The driven damped pendulum obeys + + _0^2 =A_d ( _d t) where is the angle from the…

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 driven damped pendulum obeys

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

where

\theta

is the angle from the vertical in radians,

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

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

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

and

A_d=0.9\,\mathrm{s^{-2}}

. Air resistance beyond the stated linear damping is ignored. (a) Write this second-order equation as a first-order system in the three variables

\theta

\omega\equiv\dot\theta

and

t

, giving each derivative. (b) The drive period is

T_d=2\pi/\omega_d

. The pendulum is released from

\theta=0.2\,\mathrm{rad}

at rest; after the first

100

drive periods are discarded as transient, the state is sampled once every drive period (a Poincare section) for six further periods, with the angle wrapped to

(-\pi,\pi]

. All six samples come out at the same

(\theta,\omega)

, in rad and rad s

^{-1}

(-0.5602,1.9336)

, unchanged to the four decimals quoted. State what these six sampled points reveal about the periodicity of the motion, give the numerical value of

T_d

, and describe the shape traced out in the

(\theta,\dot\theta)

phase portrait once the transient has been discarded.

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