Math 6410 • ODE

Consider the nonlinear system on \(\mathbb{R}^2\):
\[
\begin{cases}
\dot{x}=-y+x(1-x^2-y^2),\\
\dot{y}=x+y(1-x^2-y^2).
\end{cases}
\]

\begin{enumerate}

\item
Show that the unit circle \(S^1=\{(x,y):x^2+y^2=1\}\) is an invariant set for the flow.


\item
Prove that every trajectory starting near \(S^1\) spirals toward it as \(t\to\infty\).


\item
Define a Poincar\'e section \(\Sigma=\{(x,y):y=0,\ x>0\}\) and compute the Poincar\'e map
\(P:\Sigma\to\Sigma\) for points near the unit circle.


\item
Linearize the Poincar\'e map near the fixed point corresponding to the periodic orbit on \(S^1\),
and determine its stability.

\end{enumerate}