README.md

Damped, Oscillatorily Driven Pendulum Motion

A ball of mass $m$ is attached to a stiff, but massless rod of length $l$. Gravity acts on it, providing a force $-mg\sin(\theta)$ perpendicular to the rod. The pendulum is also damped by a force $-\gamma\omega$ and driven by an oscillatory force $F_0\cos(\omega_dt)$. Thus, the equations of motion are $ml\alpha = -mg\sin(\theta) - \gamma\omega + F_0\cos(\omega_dt)$. The constants used are $l = g = m = 1$, $\gamma = 0.05$, $F_0 = 0.7$, and $\omega_d = 0.7$. Poincare plots and phase plots of this system are respectively shown below.

Poincare plots of three initial conditions. Blue: $\theta = 0$, $\omega = 1$. Red: $\theta = 0.6$, $\omega = 0.8$. Purple: $\theta = 1$, $\omega = 0$. (above)

Modded phase plot of initial condition $\theta = 1$, $\omega = 0$; $\phi$ is time $t$ modulo $2\pi/\omega_d$. (above)

Phase plots over time for the three initial conditions. Blue: $\theta = 0$, $\omega = 1$. Red: $\theta = 0.6$, $\omega = 0.8$. Purple: $\theta = 1$, $\omega = 0$. (above)

Phase plots over time for the three initial conditions. Blue: $\theta = 0$, $\omega = 0.9999$. Red: $\theta = 0$, $\omega = 1.0001$. Purple: $\theta = 0$, $\omega = 1$. (above)

As seen, close initial conditions can diverge after a short amount of time. However, as the paths do not interesect, the motion is deterministic.