I am looking at exercise 4.4.2 in Strogatz's Nonlinear Dynamics and Chaos book, but I am working through the book as a self-study. One of the questions deals with the equation for an overdamped pendulum in section 4.4 of the book. The equation makes an assumption that the inertia term is negligible, so equation is:
$$ b\dot{\theta} + mgL\sin{\theta} = \Gamma $$
Where $b, m, g, L, \Gamma$ are constants. The equation is further nondimensionalized to form
$$ \theta' = \gamma - \sin{\theta} $$
where $\theta' = \frac{d\theta}{d\tau}$, where $\tau$ is dimensionless time. The actual constants are $\tau = \frac{mgL}{b}t$, and $\gamma = \frac{\Gamma}{mgL}$.
The question that is asked in the book is below.
I understand the equation well enough, but I was not clear on what is being asked for when they want a plot of $\sin{\theta}(t)$ versus $t$. Seems like this is a nonuniform oscillator, so when $\gamma < 1$ then there will be some stable and unstable fixed points, and the time behavior of the system will be fast and then slow progress towards the stable fixed points. If $\gamma = 1$ then you have some half-stable fixed points, so you have progress towards these fixed points over time with slight perturbations sending the system back into oscillation. But I was not sure if there was a more precise way to characterize these time behaviors using the equations? Would I need to reintroduce the dimensional time variable back into the equation to demonstrate the time behavior on the $t$ scale, or is there a better way?