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

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