91Ó°ÊÓ

(a) Draw the (local) phase portrait for the system $$ \begin{aligned} &\dot{x}=x(1-x) \\ &\dot{y}=-y(1-2 x) \end{aligned} $$ (b) Show that this system (which has a saddle-saddle connection) is not structurally stable. Hint: For \(\mu=\varepsilon /(d+2)\), show that the system $$ \begin{aligned} &\dot{x}=x(1-x) \\ &\dot{y}=-y(1-2 x)+\mu x \end{aligned} $$ is \(\varepsilon\)-close to the system in part (a) on any compact set \(K \subset \mathbf{R}^{2}\) of diameter \(d\). Sketch the (local) phase portrait for this system with \(\mu>0\) and assuming that the systems in (a) and (b) are topologically equivalent for \(\mu \neq 0\), arrive at a contradiction as in Problems 1-3.

Find the stable, unstable and center subspaces, \(E^{*}, E^{u}\) and \(E^{c}\) for the linear maps \(L(x)=A x\) where the matrix (a) \(A=\left[\begin{array}{rr}2 & 0 \\ 1 & -1\end{array}\right]\) (b) \(A=\left[\begin{array}{cc}1 / 2 & 1 \\ 0 & 1 / 2\end{array}\right]\) (c) \(A=\left[\begin{array}{rr}1 & -1 \\ 1 & 1\end{array}\right]\) (d) \(A=\left[\begin{array}{ll}1 & 1 \\ 1 & 2\end{array}\right]\).