I am trying to find a proof for the Theorem for Nonlinear centers for conservative systems (Strogatz, Theorem 6.5.1).
“Consider the system $\dot {\vec x}= f(\vec x)$, where $\vec x = (x,y) \in {\mathbb{R}^2}$, and f is continuously differentiable. Suppose there exists a conserved quantity $E(\vec x)$ and suppose that is an isolated fixed point $\vec x*$(i.e., there are no other fixed points in a small neighborhood surrounding $\vec x*$). If $\vec x*$ is a local minimum of E, then all trajectories sufficiently close to are closed. ”
But I found only “Intuitive justifications” or plausibility arguments.
One of the key ideas of the proof is the proposition that
“Near a local maximum or minimum of a continuously differentiable real function $E(\vec x)$, where $\vec x = (x,y) \in {\mathbb{R}^2}$, the contours are closed curves”
which though it seems obvious, again I failed to find a strict justification (by myself or in the literature)
a. Is there a rigorous proof of the theorem?
or, at least,
b. for the proposition about the closed contours near to the extremum of a function?