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.

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.

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 Qualitative Analysis of Class of Fractional-Order Chaotic System via Bifurcation and Lyapunov Exponents Notions

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Ndolane Sene
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Chaotic System ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Fractional Chaotic System ◽  
The Stability ◽  
Predictor Corrector

This paper presents a modified chaotic system under the fractional operator with singularity. The aim of the present subject will be to focus on the influence of the new model’s parameters and its fractional order using the bifurcation diagrams and the Lyapunov exponents. The new fractional model will generate chaotic behaviors. The Lyapunov exponents’ theories in fractional context will be used for the characterization of the chaotic behaviors. In a fractional context, the phase portraits will be obtained with a predictor-corrector numerical scheme method. The details of the numerical scheme will be presented in this paper. The numerical scheme will be used to analyze all the properties addressed in this present paper. The Matignon criterion will also play a fundamental role in the local stability of the presented model’s equilibrium points. We will find a threshold under which the stability will be removed and the chaotic and hyperchaotic behaviors will be generated. An adaptative control will be proposed to correct the instability of the equilibrium points of the model. Sensitive to the initial conditions, we will analyze the influence of the initial conditions on our fractional chaotic system. The coexisting attractors will also be provided for illustrations of the influence of the initial conditions.

Hidden Attractor and Multistability in a Novel Memristor-Based System Without Symmetry

2021 ◽  
Vol 31 (11) ◽  
pp. 2150168
Author(s):  
Musha Ji’e ◽  
Dengwei Yan ◽  
Lidan Wang ◽  
Shukai Duan
Keyword(s):  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Control Unit ◽  
Basins Of Attraction ◽  
Phase Portraits ◽  
Dynamic Behaviors ◽  
Potential Applications

Memristor, as a typical nonlinear element, is able to produce chaotic signals in chaotic systems easily. Chaotic systems have potential applications in secure communications, information encryption, and other fields. Therefore, it is of importance to generate abundant dynamic behaviors in a single chaotic system. In this paper, a novel memristor-based chaotic system without equilibrium points is proposed. One of the essential features is the absence of symmetry in this system, which increases the complexity of the new system. Then, the nonlinear dynamic behaviors of the system are analyzed in terms of chaos diagrams, bifurcation diagrams, Poincaré maps, Lyapunov exponent spectra, the sum of Lyapunov exponents, phase portraits, 0–1 test, recurrence analysis and instantaneous phase. The results of the sum of Lyapunov exponents show that the given system is a quasi-Hamiltonian system with certain initial conditions (IC) and parameters. Next, other critical phenomena, such as hidden multi-scroll attractors, abundant coexistence characteristics, are found characterized through basins of attraction and others. Especially, it reveals some rare phenomena in other systems that multiple hidden hyperchaotic attractors coexist. Finally, the circuit implementation based on Micro Control Unit (MCU) confirms theoretical analysis and the numerical simulation.

Detecting Parameter Changes Using Experimental Nonlinear Dynamics and Chaos

1996 ◽  
Vol 118 (3) ◽  
pp. 375-383 ◽  
Author(s):  
R. S. Chancellor ◽  
R. M. Alexander ◽  
S. T. Noah
Keyword(s):  
Nonlinear Dynamics ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Phase Portraits ◽  
Unstable Periodic Orbits ◽  
Bifurcation Diagrams ◽  
Frequency Spectra ◽  
Nonlinear Dynamics And Chaos ◽  
Chaotic Nature ◽  
Parameter Values

A method of detecting parameter changes using analytical and experimental nonlinear dynamics and chaos is applied to a piecewise-linear oscillator. Experimental data show the chaotic nature of the system through phase portraits, Poincare´ maps, frequency spectra and bifurcation diagrams. Unstable periodic orbits were extracted from each chaotic time series obtained from the system with six different parameter values. Movement of the unstable periodic orbits in phase space is used to detect parameter changes in the system.

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.

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.

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.

scholarly journals Parameter-Independent Dynamical Behaviors in Memristor-Based Wien-Bridge Oscillator

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Ning Wang ◽  
Bocheng Bao ◽  
Tao Jiang ◽  
Mo Chen ◽  
Quan Xu
Keyword(s):  
Chaotic Attractor ◽  
Initial Conditions ◽  
System Model ◽  
Phase Portraits ◽  
Transient Chaos ◽  
Chaotic Oscillator ◽  
Initial Condition ◽  
Bifurcation Diagrams ◽  
System Parameters ◽  
Dynamical Behaviors

This paper presents a novel memristor-based Wien-bridge oscillator and investigates its parameter-independent dynamical behaviors. The newly proposed memristive chaotic oscillator is constructed by linearly coupling a nonlinear active filter composed of memristor and capacitor to a Wien-bridge oscillator. For a set of circuit parameters, phase portraits of a double-scroll chaotic attractor are obtained by numerical simulations and then validated by hardware experiments. With a dimensionless system model and the determined system parameters, the initial condition-dependent dynamical behaviors are explored through bifurcation diagrams, Lyapunov exponents, and phase portraits, upon which the coexisting infinitely many attractors and transient chaos related to initial conditions are perfectly offered. These results are well verified by PSIM circuit simulations.

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