Jacobi Stability of Simple Chaotic Systems with One Lyapunov Stable Equilibrium

2021 ◽  
Author(s):  
Changzhi Li ◽  
Biyu Chen ◽  
Aimin Liu ◽  
Huanhuan Tian
Keyword(s):  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Geometrical Structure ◽  
Chaotic Behavior ◽  
Internal Environment ◽  
Small Perturbations ◽  
Chaotic Flows ◽  
Dynamical Behaviors ◽  
Onset Of Chaos ◽  
Deviation Vector

Abstract This paper presents Jacobi stability analysis of 23 simple chaotic systems with only one Lyapunov stable equilibrium by Kosambi-Cartan-Chern (KCC) theory, and analyzes the chaotic behavior of these systems from the geometric viewpoint. Different from Lyapunov stability, the unique equilibrium for each system is always Jacobi unstable. Moreover, the dynamical behaviors of deviation vector near equilibrium are discussed to reveal the onset of chaos for these 23 systems, and show furtherly the coexistence of unique Lyapunov stable equilibrium and chaotic attractor for each system geometrically. The obtaining results show that these chaotic systems are not robust to small perturbations of the equilibrium, indicating that the systems are extremely sensitive to internal environment. This reveals that the chaotic flows generated by these systems may be related to Jacobi instability of the equilibrium. It is hoped that the study of this paper can help reveal the true geometrical structure of hidden chaotic attractors.

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.

scholarly journals Symbolic Methods for Analysing Bifurcations and Chaos of Two Five-Parameter Families of Planar Quadratic Maps

2020 ◽  
Vol 8 (2) ◽  
pp. 72-79
Author(s):  
Sarbast H. Mikaeel ◽  
Bewar H. Othman
Keyword(s):  
Stable Equilibrium ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Algorithmic Approach ◽  
Quadratic Maps ◽  
Dynamical Behaviors ◽  
Planar Maps ◽  
Symbolic Methods ◽  
The Stability

In this work, we analyze the dynamical behaviors of two five-parameter families of planar quadratic maps by utilizing strategies of symbolic computation. We are going to use computer algebra methods to clarify how to detect the stability of equilibrium points to analyze chaos and also the bifurcation of planar maps. Based on strategies for solving the systems in types of semi-algebraic and by utilizing an algorithmic approach, we obtain respectively for the two maps, sufficient conditions on the parameters to have a prescribed number of (stable) equilibrium points; necessary conditions on the parameters to undergo a certain type of bifurcation or to have chaotic behavior induced by snapback repeller.

Jacobi analysis of a segmented disc dynamo system

2020 ◽  
Vol 17 (14) ◽  
pp. 2050205
Author(s):  
Aimin Liu ◽  
Biyu Chen ◽  
Yuming Wei
Keyword(s):  
Angular Velocity ◽  
Periodic Orbit ◽  
Finsler Geometry ◽  
Small Perturbations ◽  
Stable Periodic Orbit ◽  
Onset Of Chaos ◽  
Internal Parameters ◽  
Geometric Objects ◽  
Stable Periodic ◽  
Deviation Vector

In this paper, Jacobi stability of a segmented disc dynamo system is geometrically investigated from viewpoint of Kosambi–Cartan–Chern (KCC) theory in Finsler geometry. First, the geometric objects associated to the reformulated system are explicitly obtained. Second, the Jacobi stability of equilibria and a periodic orbit are discussed in the light of deviation curvature tensor. It is shown that all the equilibria are always Jacobi unstable for any parameters, a Lyapunov stable periodic orbit falls into both Jacobi stable regions and Jacobi unstable regions. The considered system is not robust to small perturbations of the equilibria, and some fragments of the periodic orbit are included in fragile region, indicating that the system is extremely sensitive to internal parameters and environment. Finally, the dynamics of the deviation vector and its curvature near all the equilibria are presented to interpret the onset of chaos in the dynamo system. In a physical sense, magnetic fluxes and angular velocity can show irregular oscillations under some certain cases, these oscillations may reveal the irregularity of magnetic field’s evolution and reversals.

STATISTICS OF SOME LOW-DIMENSIONAL CHAOTIC FLOWS

2001 ◽  
Vol 11 (10) ◽  
pp. 2675-2682 ◽  
Author(s):  
ELENA S. DIMITROVA ◽  
OLEG I. YORDANOV
Keyword(s):  
Chaotic Systems ◽  
Lorenz System ◽  
Slow Process ◽  
Chaotic Flows ◽  
Onset Of Chaos ◽  
Controlling Parameter ◽  
The Lorenz System ◽  
Low Dimensional ◽  
Statistical Symmetry ◽  
Statistical Functions

As a result of the recent finding that the Lorenz system exhibits blurred self-affinity for values of its controlling parameter slightly above the onset of chaos, we study other low-dimensional chaotic flows with the purpose of providing an approximate description of their second-order, two-point statistical functions. The main pool of chaotic systems on which we focus our attention is that reported by Sprott [1994], generalized however to depend on their intrinsic number of parameters. We show that their statistical properties are adequately described as processes with spectra having three segments all of power-law type. On this basis we identify quasi-periodic behavior pertaining to the relatively slow process in the attractors and approximate self-affine statistical symmetry characterizing the fast processes.

SIMPLE CHAOTIC FLOWS WITH ONE STABLE EQUILIBRIUM

2013 ◽  
Vol 23 (11) ◽  
pp. 1350188 ◽  
Author(s):  
MALIHE MOLAIE ◽  
SAJAD JAFARI ◽  
JULIEN CLINTON SPROTT ◽  
S. MOHAMMAD REZA HASHEMI GOLPAYEGANI
Keyword(s):  
Equilibrium Point ◽  
Stability Criterion ◽  
Chaotic Systems ◽  
Stable Equilibrium ◽  
Computer Search ◽  
Stable Equilibrium Point ◽  
Hidden Attractors ◽  
Chaotic Flows ◽  
Hurwitz Stability ◽  
Quadratic Nonlinearities

Using the Routh–Hurwitz stability criterion and a systematic computer search, 23 simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a coexisting stable equilibrium point. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

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.

A New Category of Three-Dimensional Chaotic Flows with Identical Eigenvalues

2020 ◽  
Vol 30 (02) ◽  
pp. 2050026 ◽  
Author(s):  
Zahra Faghani ◽  
Fahimeh Nazarimehr ◽  
Sajad Jafari ◽  
Julien C. Sprott
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Special Property ◽  
Computer Search ◽  
Chaotic Flows ◽  
Stable Equilibria ◽  
Dynamical Properties ◽  
Simple Systems ◽  
First Time

In this paper, some new three-dimensional chaotic systems are proposed. The special property of these autonomous systems is their identical eigenvalues. The systems are designed based on the general form of quadratic jerk systems with 10 terms, and some systems with stable equilibria. Using a systematic computer search, 12 simple chaotic systems with identical eigenvalues were found. We believe that systems with identical eigenvalues are described here for the first time. These simple systems are listed in this paper, and their dynamical properties are investigated.

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.

scholarly journals Theoretical Design and Circuit Implementation of Integer Domain Chaotic Systems

2014 ◽  
Vol 24 (10) ◽  
pp. 1450128 ◽  
Author(s):  
Qianxue Wang ◽  
Simin Yu ◽  
Christophe Guyeux ◽  
Jacques M. Bahi ◽  
Xiaole Fang
Keyword(s):  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Random Sequence ◽  
Noise Signal ◽  
Circuit Implementation ◽  
New Approach ◽  
Uniform Noise ◽  
Definition Of ◽  
First Time ◽  
Hybrid Circuit

In this paper, a new approach for constructing integer domain chaotic systems (IDCS) is proposed, and its chaotic behavior is mathematically proven according to Devaney's definition of chaos. Furthermore, an analog-digital hybrid circuit is also developed for realizing the designed basic IDCS. In the IDCS circuit design, chaos generation strategy is realized through a sample-hold circuit and a decoder circuit so as to convert the uniform noise signal into a random sequence, which plays a key role in circuit implementation. The experimental observations further validate the proposed systematic methodology for the first time.

Classification of Chaotic Behaviors in Jerky Dynamical Systems

2021 ◽  
Vol 30 (1) ◽  
pp. 93-110
Author(s):  
Tianyi Wang ◽  
Keyword(s):  
Machine Learning ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Learning Algorithm ◽  
Model Systems ◽  
Supervised Machine Learning ◽  
Learning Program ◽  
New Methods ◽  
Over Time

Differential equations are widely used to model systems that change over time, some of which exhibit chaotic behaviors. This paper proposes two new methods to classify these behaviors that are utilized by a supervised machine learning algorithm. Dissipative chaotic systems, in contrast to conservative chaotic systems, seem to follow a certain visual pattern. Also, the machine learning program written in the Wolfram Language is utilized to classify chaotic behavior with an accuracy around 99.1±1.1%.

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