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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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  • $\begingroup$ There are a lot of interesting papers on "Isochronous center". $\endgroup$ Sep 27 '16 at 18:54
  • $\begingroup$ As @LoicTeyssier said, existence of other singularity in the boundary domain of periodic orbits, imply that the period is not bounded. $\endgroup$ Sep 27 '16 at 18:57
  • $\begingroup$ @Loic I am wonder can one produce some results on the period with the following geometric consideration: $\endgroup$ Sep 27 '16 at 19:02
  • $\begingroup$ For a vector field $\dot x= P(x,y),\;\; \dot y= Q(x,y),\;\;$ the following integral along a closed orbit with perod $\tau$ is constant $2\pi\;\;\;\;\; \int_{0}^{\tau} \frac{P^{2}Q_{x}-PQ(Q_{y}-P_{x})-Q^{2}P_{y}}{P^{2}+Q^{2}}$ $\endgroup$ Sep 27 '16 at 19:16
  • $\begingroup$ A consequence of The Gauss Bonnet Theorem $\endgroup$ Sep 27 '16 at 19:16
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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.

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  • $\begingroup$ Thanks. Could you give me a reference for this formula ? $\endgroup$ May 19 '16 at 9:46
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    $\begingroup$ Just write the curve integral as a integral in the time variable by parametrezing the curve through the flow of the vector field. $\endgroup$ May 19 '16 at 10:17
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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.

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