Math 6410 • ODE

\textbf{Limit Sets.}
Let \(\varphi_t(x_0)\) be the flow of the differential equation
\[
\begin{aligned}
  \dot{x} &= f(x),\\
  x(0) &= x_0,
\end{aligned}
\]
where \(f\in C^1(\R^n)\) satisfies a global Lipschitz condition, and \(x_0\in\R^n\).
\begin{enumerate}
\item[(a)] Given a trajectory \(\Gamma\) of \(\varphi_t\), define the \(\omega\)-limit set
\(\omega(\Gamma)\).
\item[(b)] Prove that \(\omega(\Gamma)\) is closed and invariant with respect to the flow
\(\varphi_t\).
\end{enumerate}