The same driven damped pendulum

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos · Nonlinear dynamics and chaos

Computational Physics, Ch When trajectories separate: the driven pendulum and the logistic map

Problem

The same driven damped pendulum,

\ddot\theta+\gamma\dot\theta+\omega_0^2\sin\theta=A_d\cos(\omega_d t)

with

\omega_0=1\,\mathrm{s^{-1}}

\gamma=0.5\,\mathrm{s^{-1}}

\omega_d=2/3\,\mathrm{s^{-1}}

, is now driven at

A_d=1.07\,\mathrm{s^{-2}}

. Air resistance beyond the stated linear damping is ignored. (a) The pendulum is released from

\theta=0.2\,\mathrm{rad}

at rest. After a transient of

100

drive periods is discarded, the state is sampled once per drive period,

n=101,\dots,106

, giving

(\theta,\dot\theta)

in rad and rad s

^{-1}

, with the angle wrapped to

(-\pi,\pi]

n=101

(-2.4770,1.1494)

;

n=102

(-2.2077,1.3461)

;

n=103

(-2.4770,1.1494)

;

n=104

(-2.2077,1.3461)

;

n=105

(-2.4770,1.1494)

;

n=106

(-2.2077,1.3461)

. Describe the pattern in this Poincare section and state the period of the underlying motion as a multiple of the drive period

T_d=2\pi/\omega_d

. (b) Give the period of the motion in seconds, and describe the shape traced out in the

(\theta,\dot\theta)

phase portrait once the transient is discarded.