Theorem 9. 3. 3 provides no information about the stability of a critical point of a locally linear system if that point is a center of the corresponding linear system. That this must be the case is illustrated by the system a = y + x(x² + y2), = -x +ay(x² + y2)(25) and 4* = y – x(x2 + y2), ay = -x - y(x² + y2) where a is a real-valued constant. A. Show that, for all values of a, (0,0) is a critical point of system (25) and, furthermore, is a center of the corresponding linear system. B. Show that, for all values of a, system (25) is locally linear. C. Let r2 = x + y, and note that x dx/dt+ydy/dt = r dr/dt. Show that dr/dt = ar. D. Show that, for any a < 0,r decreases to 0 as 1 0; hence the critical point is asymptotically stable. E. Show that, for any a > 0, the solution of the initial value problem for r with r = ro at t = 0 becomes unbounded as increases towards 1/(2013), and hence the critical point is unstable