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.

NOISY CHAOS IN A QUASI-INTEGRABLE HAMILTONIAN SYSTEM WITH TWO DOF UNDER HARMONIC AND BOUNDED NOISE EXCITATIONS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250117 ◽  
Author(s):  
C. B. GAN ◽  
Y. H. WANG ◽  
S. X. YANG ◽  
H. LEI
Keyword(s):  
Hamiltonian System ◽  
Integrable Hamiltonian System ◽  
Melnikov Method ◽  
Threshold Region ◽  
Mean Square ◽  
Bounded Noise ◽  
Onset Of Chaos ◽  
Extended Approach ◽  
The Mean ◽  
Quasi Integrable Hamiltonian System

This paper presents an extended form of the high-dimensional Melnikov method for stochastically quasi-integrable Hamiltonian systems. A quasi-integrable Hamiltonian system with two degree-of-freedom (DOF) is employed to illustrate this extended approach, from which the stochastic Melnikov process is derived in detail when the harmonic and the bounded noise excitations are imposed on the system, and the mean-square criterion on the onset of chaos is then presented. It is shown that the threshold of the onset of chaos can be adjusted by changing the deterministic intensity of bounded noise, and one can find the range of the parameter related to the bandwidth of the bounded noise excitation where the chaotic motion can arise more readily by investigating the changes of the threshold region. Furthermore, some parameters are chosen to simulate the sample responses of the system according to the mean-square criterion from the extended stochastic Melnikov method, and the largest Lyapunov exponents are then calculated to identify these sample responses.

scholarly journals Harmonic oscillator Brownian motion: Langevin approach revisited

2021 ◽  
Vol 18 (1) ◽  
pp. 97
Author(s):  
O. Contreras-Vergara ◽  
N. Lucero-Azuara ◽  
N. Sánchez-Salas ◽  
J. I. Jiménez-Aquino
Keyword(s):  
Harmonic Oscillator ◽  
Free Particle ◽  
Gaussian White Noise ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Brownian Movement ◽  
Deterministic Equation ◽  
Langevin Approach ◽  
The Mean ◽  
Movement Problem

The original strategy applied by Langevin to Brownian movement problem is used to solve the case of a free particle under a harmonic potential. Such straightforward strategy consists in separating the noise termin the Langevin equation in order to solve a deterministic equation associated with the Mean Square Displacement (MSD). In this work, to achieve our goal we first calculate the variance for the stochastic harmonic oscillator and then the MSD appears immediately. We study the problem in the damped and lightly damped cases and show that, for times greater than the relaxation time, Langevin's original strategy is quite consistent with the exact theoretical solutions reported by Chandrasekhar and Lemons, these latter obtained using the statistical properties of a Gaussian white noise. Our results for the MSDs are compared  with the exact theoretical solutions as well as with the numerical simulation.

Free Oscillations of a Damped Simple Pendulum: An Analog Simulation Experiment

2019 ◽  
Vol 01 (04) ◽  
pp. 1950015 ◽  
Author(s):  
Ivan Skhem Sawkmie ◽  
Mangal C. Mahato
Keyword(s):  
Harmonic Oscillator ◽  
Free Oscillation ◽  
Simulation Experiment ◽  
Free Oscillations ◽  
Analog Simulation ◽  
Simple Pendulum ◽  
Electronic Circuitry ◽  
Undergraduate Laboratory ◽  
Angular Amplitude ◽  
Theoretical Results

The frequency of free oscillation of a damped simple pendulum with large amplitude depends on its amplitude unlike the amplitude-independent frequency of oscillation of a damped simple harmonic oscillator. This aspect is not adequately emphasized in the undergraduate courses due to experimental and theoretical difficulties. We propose an analog simulation experiment to study the free oscillations of a simple pendulum that could be performed in an undergraduate laboratory. The needed sinusoidal potential is obtained approximately by using the available AD534 IC by suitably augmenting the electronic circuitry. To keep the circuit simple enough we restrict the initial angular amplitude of the simple pendulum to a maximum of [Formula: see text]. The results compare well qualitatively with the theoretical results. The small quantitative discrepancy is attributed to the inexact nature of the used “sinusoidal potential”.

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.

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