [1]. The behavior of the scaled relaxation and retardation parameter on the system is also studied. The phase portrait and time domain diagrams of the Lorenz system for suitable parameter values have been used to analyze the system.

scholarly journals Chaotic dynamics of fractional Vallis system for El-Niño

2019 ◽  
Vol 22 (3) ◽  
pp. 825-842
Author(s):  
Amey Deshpande ◽  
Varsha Daftardar-Gejji
Keyword(s):  
El Niño ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
El Nino ◽  
Critical Value ◽  
Additional Parameter ◽  
Weather Phenomenon ◽  
The Lorenz System ◽  
Range Of Values

Abstract Vallis proposed a simple model for El-Niño weather phenomenon (referred as Vallis system) by adding an additional parameter p to the Lorenz system. He showed that the chaotic behavior of the Vallis system is related to the El-Niño effect. In the present article we study fractional version of Vallis system in detail. We investigate bifurcations and chaos present in the fractional Vallis system and the effect of variation of system parameter p. It is observed that the range of values of parameter p for which the Vallis system is chaotic, reduces with the reduction of the fractional order. Further we analyze the incommensurate fractional Vallis system and find the critical value below which the system loses chaos. We also synchronize Vallis system with Bhalekar-Gejji system.

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.

LORENZ EQUATION AND CHUA’S EQUATION

1996 ◽  
Vol 06 (12b) ◽  
pp. 2443-2489 ◽  
Author(s):  
LADISLAV PIVKA ◽  
CHAI WAH WU ◽  
ANSHAN HUANG
Keyword(s):  
Lorenz System ◽  
Chaotic Behavior ◽  
Nonlinear Function ◽  
Mathematical Structure ◽  
Nonlinear Functions ◽  
Lorenz Equation ◽  
Functions Of Two Variables ◽  
The Lorenz System ◽  
Theoretical Results ◽  
Scalar Nonlinearity

The dynamical properties of two classical paradigms for chaotic behavior are reviewed—the Lorenz and Chua’s Equations—on a comparative basis. In terms of the mathematical structure, the Lorenz Equation is more complicated than Chua’s Equation because it requires two nonlinear functions of two variables, whereas Chua’s Equation requires only one nonlinear function of one variable. It is shown that most standard routes to cbaos and dynamical phenomena previously observed from the Lorenz Equation can be produced in Chua’s system with a cubic nonlinearity. In addition, we show other phenomena from Chua’s system which are not observed in the Lorenz system so far. Some differences in the topological geometric models are also reviewed. We present some theoretical results regarding Chua’s system which are absent for the Lorenz system. For example, it is known that Chua’s system is topologically conjugate to the class of systems with a scalar nonlinearity (except for a measure zero set) and is therefore canonical in this sense. We conclude with some reasons why Chua’s system can be considered superior or more suitable than the Lorenz system for various applications and studies.

scholarly journals Stability Analysis of the Chaotic Lorenz System with a State-Feedback Controller

2021 ◽  
Author(s):  
Dan Jones
Keyword(s):  
Strange Attractor ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Geometric Approach ◽  
Stable Equilibrium Point ◽  
The Lorenz System ◽  
The Stability ◽  
Parameter Values

The Lorenz model is considered a benchmark system in chaotic dynamics in that it displays extraordinary sensitivity to initial conditions and the strange attractor phenomenon. Even though the system tends to amplify perturbations, it is indeed possible to convert a strange attractor to a non-chaotic one using various control schemes. In this work it is shown that the chaotic behavior of the Lorenz system can be suppressed through the use of a feedback loop driven by a quotient controller. The stability of the controlled Lorenz system is evaluated near its equilibrium points using Routh-Hurwitz testing, and the global stability of the controlled system is established using a geometric approach. It is shown that the controlled Lorenz system has only one globally stable equilibrium point for the set of parameter values under consideration.

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