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 $F_0\cos ({\omega }_{d}t)$. Thus, the equations of motion are $ml\alpha =-mg\sin (\theta )-\gamma \omega +F_0\cos ({\omega }_{d}t)$. 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 $\theta =1$, $\omega =0$; $\varphi$ 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.