The system x = 3(x + y 1 3 x3 k), y = 1 3 (x + 0.8y 0.7) is a special case of the

Chapter 9, Problem 21

(choose chapter or problem)

The system x = 3(x + y 1 3 x3 k), y = 1 3 (x + 0.8y 0.7) is a special case of the FitzhughNagumo15 equations, which model the transmission of neural impulses along an axon. The parameter k is the external stimulus. (a) Show that the system has one critical point regardless of the value of k. (b) Find the critical point for k = 0 and show that it is an asymptotically stable spiral point. Repeat the analysis for k = 0.5 and show that the critical point is now an unstable spiral point. Draw a phase portrait for the system in each case. (c) Find the value k0 where the critical point changes from asymptotically stable to unstable. Find the critical point and draw a phase portrait for the system for k = k0. (d) For k > k0 the system exhibits an asymptotically stable limit cycle; the system has a Hopf bifurcation point at k0. Draw a phase portrait for k = 0.4, 0.5, and 0.6; observe that the limit cycle is not small when k is near k0. Also plot x versus t and estimate the period T in each case. (e) As k increases further, there is a value k1 at which the critical point again becomes asymptotically stable and the limit cycle vanishes. Find k1.