# Periodic solutions with period converging to $0$ converge to equilibrium

Let's assume $$f: \mathbb{R}^d \to \mathbb{R}^d$$ is continuous and locally lipschitz. Consider the autonomous ODE $$x' = f(x)$$. Assume $$\exists R > 0. \forall n \in \mathbb{N}. \exists \varphi_n: \mathbb{R} \to \mathbb{R}^d$$ which is a $$\frac 1 n$$-periodic solution and such that $$\forall t \in \mathbb{R}. \varphi_n(t) \in \stackrel{-}{B}(0,R)$$. Proof that $$\exists p \in \stackrel{-}{B}(0,R). f(p) = 0$$.

I have the intuition that maybe I could work in $$(C[0,1],\|\cdot\|_{\infty})$$ extract a converging partial $$\varphi_n \to \varphi$$ and prove that the limit has to be $$0$$-periodic. In the language of dynamical systems, we have a sequence of cycles with radius converging to $$0$$ and have to show that they have a equilibrium as a limit point. What is a nice way of proving this?

Edit

I think that using a Poincaré map can be of use here.

• Idea: Take the centers of gravity or some other centers of these cycles and use compactness to find an accumulation point. Use continuity and thus boundedness of $f$ to show that this point satisfies the properties for $p$. – LutzL Jun 8 at 13:48
• @LutzL so I take an instant of time and observe the solution there. Since it is confined in a compact set there is a partial that converges to some $p$. But how would continuity of $f$ give me that $f(p) = 0$? – Javier Jun 9 at 8:08
• I'm afraid that I'm myself not clear on. If $\|f\|<M$ in $\bar B(0,R)$, then the length of the cycle with period $\frac1n$ can not be more than $\frac{M}n$. As a cycle, it must has opposing directions, as they sum to zero. Now apply some intermediate value argument... The problem is that standard examples like $\ddot x+x=0$ have constant period around the center, perhaps $\ddot x+x^3=0$ is an illustration for the situation in the claim. – LutzL Jun 9 at 8:25
• @LutzL I actually found the solution in Verhulst's "Nonlinear Differential Equations and Dynamical Systems", theorem 4.8, it is quite elegant. Thank you, I will check your example – Javier Jun 9 at 8:57