Could anyone comment on the following ODE problem? Thank you!
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be $C^{1}$ and let $X^{(n)}(t)$ be a sequence of periodic solutions of $\frac{dX}{dt}(t)=f(X(t)).$ Assume that $X^{(n)}(0)$ converges and let $X(t)$ be the solution with $X(0)=\lim\limits_{n\rightarrow \infty}X^{(n)}(0)$.
Prove of disprove that $X(t)$ is periodic.