Math 6410 • ODE

\begin{enumerate}

\item
[2] Let \(f:\mathbb{R}^d\to\mathbb{R}^d\) be a continuously differentiable function such that \(f(0)=0\).
State the definition: \(z(t)=0\) is a Liapunov stable solution of \(\dot{u}=f(u)\) on \(0\le t<\infty\).


\item
[15] State conditions on the eigenvalues and eigenvectors of a real \(d \times d\) matrix \(A\) that
hold if and only if \(z(t)=0\) is Liapunov stable solution for \(\dot{x}=Ax\) on
\(0\le t<\infty\).
Prove your conditions.


\item
[8] Let \(A(t)\) be a \(T>0\) periodic \(d\times d\) real continuous matrix function. State but
don't prove necessary and sufficient conditions on \(A(t)\) so that the solution \(z(t)\) \(= 0\) is
Liapunov stable for \(\dot{x}=A(t)x\) on \(0\le t<\infty\).


\end{enumerate}