Periodic orbits in the plane Consider a vector field $F:\mathbb{R}^2\rightarrow \mathbb{R}^2$ of the following form $F(y_1,y_2)=(y_2,\mu(y_1))$, where $\mu\in\mathscr{C}^1(\mathbb{R})$ has appropriate growth so that the solutions of $Y'=F(Y)$ exists globally for any initial condition. Assume for some $a\in\mathbb{R}$ that $\mu(a)=0>\mu'(a)$.
The jacobian matrix of $F$ at $a$ is $A:=\begin{pmatrix} 0&1\\\mu'(a)&0\end{pmatrix}$ and has two conjugate pure imaginary eigenvalues : the dynamic of the linearized system $Y'=AY$ reduces to periodic solutions, the flow taking its values on ellipses.
I know that if $Y_0$ is taken sufficiently close to $(0,a)$, the corresponding orbit of $Y'=F(Y)$ is also periodic (the latter fact seems rather standard). I wonder what is the possible values for the periods of such orbits. I have the feeling that this set should (at least) contain an interval $I$, which would be lower bounded by a positive constant depending on the period of the linearized system. I wonder underwich condition this set of period is unbounded.
Is there any reference concerning these results ? 
Best,
Ayman
 A: The period of the compact, non-singular orbit $\gamma$ is given by $$T(\gamma)=\oint_\gamma \tau$$ where $\tau$ is any differential $1$-form such that $\tau(F)=1$, e.g. $\tau:=\frac{\mathrm{d}y_2}{\mu(y_1)}$. From this expression you may be able to study the boundedness of $T$ as $\gamma$ closes on another stationary point.
A: For convenience, let's take $a=0$ and $\mu'(a) = -1$, so the linearized system has solutions $y_1 = r \sin(t)$, $y_2 = r \cos(t)$.  Write a periodic  solution of the
nonlinear system as $y_1 = r(t) \sin(\theta(t))$, $y_2 = r(t) \cos(\theta(t))$.  We then get
$$ \dot{\theta} = \frac{\dot{y_1} \cos(\theta) - \dot{y_2} \sin(\theta)}{r} = \cos^2(\theta) - \frac{\mu(r \sin(\theta))}{r} \sin(\theta)$$
and (assuming this is always positive) the period is
$$ T = \int_0^{2\pi} \dfrac{d\theta}{\dot{\theta}}$$
You get very long periods if $\dot{\theta}$ gets close to $0$ at some point.
On the other hand, very short periods will require $\dot{\theta}$ to be large except on very small intervals; that won't happen for trajectories close to the origin, but might for trajectories very far away.
