Theorem 9.3.3 provides no information about the stability of a critical point of a

Chapter 9, Problem 23

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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 dx dt = y + x( x2 + y2), dy (25) dt = x + y( x2 + y2) and dx dt = y x( x2 + y2), dy (26) dt = x y( x2 + y2) where is a real-valued constant. a. Show that, for all values of , (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 , system (25) is locally linear. c. Let r2 = x2+y2, and note that x dx/dt+y dy/dt = r dr/dt. Show that dr/dt = r3. d. Show that, for any < 0, r decreases to 0 as t ; hence the critical point is asymptotically stable. e. Show that, for any > 0, the solution of the initial value problem for r with r = r0 at t = 0 becomes unbounded as t increases towards 1/(2r2 0 ), and hence the critical point is unstable. 2