Dynamical system $f(x)=x^4\sin(1/x)$ (for x not equal to zero) and $f(x)=0$ (for x=0). How to find equilibrium points, and determining the stability of each equilibrium point?
I found the equilibrium points x=0 and $x=1/kπ$, where k is integer.
$f'(x)=4x^3\sin(1/x)-x^4\cos(1/x)$.
When $k$ is 'even', $f'(x=1/kπ)=1/π^4$ which is positive, therefore not asymptotically stable.
And when $k$ is 'odd', $f'(x=1/kπ)=-1/π^4$ which is negative, therefore asymptotically stable.
But $f'(x=0)=0$, which is not hyperbolic. And this is the part that I can not solve.
Hint: Consider a small perturbation $x(t)$ of the zero solution, say with $x(0)=\epsilon \neq 0$. This solution will be caught in between two of the other equilibrium points (or perhaps exactly at one of them), and this lets you permit what happens to the solution $x(t)$ in the long run.