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.

Multiscroll Chaotic Sea Obtained from a Simple 3D System Without Equilibrium

2016 ◽  
Vol 26 (02) ◽  
pp. 1650031 ◽  
Author(s):  
Sajad Jafari ◽  
Viet-Thanh Pham ◽  
Tomasz Kapitaniak
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Chaotic Systems ◽  
Poincaré Map ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Equilibrium System ◽  
Frequency Spectra ◽  
Chaotic Flows ◽  
New System

Recently, many rare chaotic systems have been found including chaotic systems with no equilibria. However, it is surprising that such a system can exhibit multiscroll chaotic sea. In this paper, a novel no-equilibrium system with multiscroll hidden chaotic sea is introduced. Besides having multiscroll chaotic sea, this system has two more interesting properties. Firstly, it is conservative (which is a rare feature in three-dimensional chaotic flows) but not Hamiltonian. Secondly, it has a coexisting set of nested tori. There is a hidden torus which coexists with the chaotic sea. This new system is investigated through numerical simulations such as phase portraits, Lyapunov exponents, Poincaré map, and frequency spectra. Furthermore, the feasibility of such a system is verified through circuital implementation.

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.

Chaotic Dynamics in Duffing System with Two External Forcings

2011 ◽  
Vol 216 ◽  
pp. 777-781
Author(s):  
Shu Guang Zhang ◽  
Zhi Yong Zhu ◽  
Zhi Guo
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Homoclinic Bifurcation ◽  
External Forcing ◽  
Bifurcation Diagrams ◽  
Duffing System ◽  
External Forcings ◽  
Maximum Lyapunov Exponents

Duffing system with two external forcing terms is investigated in detail. The criterion of existence of chaos under the periodic perturbation is given by using Melnikov's method. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, maximum lyapunov exponents and Poincare map are given to illustrate the theoretical analysis.

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.

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.

scholarly journals A Meminductor-based Chaotic System

2020 ◽  
Vol 49 (2) ◽  
pp. 317-332
Author(s):  
Aixue Qi ◽  
Lei Ding ◽  
Wenbo Liu
Keyword(s):  
Theoretical Analysis ◽  
Chaotic System ◽  
Phase Trajectory ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Circuit Parameter ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Simulation Results

We propose a meminductor-based chaotic system. Theoretical analysis and numerical simulations reveal complex dynamical behaviors of the proposed meminductor-based chaotic system with five unstable equilibrium points and three different states of chaotic attractors in its phase trajectory with only a single change in circuit parameter. Lyapunov exponents, bifurcation diagrams, and phase portraits are used to investigate its complex chaotic and multi-stability behaviors, including its coexisting chaotic, periodic and point attractors. The proposed meminductor-based chaotic system was implemented using analog integrators, inverters, summers, and multipliers. PSPICE simulation results verified different chaotic characteristics of the proposed circuit with a single change in a resistor value.

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.

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.

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