The same driven damped pendulum

PHYSICS · Multi-part · 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=1.2\,\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 and integrated by fourth-order Runge-Kutta with

942

steps to the drive period,

\Delta t=0.0100\,\mathrm{s}

. After discarding a transient of

100

drive periods, the state is sampled once per drive period for ten further periods, with the angle wrapped to

(-\pi,\pi]

, giving

\theta

(rad):

1.161

-0.954

-0.116

0.429

-2.512

1.902

0.244

2.775

0.258

1.154

, with no two values repeating and no obvious cycle among them. These ten numbers belong to that one run: a hundred drive periods is

942\,\mathrm{s}

, far beyond this pendulum's predictability horizon, so a different step or a different starting angle gives ten different numbers, and it is their pattern rather than their values that the question is about. Repeating this for

10^5

drive periods gives

10^5

points that never coincide but stay confined to a bounded region of the

(\theta,\dot\theta)

plane. What do these sampled points show about the periodicity of the motion? ((b)) Describe the shape traced out in the

(\theta,\dot\theta)

phase portrait once the transient is discarded, and contrast it with the phase portraits of period-1 and period-2 motion.

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