find the fixed points of $\dot{\theta} = \omega - a\sin{\theta}$ for $a > \omega $

Question

I am working through an example in Strogatz's dynamical systems book (second edition, example 4.3.1, page 99). and came across something in one of his examples. The problem is to find the fixed points in the flow of a system given below and then find the stability at both of the fixed points.

The equation is given by:

$$ \dot{\theta} = \omega - a\sin{\theta} \ $$

Now to find the fixed points in the flow I would set $\dot{\theta} = 0$, and solve for $\theta$, so:

$$ 0 = \omega - a\sin{\theta} \\ sin{\theta}^* = \frac{\omega}{a}\\ $$

But the book also gives an additional solution:

$$ \cos{\theta}^* = \pm \sqrt{1 - (\omega/a)^2} $$

The question is how did Strogatz arrive at this second solution for $\cos{\theta}^*$? From the plot of the system it is obvious that there are 2 fixed points in the domain of the equation, but I was not sure how he arrived at this second fixed point. Any suggestions would help. Thanks.

Answer 1

The fixed points are given by the condition $$ \sin \theta^* = \omega/a , $$ nothing else. (And this equation has two solution per period of the sine function, if $\omega<a$.)

The second formula, $$ \cos \theta^* = \pm \sqrt{1-(\omega/a)^2} $$ is not an additional solution, but just a consequence of the identity $$ \cos^2 x + \sin^2 x = 1 . $$ (He needs that formula in order to evaluate $f'(\theta^*)$ just below.)