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I was wondering if someone could help me with an exercise from Hirsch, Smale, and Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos.

Let γ be a closed orbit of a planar system. Let λ be the period of γ . Let {$γ_n$} be a sequence of closed orbits. Suppose the period of $γ_n$ is λn. If there are points $X_n \in γ_n$ such that $X_n$ → X ∈ γ , prove that $λ_n$ → λ.

JAB
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  • I was thinking we could take a transverse section that intersected each of the $\gamma_n$ and $\gamma$ at one point(i.e., the $X_n$ and $X$). Then if i denoted the system by $f$ and assumed $f(X_n,t)=X_n$, then we would have $f(Xn,t+λn)=f(Xn,t)=X_n \rightarrow X=f(X,t)=f(X,t+\lambda)$. Would this imply $\lambda_n \rightarrow \lambda$ – JAB Apr 18 '16 at 03:15

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Hint: Notice that since there are Poincaré sections, the whole orbit $\gamma_n$ converges to the orbit $\gamma$ when $X_n\to X$, in the sense that $$ \max_{p\in\gamma}\min_{p_n\in\gamma_n}d(p,p_n)\to0. $$

John B
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