phase flow of a non-uniform oscillator $\dot{\theta} = \mu + \cos{\theta} + \cos{2\theta}$ and possible error in posted solution

Question

I am looking at one of the problems in Strogatz's book on Dynamical Systems and Chaos. The problem is number 4.3.5 on page 117 of the second edition of the book. I am trying to plot the phase flow of the equation as well as find the fixed points.

The problem is to find the phase flow of a non-uniform oscillator for the equation:

$$ \dot{\theta} = \mu + \cos{\theta} + \cos{2\theta} $$

Now in the posted solution from the solution manual, the phase flow seems to be going in the wrong direction. Here is the picture posted in the solution manual.

This is for $\mu = -2$.

Here is the issue. Since the curve for the derivative is less than zero, shouldn't the phase flow go from right to left. But in the image the arrow is pointing from left to right. Similarly the arrow for the phase flow on the circle should go clockwise corresponding to a negative derivative, right? It seems like the arrows are reversed, but I just wanted to make sure I was not missing anything.

Second, I was hoping someone could indicate how to find the fixed points for an equation like this. Should I substitute $\cos{2}\theta$ for something like $2cos^2{\theta} - 1$ and then try to find the roots of the polynomial? But I am not sure if that works with sin and cosines. Or is there a better substitution to make?

Thanks.

Answer 1

Use that $$\cos(x)+\cos(y)=2 \cos \left(\frac{x}{2}-\frac{y}{2}\right) \cos \left(\frac{x}{2}+\frac{y}{2}\right)$$

Answer 2

No solution has yet been posted here. Integration has not so far been done.

It re-states the given problem input simply in another graphical form for sake of clarity (before arriving at a solution) as:

$$ \dot \theta = \cos \theta + \cos 2\theta -2 $$

Plotted values at $ \theta =(0,\pi, 2 \pi) $ are the angular velocities $ \dot \theta = (0, \mu=-2,0) $ respectively.