Math 6410 • ODE

\begin{enumerate}
\item[(a)] Show that the only equilibrium point of the system
\[
x'=-2y-4x^3-4xy^2,\qquad y'=x-2x^2y-2y^3
\]
is the origin.

\item[(b)] Draw what conclusions you can about stability, asymptotic stability, and
regions of attraction, using first i) linearization and then ii) Lyapunov functions.
Make clear the reasons for your conclusions.

\item[(c)] What if the signs of the nonlinear terms in a) were changed?

\item[(d)] What kinds of functions \(f\) and \(g\) would give similar results for systems
of the form \(x'=-2y+f(x,y)\), \(y'=x+g(x,y)\)?
\end{enumerate}