Math 6410 • ODE

\begin{enumerate}
\item[(a)] Let \(f(x)\in C^1(\R^n,\R^n)\) satisfy \(f(0)=0\). State the definitions:
the zero solution of
\[
  \begin{cases}
    \dfrac{dx}{dt}=f(x),\\
    x(0)=x_0,
  \end{cases}
\]
is \textit{Stable}, \textit{Asymptotically Stable} and \textit{Unstable}.
\item[(b)] Show that the zero solution is asymptotically stable.
\[
  \begin{cases}
    \dot{x}=xy-x^3\\
    \dot{y}=-y+x^3y
  \end{cases}
\]
\item[(c)] Suppose that the real \(n\times n\) matrix \(A\) has an eigenvalue with
\(\Reop\,\lambda>0\). Prove that the zero solution of \(\dot{x}=Ax\) is unstable.
\end{enumerate}