Homoclinic bifurcation and chaos in Duffing oscillator driven by an amplitude-modulated force

2007 ◽  
Vol 376 ◽  
pp. 223-236 ◽  
Author(s):  
V. Ravichandran ◽  
V. Chinnathambi ◽  
S. Rajasekar
Keyword(s):  
Duffing Oscillator ◽  
Homoclinic Bifurcation ◽  
Bifurcation And Chaos

scholarly journals Effect of Nonlinear Dissipation on the Basin Boundaries of a Driven Two-Well Modified Rayleigh–Duffing Oscillator

2015 ◽  
Vol 25 (02) ◽  
pp. 1550024 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
J. B. Chabi Orou
Keyword(s):  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Amplitude ◽  
Nonlinear Damping ◽  
Homoclinic Bifurcation ◽  
Nonlinear Dissipation ◽  
Transition To Chaos ◽  
Melnikov Criterion ◽  
Quadratic Nonlinearities ◽  
Basin Boundaries

This paper considers the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh–Duffing oscillator where pure cubic, unpure cubic, pure quadratic and unpure quadratic nonlinearities are considered. By analyzing the potential, an analytic expression is found for the homoclinic orbit. The Melnikov criterion is used to examine a global homoclinic bifurcation and transition to chaos. Unpure quadratic parameter and parametric excitation amplitude effects are found on the critical Melnikov amplitude μ cr . Finally, the phase space of initial conditions is carefully examined in order to analyze the effect of the nonlinear damping, and particularly how the basin boundaries become fractalized.

HOMOCLINIC BIFURCATION AND CHAOS IN COUPLED SIMPLE PENDULUM AND HARMONIC OSCILLATOR UNDER BOUNDED NOISE EXCITATION

2005 ◽  
Vol 15 (01) ◽  
pp. 233-243 ◽  
Author(s):  
W. Q. ZHU ◽  
Z. H. LIU
Keyword(s):  
Harmonic Oscillator ◽  
Random Motion ◽  
Homoclinic Bifurcation ◽  
Mean Square ◽  
Bounded Noise ◽  
Bifurcation And Chaos ◽  
Simple Pendulum ◽  
Onset Of Chaos ◽  
Weakly Coupled ◽  
Threshold Amplitude

The homoclinic bifurcation and chaos in a system of weakly coupled simple pendulum and harmonic oscillator subject to light dampings and weakly external and (or) parametric excitation of bounded noise is studied. The random Melnikov process is derived and mean-square criteria is used to determine the threshold amplitude of the bounded noise for the onset of chaos in the system. The threshold amplitude is also determined by vanishing the numerically calculated maximal Lyapunov exponent. The threshold amplitudes are further confirmed by using the Poincaré maps, which indicate the path from periodic motion to chaos or from random motion to random chaos in the system as the amplitude of bounded noise increases.

scholarly journals Bifurcation and Chaos Prediction in Nonlinear Gear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Anooshirvan Farshidianfar ◽  
Amin Saghafi
Keyword(s):  
Numerical Simulation ◽  
Phase Plane ◽  
Homoclinic Bifurcation ◽  
Threshold Values ◽  
Bifurcation And Chaos ◽  
System Parameters ◽  
Transition To Chaos ◽  
Time Histories ◽  
Fourier Spectra ◽  
Gear Systems

The homoclinic bifurcation and transition to chaos in gear systems are studied both analytically and numerically. Applying Melnikov analytical method, the threshold values for the occurrence of chaotic motion are obtained. The influence of system parameters on the character of vibration is studied. The numerical simulation of the system including bifurcation diagram, phase plane portraits, Fourier spectra, and time histories is considered to confirm the analytical predictions for the occurrence of homoclinic bifurcation and chaos in nonlinear gear systems.

scholarly journals The Focus Case of a Nonsmooth Rayleigh-Duffing Oscillator

2021 ◽  
Author(s):  
Zhaoxia Wang ◽  
Hebai Chen ◽  
Yilei Tang
Keyword(s):  
Limit Cycle ◽  
Duffing Oscillator ◽  
Pitchfork Bifurcation ◽  
Homoclinic Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Bifurcation Curve ◽  
Limit Cycle Bifurcation ◽  
Theoretical Results ◽  
Double Limit

Abstract In this paper, we study the global dynamics of a nonsmooth Rayleigh-Duffing equation x¨ + ax˙ + bx˙|x˙| + cx + dx3 = 0 for the case d > 0, i.e., the focus case. The global dynamics of this nonsmooth Rayleigh-Duffing oscillator for the case d < 0, i.e., the saddle case, has been studied completely in the companion volume [Int. J. Non-Linear Mech., 129 (2021) 103657]. The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent occurs. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

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.

HOMOCLINIC BIFURCATION AND CHAOS IN Φ6-RAYLEIGH OSCILLATOR WITH THREE WELLS DRIVEN BY AN AMPLITUDE MODULATED FORCE

2011 ◽  
Vol 21 (06) ◽  
pp. 1583-1593 ◽  
Author(s):  
M. SIEWE SIEWE ◽  
C. TCHAWOUA ◽  
S. RAJASEKAR
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Diagram ◽  
Poincaré Map ◽  
Chaotic Behavior ◽  
Homoclinic Bifurcation ◽  
Maximal Lyapunov Exponent ◽  
Poincare Map ◽  
Bifurcation And Chaos

With amplitude modulated excitation, the effect on chaotic behavior of Φ6-Rayleigh oscillator with three wells is investigated in this paper. The Melnikov theorem is used to detect the conditions for possible occurrence of chaos. The results show that the domain of the appearance of chaos is enlarged as both amplitudes of modulated and unmodulated forces increase. The effect of these two amplitudes, when both frequencies of modulated and unmodulated forces are different, on bifurcation diagram and Poincaré map is also investigated, in addition to the surface of Maximal Lyapunov exponent versus modulated and unmodulated parameters for suppressing chaos being shown.

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