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.

CHAOS IN PIECEWISELY INTEGRABLE TRAIN MODEL FOR TWO BLOCKS

2002 ◽  
Vol 13 (01) ◽  
pp. 41-48 ◽  
Author(s):  
J. SZKUTNIK ◽  
K. KUŁAKOWSKI
Keyword(s):  
Lyapunov Exponents ◽  
Friction Force ◽  
Analytical Solutions ◽  
Equations Of Motion ◽  
Power Spectra ◽  
Phase Portraits ◽  
Velocity Dependence ◽  
Bifurcation Diagrams ◽  
Stick Slip ◽  
Block System

The train model of two blocks with stick-slip dynamics (M. de Sousa Vieira, 1995) is believed to be the simplest spring-block system, which displays chaos. Here we simplify it even more by linearizing the velocity dependence of the friction force. In this way, the nonlinearity of the equations of motion is reduced to the time moments, when a block starts to move or stops, and when the analytical solutions are to be sewn together. We demonstrate, that for small values of the velocity of blocks, the character of motion is not changed. This is observed on the bifurcation diagrams, the Lyapunov exponents, the phase portraits and the power spectra.

Nonlinear Dynamics of a Pendulum Excited by a Crank-Shaft-Slider Mechanism

2014 ◽  
Author(s):  
Rafael H. Avanço ◽  
Hélio A. Navarro ◽  
Reyolando M. L. R. F. Brasil ◽  
José M. Balthazar
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponents ◽  
Nonlinear Model ◽  
Special Interest ◽  
Initial Conditions ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Numerical Techniques ◽  
Vertical Excitation ◽  
Time Histories

In this analysis, we consider the dynamics of a pendulum under vertical excitation of a crank-shaft-slider mechanism. The nonlinear model approaches that of a classical parametrically excited pendulum when the ratio of the length of the shaft to the radius of the crank is very large. Numerical techniques are employed to investigate the results for different parameters and initial conditions. Lyapunov exponents, bifurcation diagrams, time histories and phase portraits are presented to explore conditions when the pendulum performs or not full rotations. Of special interest are the resonance regions. Rotations together with oscillations and chaos were observed in some resonance zones.

CHUA SYSTEMS WITH DISCONTINUITIES

1999 ◽  
Vol 09 (04) ◽  
pp. 591-616 ◽  
Author(s):  
C.-H. LAMARQUE ◽  
O. JANIN ◽  
J. AWREJCEWICZ
Keyword(s):  
Lyapunov Exponents ◽  
Existence And Uniqueness ◽  
Special Class ◽  
Equilibrium Solution ◽  
Phase Portraits ◽  
Global Behavior ◽  
Bifurcation Diagrams ◽  
Trivial Equilibrium ◽  
Analytical Expressions ◽  
Special Case

We present a special class of mechanical systems that could be written as Chua circuits with discontinuities. We recall the general frame for the study of such models. Results of existence and uniqueness are given. Then numerical results obtained via piecewise analytical expressions are presented. We discuss some bifurcation diagrams, phase portraits. Chaos is characterized by computing Lyapunov exponents. We analyze the global behavior in a special case where discontinuity stabilizes the trivial equilibrium solution.

On the Dynamics and Control of a Fractional Form of the Discrete Double Scroll

2019 ◽  
Vol 29 (06) ◽  
pp. 1950078 ◽  
Author(s):  
Adel Ouannas ◽  
Amina-Aicha Khennaoui ◽  
Samir Bendoukha ◽  
Giuseppe Grassi
Keyword(s):  
Fractional Order ◽  
Numerical Results ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Complete Synchronization ◽  
General Behavior ◽  
Control Schemes ◽  
Dynamics And Control ◽  
And Control

This paper is concerned with the dynamics and control of the fractional version of the discrete double scroll hyperchaotic map. Using phase portraits and bifurcation diagrams, we show that the general behavior of the proposed map depends on the fractional order. We also present two control schemes for the proposed map, one that adaptively stabilizes the map, and another to achieve the complete synchronization of a pair of maps. Numerical results are presented to illustrate the findings.

scholarly journals Numerical Analysis of Nonlinear Dynamics Based on Spin-VCSELs with Optical Feedback

Photonics ◽  
2021 ◽  
Vol 8 (1) ◽  
pp. 10
Author(s):  
Tingting Song ◽  
Yiyuan Xie ◽  
Yichen Ye ◽  
Bocheng Liu ◽  
Junxiong Chai ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Optical Feedback ◽  
External Disturbance ◽  
Power Spectra ◽  
Phase Portraits ◽  
Cavity Surface ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Spin Flip ◽  
Emitting Lasers

In this paper, the nonlinear dynamics of a novel model based on optically pumped spin-polarized vertical-cavity surface-emitting lasers (spin-VCSELs) with optical feedback is investigated numerically. Due to optical feedback being the external disturbance component, the complex nonlinear dynamical behaviors can be enhanced and the regions of different nonlinear dynamics in size can be extended with appropriate parameters of spin-VCSELs. According to the equations of the modified spin-flip model (SFM), the comparison of bifurcation diagrams is first presented for the clear presentation of different routes to chaos. Meanwhile, numerous bifurcation diagrams in color are illustrated to demonstrate the rich dynamical regimes intuitively, and the crucial effects of optical feedback strength, feedback delay, linewidth enhancement factor, and spin-flip relaxation rate on the region evolvement of complex dynamics of the proposed model are revealed to investigate the dependence of dynamical behaviors on external and internal parameters when the optical feedback scheme is introduced. These parameters play a remarkable role in enhancing the mechanism of complex dynamic oscillations. Furthermore, utilizing combination with time series, power spectra, and phase portraits, the various dynamical behaviors observed in the bifurcation diagram are simulated numerically. Correspondingly, the powerful measure 0–1 test is employed to distinguish between chaos and non-chaos.

Conservative Chaos in a Class of Nonconservative Systems: Theoretical Analysis and Numerical Demonstrations

2018 ◽  
Vol 28 (07) ◽  
pp. 1850087 ◽  
Author(s):  
Shijian Cang ◽  
Aiguo Wu ◽  
Ruiye Zhang ◽  
Zenghui Wang ◽  
Zengqiang Chen
Keyword(s):  
Theoretical Analysis ◽  
Lyapunov Exponents ◽  
Time Reversal ◽  
Dynamical Evolution ◽  
Phase Portraits ◽  
Time Reversal Symmetry ◽  
Bifurcation Diagrams ◽  
Continuous Curve ◽  
Nonconservative Systems ◽  
Chaotic Flows

This paper proposes a class of nonlinear systems and presents one example system to illustrate its interesting dynamics, including quasiperiodic motion and chaos. It is found that the example system is a subsystem of a non-Hamiltonian system, which has a continuous curve of equilibria with time-reversal symmetry. In this study, the dynamical evolution of the example system with three different kinds of external excitations are fully investigated by using general chaotic analysis methods such as Poincaré sections, phase portraits, Lyapunov exponents and bifurcation diagrams. Both theoretical analysis and numerical simulations show that the example system is nonconservative but has conservative chaotic flows, which are numerically verified by the sum of its Lyapunov exponents. It is also found that the example system has time-reversal symmetry.

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.

scholarly journals A Nonlinear Five-Term System: Symmetry, Chaos, and Prediction

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 865
Author(s):  
Vo Phu Thoai ◽  
Maryam Shahriari Kahkeshi ◽  
Van Van Huynh ◽  
Adel Ouannas ◽  
Viet-Thanh Pham
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Lyapunov Exponents ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Phase Portraits ◽  
Learning Approach ◽  
Bifurcation Diagrams ◽  
Machine Learning Approach ◽  
Simple Systems

Chaotic systems have attracted considerable attention and been applied in various applications. Investigating simple systems and counterexamples with chaotic behaviors is still an important topic. The purpose of this work was to study a simple symmetrical system including only five nonlinear terms. We discovered the system’s rich behavior such as chaos through phase portraits, bifurcation diagrams, Lyapunov exponents, and entropy. Interestingly, multi-stability was observed when changing system’s initial conditions. Chaos of such a system was predicted by applying a machine learning approach based on a neural network.

scholarly journals Chaotic Vibrations of Nonlinearly Supported Tubes in Crossflow

1993 ◽  
Vol 115 (2) ◽  
pp. 128-134 ◽  
Author(s):  
Y. Cai ◽  
S. S. Chen
Keyword(s):  
Mathematical Model ◽  
Instability Region ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Support Plate ◽  
Chaotic Vibrations ◽  
Power Spectral ◽  
Fluid Damping ◽  
Inactive Mode

Chaotic vibrations associated with the fluidelastic instability of nonlinearly supported tubes in a crossflow is studied theoretically on the basis of the unsteady-flow theory and a bilinear mathematical model. Effective tools, including phase portraits, power spectral density, Poincare maps, Lyapunov exponent, fractal dimension, and bifurcation diagrams, are utilized to distinguish periodic and chaotic motions when the tubes vibrate in the instability region. The results show periodic and chaotic motions in the region corresponding to fluid-damping-controlled instability. Nonlinear supports, with symmetric or asymmetric gaps, significantly affect the distribution of periodic, quasi-periodic, and chaotic motions of a tube exposed to various flow velocities in the instability region of the tube-support-plate-inactive mode.

scholarly journals Coexisting infinitely many attractors in a new chaotic system with a curve of equilibria: Its extreme multi-stability and Kolmogorov–Sinai entropy computation

2019 ◽  
Vol 11 (11) ◽  
pp. 168781401988804
Author(s):  
Atefeh Ahmadi ◽  
Xiong Wang ◽  
Fahimeh Nazarimehr ◽  
Fawaz E Alsaadi ◽  
Fuad E Alsaadi ◽  
...  
Keyword(s):  
Dynamical System ◽  
Mathematical Model ◽  
Lyapunov Exponent ◽  
Chaotic System ◽  
Phase Portraits ◽  
Initial Condition ◽  
Bifurcation Diagrams ◽  
Hidden Attractors ◽  
New Chaotic System ◽  
The Mathematical Model

A new five-dimensional chaotic system with extreme multi-stability is introduced in this article. The mathematical model is established, and numerical simulations are done. This dynamical system complicates incident of extreme multi-stability. Most significantly, relied on the mathematical model, the recently proposed system has a curve of equilibria that ends in the occurrence of hidden attractors. We examine the initial-condition-dependent dynamics of this system. We inspect that there is an unrestricted number of coexistent attractors, which signifies the occurrence of extreme multi-stability strictly. In addition, the extreme multi-stability according to initial condition is investigated consuming the Lyapunov exponent spectra and bifurcation diagrams. The existence of coexisting infinitely many attractors is displayed with phase portraits. In the end, we calculate and debate Kolmogorov–Sinai entropy in the chaotic system. We direct trying the Kolmogorov–Sinai technique of entropic inspection for the dynamics of the system.

Sign in / Sign up

Export Citation Format

Share Document