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



Periodic orbits in the plane
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.


