Existence, Uniqueness & Bifurcations · Challenge problem

For a constant $a$, consider $y' = \sin y - y + a y^3$. Since $f'(0) = 0$ for every $a$, linearisation cannot decide the stability of the equilibrium $y = 0$. It turns out that $y = 0$ is asymptotically stable exactly when $a < c$. Find $c$.