Suppose the period of γn is λn. If there are points Xn ∈ γn such that Xn → X ∈ γ , prove that λn → λ.

Question

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.

Answer 1

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 $$ \max_{p\in\gamma}\min_{p_n\in\gamma_n}d(p,p_n)\to0. $$