scholarly journals Chaotic Motion in Forced Duffing System Subject to Linear and Nonlinear Damping

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Tai-Ping Chang
Keyword(s):  
Lyapunov Exponent ◽  
Nonlinear System ◽  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Nonlinear Damping ◽  
Critical Value ◽  
Duffing System ◽  
Linear And Nonlinear

This paper investigates the chaotic motion in forced Duffing oscillator due to linear and nonlinear damping by using Melnikov technique. In particular, the critical value of the forcing amplitude of the nonlinear system is calculated by Melnikov technique. Further, the top Lyapunov exponent of the nonlinear system is evaluated by Wolf’s algorithm to determine whether the chaotic phenomenon of the nonlinear system actually occurs. It is concluded that the chaotic motion of the nonlinear system occurs when the forcing amplitude exceeds the critical value, and the linear and nonlinear damping can generate pronounced effects on the chaotic behavior of the forced Duffing oscillator.

Three Conditions Lyapunov Exponents Should Satisfy

2003 ◽  
Author(s):  
Jingjun Lou ◽  
Shijian Zhu
Keyword(s):  
Dynamical System ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Chaotic Motion ◽  
The Other ◽  
Cubic Nonlinearity ◽  
3 Dimensional ◽  
Duffing System ◽  
Dissipative Dynamical System

In contrast to the unilateral claim in some papers that a positive Lyapunov exponent means chaos, it was claimed in this paper that this is just one of the three conditions that Lyapunov exponent should satisfy in a dissipative dynamical system when the chaotic motion appears. The other two conditions, any continuous dynamical system without a fixed point has at least one zero exponent, and any dissipative dynamical system has at least one negative exponent and the sum of all of the 1-dimensional Lyapunov exponents id negative, are also discussed. In order to verify the conclusion, a MATLAB scheme was developed for the computation of the 1-dimensional and 3-dimensional Lyapunov exponents of the Duffing system with square and cubic nonlinearity.

scholarly journals Identification and Machine Learning Prediction of Nonlinear Behavior in a Robotic Arm System

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1445
Author(s):  
Cheng-Chi Wang ◽  
Yong-Quan Zhu
Keyword(s):  
Neural Network ◽  
Lyapunov Exponent ◽  
Chaotic Motion ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Robotic Arm ◽  
Maximum Lyapunov Exponent ◽  
Double Pendulum ◽  
Chaotic Motions ◽  
Robotic Arms

In this study, the subject of investigation was the dynamic double pendulum crank mechanism used in a robotic arm. The arm is driven by a DC motor though the crank system and connected to a fixed side with a mount that includes a single spring and damping. Robotic arms are now widely used in industry, and the requirements for accuracy are stringent. There are many factors that can cause the induction of nonlinear or asymmetric behavior and even excite chaotic motion. In this study, bifurcation diagrams were used to analyze the dynamic response, including stable symmetric orbits and periodic and chaotic motions of the system under different damping and stiffness parameters. Behavior under different parameters was analyzed and verified by phase portraits, the maximum Lyapunov exponent, and Poincaré mapping. Firstly, to distinguish instability in the system, phase portraits and Poincaré maps were used for the identification of individual images, and the maximum Lyapunov exponents were used for prediction. GoogLeNet and ResNet-50 were used for image identification, and the results were compared using a convolutional neural network (CNN). This widens the convolutional layer and expands pooling to reduce network training time and thickening of the image; this deepens the network and strengthens performance. Secondly, the maximum Lyapunov exponent was used as the key index for the indication of chaos. Gaussian process regression (GPR) and the back propagation neural network (BPNN) were used with different amounts of data to quickly predict the maximum Lyapunov exponent under different parameters. The main finding of this study was that chaotic behavior occurs in the robotic arm system and can be more efficiently identified by ResNet-50 than by GoogLeNet; this was especially true for Poincaré map diagnosis. The results of GPR and BPNN model training on the three types of data show that GPR had a smaller error value, and the GPR-21 × 21 model was similar to the BPNN-51 × 51 model in terms of error and determination coefficient, showing that GPR prediction was better than that of BPNN. The results of this study allow the formation of a highly accurate prediction and identification model system for nonlinear and chaotic motion in robotic arms.

scholarly journals Vibration of the Duffing Oscillator: Effect of Fractional Damping

2007 ◽  
Vol 14 (1) ◽  
pp. 29-36 ◽  
Author(s):  
Marek Borowiec ◽  
Grzegorz Litak ◽  
Arkadiusz Syta
Keyword(s):  
Lyapunov Exponent ◽  
Perturbation Methods ◽  
Duffing Oscillator ◽  
Homoclinic Bifurcation ◽  
Sufficient Condition ◽  
External Excitation ◽  
Fractional Damping ◽  
Duffing System ◽  
Transition To Chaos ◽  
Melnikov Criterion

We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of the Duffing system with nonlinear fractional damping and external excitation. Using perturbation methods we have found a critical forcing amplitude above which the system may behave chaotically. The results have been verified by numerical simulations using standard nonlinear tools as Poincare maps and a Lyapunov exponent. Above the critical Melnikov amplitude μ_c, which a sufficient condition of a global homoclinic bifurcation, we have observed the region with a transient chaotic motion.

scholarly journals Confusion threshold study of the Duffing oscillator with a nonlinear fractional damping term

2020 ◽  
pp. 146134842092268
Author(s):  
Wang Mei-Qi ◽  
Ma Wen-Li ◽  
Chen En-Li ◽  
Yang Shao-Pu ◽  
Chang Yu-Jian ◽  
...  
Keyword(s):  
Chaotic Motion ◽  
Numerical Solutions ◽  
Duffing Oscillator ◽  
Time History ◽  
Nonlinear Damping ◽  
Critical Conditions ◽  
Fractional Damping ◽  
Nonlinear Parameters ◽  
Smale Horseshoe ◽  
Melnikov Theory

In this study, the critical conditions for generating chaos in a Duffing oscillator with nonlinear damping and fractional derivative are investigated. The Melnikov function of the Duffing oscillator is established based on Melnikov theory. The necessary analytical conditions and critical value curves of chaotic motion in the sense of Smale horseshoe are obtained. The numerical solutions of chaotic motion, including time history diagram, frequency spectrum diagram, phase diagram, and Poincare map, are studied. The correctness of the analytical solution is verified through a comparison of numerical and analytical calculations. The effects of linear and nonlinear parameters on chaotic motion are also analyzed. These results are relevant to the study of system dynamics.

scholarly journals Dynamic Analysis of Modified Duffing System via Intermittent External Force and Its Application

2019 ◽  
Vol 9 (21) ◽  
pp. 4683 ◽  
Author(s):  
Jianbin He ◽  
Jianping Cai
Keyword(s):  
Lyapunov Exponent ◽  
External Force ◽  
Control Method ◽  
Duffing Oscillator ◽  
Pseudorandom Sequence ◽  
Past Century ◽  
Continuous Control ◽  
Duffing System ◽  
Numerical Solution Methods ◽  
Solution Methods

Over the past century, a tremendous amount of work on the Duffing system has been done with continuous external force, including analytical and numerical solution methods, and the dynamic behavior of physical systems. However, hows does the Duffing oscillator behave if the external force is intermittent? This paper investigates the Duffing oscillator with intermittent external force, and a modified Duffing chaotic system is proposed. Different from the continuous-control method, an intermittent external force of cosine function was designed to control the Duffing oscillator, such that the modified Duffing (MD) system could behave chaotically. The dynamic characteristics of MD system, such as the strange attractors, Lyapunov exponent spectra, and bifurcation diagram spectra were outlined with numerical simulations. Numerical results showed that there existed a positive Lyapunov exponent in some parameter intervals. Furthermore, by combining it with chaos scrambling and chaos XOR encryption, a chaos-based encryption algorithm was designed via the pseudorandom sequence generated from the MD. Finally, feasibility and validity were verified by simulation experiments of image encryption.

QUANTIFYING SMOOTH AND NONSMOOTH REGULAR AND CHAOTIC DYNAMICS

2005 ◽  
Vol 15 (06) ◽  
pp. 2041-2055 ◽  
Author(s):  
J. AWREJCEWICZ ◽  
L. DZYUBAK
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Numerical Method ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Stick Slip ◽  
Excited System ◽  
Two Degree Of Freedom

This paper addresses two main paths of investigations. First, a new numerical method to trace regular and chaotic domains of any nonlinear system governed by ordinary differential equations is proposed. Second, the introduced approach is first testified using the well-known chaotic behavior of a Duffing oscillator and Lorenz system, and is then applied to analysis of discontinuous two-degree-of-freedom self-excited system with friction. Stick-slip and slip-slip chaos is reported, among others.

Effect of Different Forms of Periodic Piecewise Linear Forces on Duffing Oscillator

2021 ◽  
Vol 13 (2) ◽  
pp. 361-375
Author(s):  
S. V. Priyatharsini ◽  
B. Bhuvaneshwari ◽  
V. Chinnathambi ◽  
S. Rajasekar
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponent ◽  
Piecewise Linear ◽  
Duffing Oscillator ◽  
Control Force ◽  
Maximal Lyapunov Exponent ◽  
Simulation Tools ◽  
Duffing System ◽  
Parametric Control ◽  
Applied Forces

The paper highlights the effect of different forms of   periodic   piecewise linear forces in the ubiquitous Duffing oscillator equation. The external periodic piecewise linear forces considered are Triangular, Hat, Trapezium, Quadratic and Rectangular. With the aid of some numerical simulation tools such as bifurcation diagram, phase portrait and Poincare´ map, the different routes to chaos and various strange attractors are found to occur due to the applied forces. The effect of an ε-parametric control force in the Duffing system is also analyzed. To characterize the regular and chaotic behaviours of this system, the maximal Lyapunov exponent is employed.

scholarly journals Melnikov Chaos in a Modified Rayleigh–Duffing Oscillator with ϕ6 Potential

2016 ◽  
Vol 26 (05) ◽  
pp. 1650085 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
L. A. Hinvi ◽  
A. A. Koukpemedji ◽  
C. Ainamon ◽  
...  
Keyword(s):  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Basin Of Attraction ◽  
Melnikov Method ◽  
Nonlinear Damping ◽  
External Excitation ◽  
Melnikov Criterion ◽  
Single Well ◽  
Quadratic Damping ◽  
The Basin Of Attraction

The chaotic behavior of the modified Rayleigh–Duffing oscillator with [Formula: see text] potential and external excitation is investigated both analytically and numerically. The so-called oscillator models, for example, ship rolling motions. The single well and triple well potential cases are considered. Melnikov method is applied and the conditions for the existence of homoclinic and heteroclinic chaos are obtained. The effects of nonlinear damping on roll motion of ships are analyzed in detail. As it is known, nonlinear roll damping is a very important parameter in estimating ship responses. It is noted that the pure and unpure quadratic damping parameters affect the Melnikov criterion in the heteroclinic and homoclinic cases respectively while the pure cubic parameter affects the amplitude in both cases. The predictions have been tested with numerical simulations based on the basin of attraction. It is pointed out that certain quadratic damping effects are contrary to cubic damping effect.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

2020 ◽  
Vol 22 (4) ◽  
pp. 983-990
Author(s):  
Konrad Mnich
Keyword(s):  
Dynamical System ◽  
Duffing Oscillator ◽  
Probabilistic Approach ◽  
Nonlinear Dynamical System ◽  
Tuned Mass Damper ◽  
Nonlinear Dynamical ◽  
Duffing System ◽  
Mass Damper ◽  
Basin Stability ◽  
Stability Concept

AbstractIn this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

Non-Linearity and Chaotic Behaviour in Cyprus Stock Market

2015 ◽  
Vol 5 (4) ◽  
Author(s):  
Athina Bougioukou
Keyword(s):  
Lyapunov Exponent ◽  
Stock Market ◽  
Chaos Theory ◽  
Linear Dependence ◽  
Chaotic Behavior ◽  
Efficient Market Hypothesis ◽  
Chaotic Behaviour ◽  
Maximum Lyapunov Exponent ◽  
Efficient Market ◽  
General Index

The intention of this research is to investigate the aspect of non-linearity and chaotic behavior of the Cyprus stock market. For this purpose, we use non-linearity and chaos theory. We perform BDS, Hinich-Bispectral tests and compute Lyapunov exponent of the Cyprus General index. The results show that existence of non-linear dependence and chaotic features as the maximum Lyapunov exponent was found to be positive. This study is important because chaos and efficient market hypothesis are mutually exclusive aspects. The efficient market hypothesis which requires returns to be independent and identically distributed (i.i.d.) cannot be accepted.

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