This problem deals with the almost linear system dx dt = y + hx(x2 + l), dy dt = -x +

Chapter 7, Problem 7.3.35

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This problem deals with the almost linear system dx dt = y + hx(x2 + l), dy dt = -x + hY(X2 + l), in illustration of the sensitive case of Theorem 2, in which the theorem provides no information about the stability of the critical point (0, 0). (a) Show that (0, 0) is a center of the linear system obtained by setting h = O. (b) Suppose that h =1= O. Let r 2 = X2 + y2, then apply the fact that dx dy dr x dt + y dt = r dt to show that dr/dt = hr3. (c) Suppose that h = -1. Integrate the differential equation in (b); then show that r -+ 0 as t -+ +00. Thus (0, 0) is an asymptotically stable critical point of the almost linear system in this case. (d) Suppose that h = + 1. Show that r -+ +00 as t increases, so (0, 0) is an unstable critical point in this case.