Chapter 7.1, Problem 26P

Assuming that the trajectory corresponding to a solution $x=\varphi \left(t\right),y=\psi \left(t\right),-\infty <t<\infty$ of an autonomous system is closed, show that the solution is periodic.

Hint: Since the trajectory is closed, there exists at least one point $\left(x_0,y_0\right)$ such that $\varphi \left(t_0\right)=x_0,\psi \left(t_0\right)=y_0$, and a number $T>0$ such that $\varphi \left(t_0+T\right)=x_0,\psi \left(t_0+T\right)=y_0$. Show that $x=\Phi \left(t\right)=\varphi \left(t+T\right)$ and $y=\Psi \left(t\right)=\psi \left(t+T\right)$ is a solution, and then use the existence and uniqueness theorem to show that $\Phi \left(t\right)=\varphi \left(t\right)$ and $\Psi \left(t\right)=\psi \left(t\right)$ for all $t$.