Homoclinic bifurcation and chaos in simple pendulum under bounded noise excitation

2004 ◽  
Vol 20 (3) ◽  
pp. 593-607 ◽  
Author(s):  
Z.H. Liu ◽  
W.Q. Zhu
Keyword(s):  
Homoclinic Bifurcation ◽  
Bounded Noise ◽  
Bifurcation And Chaos ◽  
Simple Pendulum

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.

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.

Action-Minimizing Curves for Tonelli Lagrangians

2017 ◽  
Author(s):  
Alfonso Sorrentino
Keyword(s):  
Critical Value ◽  
The Other ◽  
Invariant Sets ◽  
Simple Pendulum ◽  
New Concepts ◽  
Peierls Barrier ◽  
Average Action

This chapter discusses the notion of action-minimizing orbits. In particular, it defines the other two families of invariant sets, the so-called Aubry and Mañé sets. It explains their main dynamical and symplectic properties, comparing them with the results obtained in the preceding chapter for the Mather sets. The relation between these new invariant sets and the Mather sets is described. As a by-product, the chapter introduces the Mañé's potential, Peierls' barrier, and Mañé's critical value. It discusses their properties thoroughly. In particular, it highlights how this critical value is related to the minimal average action and describes these new concepts in the case of the simple pendulum.

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