This is supposed to be an ordinary differential equations class exercise:
Let $g\in \scr{C}^2(\Bbb{R}^2;\Bbb{R})$ and consider $$H_g(x,y)=\begin{pmatrix}\partial_yg(x,y)\\ -\partial_xg(x,y)\end{pmatrix}.$$
Show that, if $g$ admits a strict local maximum point (non-degenerated) at $p_0=(x_0,y_0)$ then near to $p_0$ there are infinitely many periodic orbits of the o.d.e. $p'(s)=H_g(p(s))$
I know facts like:
- the level curves of $g$ are integral curves of the o.d.e. $p'(s)=H_g(p(s))$;
- $\langle \nabla g,H_g\rangle=0$;
- basic facts about $\nabla g$, etc;
- basic facts about o.d.e.'s like existence and uniqueness theorem, o.d.e.'s systems, etc.
Furthermore, I can geometrically visualize that, if $p_0$ is a point of maximum isolated, then "around it", there is infinitely many closed level curves (imagine the surface given by the graph of $g$). But I don't have any idea of how to prove it formally...
I'm really stuck!