type of bifurcation occurs

Question

What are the fixed points of

$$ θ'=1-a\sinθ $$

what type of bifurcation occurs at $$ a=1, \;\;\;θ=π/2 $$

Solution:

$$ 1/a=\sinθ $$

or $$θ=\arcsin(1/a) $$

I cant seem to find the proper fixed points after this step

Answer 1

You don't need to find any fixed points analytically to get the answer you are seeking.

Plot the function $$ a=\frac{1}{\sin \theta} $$ in the interval $(\pi/2-\varepsilon,\pi/2+\varepsilon)$. Fixing $a=\hat{a}$ gets you an idea about the number of fixed points and their types (this is called bifurcation diagram).

Note that if $a>1$ then you have two fixed points, one is stable and another is unstable. If $a<1$ then you have no fixed points, the two above approached each other and collapsed. This should be known to you as a saddle-node, or tangent, or fold bifurcation.