scholarly journals The Fractional Form of the Tinkerbell Map Is Chaotic

2018 ◽  
Vol 8 (12) ◽  
pp. 2640 ◽  
Author(s):  
Adel Ouannas ◽  
Amina-Aicha Khennaoui ◽  
Samir Bendoukha ◽  
Thoai Vo ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Lyapunov Exponents ◽  
Stability Theory ◽  
Chaotic Map ◽  
Discrete Systems ◽  
Numerical Results ◽  
Asymptotic Convergence ◽  
Bifurcation Diagrams ◽  
Difference Form ◽  
The Stability

This paper is concerned with a fractional Caputo-difference form of the well-known Tinkerbell chaotic map. The dynamics of the proposed map are investigated numerically through phase plots, bifurcation diagrams, and Lyapunov exponents considered from different perspectives. In addition, a stabilization controller is proposed, and the asymptotic convergence of the states is established by means of the stability theory of linear fractional discrete systems. Numerical results are employed to confirm the analytical findings.

On the dynamics and control of a new fractional difference chaotic map

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Samir Bendoukha
Keyword(s):  
Chaotic Map ◽  
Discrete Systems ◽  
Fractional Difference ◽  
Difference Map ◽  
Dynamics And Control ◽  
Generalized Hénon Map ◽  
Use Phase ◽  
The Rich ◽  
The Stability ◽  
And Control

Abstract In this paper, we propose and study a fractional Caputo-difference map based on the 2D generalized Hénon map. By means of numerical methods, we use phase plots and bifurcation diagrams to investigate the rich dynamics of the proposed map. A 1D synchronization controller is proposed similar to that of Pecora and Carrol, whereby we assume knowledge of one of the two states at the slave and replicate the second state. The stability theory of fractional discrete systems is used to guarantee the asymptotic convergence of the proposed controller and numerical simulations are employed to confirm the findings.

Experimental Observations of Propagating Waves and Switching Phenomena in a Coupled Bistable Oscillator System

2014 ◽  
Vol 24 (12) ◽  
pp. 1450157 ◽  
Author(s):  
Kuniyasu Shimizu
Keyword(s):  
Invariant Subspace ◽  
Spectral Distribution ◽  
Numerical Results ◽  
Periodic Waves ◽  
Bifurcation Diagrams ◽  
Propagating Waves ◽  
Different Types ◽  
The Stability ◽  
The One ◽  
Parameter Bifurcation

In this study, we construct a circuit composed of bistable oscillators and we report the experimental observations of quasi-periodic waves propagating in the circuit and compare them with the associated numerical results. Two different types of propagating quasi-periodic waves with identical parameter sets are experimentally verified. The associated numerical results are distinguished by comparing trajectories on the phase planes and by analyzing the one-parameter bifurcation diagrams. Furthermore, the experiments reveal five different types of switching oscillations. The associated numerical results are also presented, and the stability of the switching oscillations (when constrained to an invariant subspace) is numerically investigated. We also calculate spectral distribution for one type of the switching oscillation.

NUMERICAL TREATMENT OF EDUCATIONAL CHAOS OSCILLATOR

2007 ◽  
Vol 17 (10) ◽  
pp. 3657-3661 ◽  
Author(s):  
ARÜNAS TAMAŠEVIČIUS ◽  
TATJANA PYRAGIENĖ ◽  
KȨSTUTIS PYRAGAS ◽  
SKAIDRA BUMELIENĖ ◽  
MANTAS MEŠKAUSKAS
Keyword(s):  
Mathematical Model ◽  
Lyapunov Exponents ◽  
Power Spectra ◽  
Numerical Results ◽  
Phase Portraits ◽  
Numerical Treatment ◽  
Bifurcation Diagrams

A mathematical model of a recently suggested chaos oscillator for educational purposes is treated and numerical results are presented. Bifurcation diagrams, phase portraits, power spectra, Lyapunov exponents are simulated. In addition, the Feigenbaum number is estimated.

Coexisting Multi-Dynamics of a Physical SBT Memristor-Based Chaotic Circuit

2020 ◽  
Vol 30 (11) ◽  
pp. 2030043
Author(s):  
Gang Dou ◽  
Hai Yang ◽  
Zhenhao Gao ◽  
Peng Li ◽  
Minglong Dou ◽  
...  
Keyword(s):  
Equilibrium Point ◽  
Lyapunov Exponents ◽  
Phase Portraits ◽  
Circuit Parameter ◽  
Bifurcation Diagrams ◽  
Chaotic Circuit ◽  
Initial State ◽  
Stable Point ◽  
The Stability ◽  
Physical Formula

This paper presents a new physical [Formula: see text] (SBT) memristor-based chaotic circuit. The equilibrium point and the stability of the chaotic circuit are analyzed theoretically. This circuit system exhibits multiple dynamics such as stable point, periodic cycle and chaos by means of Lyapunov exponents spectra, bifurcation diagrams, Poincaré maps and phase portraits, when the initial state or the circuit parameter changes. Specially, the circuit system exhibits coexisting multi-dynamics. This study provides insightful guidance for the design and analysis of physical memristor-based circuits.

Basic Concepts in the Stability Theory of Thin-walled Structures

2010 ◽  
Author(s):  
A. Guran ◽  
L. Lebedev ◽  
Michail D. Todorov ◽  
Christo I. Christov
Keyword(s):  
Stability Theory ◽  
Thin Walled Structures ◽  
Thin Walled ◽  
Basic Concepts ◽  
The Stability

scholarly journals Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

2010 ◽  
Vol 2010 ◽  
pp. 1-23 ◽  
Author(s):  
Josef Diblík ◽  
Denys Ya. Khusainov ◽  
Irina V. Grytsay ◽  
Zdenĕk Šmarda
Keyword(s):  
Lyapunov Functions ◽  
Critical Case ◽  
Autonomous Systems ◽  
Discrete Systems ◽  
Simple Eigenvalue ◽  
Discrete Equations ◽  
The Matrix ◽  
The Stability ◽  
Important Characterization

Many processes are mathematically simulated by systems of discrete equations with quadratic right-hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalueλ=1of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

1976 ◽  
Vol 78 (2) ◽  
pp. 355-383 ◽  
Author(s):  
H. Fasel
Keyword(s):  
Finite Difference ◽  
Stability Theory ◽  
Stokes Equations ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Small Periodic

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

scholarly journals Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Bin Wang ◽  
Yuangui Zhou ◽  
Jianyi Xue ◽  
Delan Zhu
Keyword(s):  
Sliding Mode Control ◽  
Fractional Order ◽  
Wide Class ◽  
Stability Theory ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Order System ◽  
Fractional Order Calculus ◽  
Simulation Results ◽  
The Stability

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

The stability analysis using Lyapunov exponents for high-DOF nonlinear vehicle plane motion

2021 ◽  
pp. 095440702110433
Author(s):  
Shuming Shi ◽  
Fanyu Meng ◽  
Minghui Bai ◽  
Nan Lin
Keyword(s):  
Stability Analysis ◽  
Quantitative Analysis ◽  
Lyapunov Exponents ◽  
Plane Motion ◽  
Vehicle Model ◽  
Motion Stability ◽  
Motion System ◽  
Nonlinear Vehicle Model ◽  
The Stability ◽  
Different Characteristics

The Lyapunov exponents method is an excellent approach for analyzing the vehicle plane motion stability, and the researchers demonstrated the effectiveness under 2-DOF vehicle model. However, whether the Lyapunov exponents approach can effectively reveal the characteristics of high-DOF nonlinear vehicle model is the key problem at present. In this paper, the Lyapunov exponents is applied to quantitatively analyze the stability of the nonlinear three and five degree of freedom vehicle plane motion system. The different characteristics between 2-DOF and high-DOF model are revealed and explained by using Lyapunov exponents. It illustrates the feasibility of using Lyapunov exponents to analyze the stability of high-DOF vehicle models, which supplements and perfects the existing quantitative analysis conclusion.

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