(a) Consider system (8.10) with k = 0.1 and = 1.25. Plot a bifurcation diagram for...

(a) Consider system (8.10) with k = 0.1 and ω = 1.25. Plot a bifurcation diagram for 0 ≤ Ŵ ≤ 0.12 and show that there is a clockwise hysteresis loop at approximately 0.04 < Ŵ < 0.08. Note that there is ringing

(oscillation) at the ramp-up and ramp-down parts of the bistable region.

194 8. Poincaré Maps and Nonautonomous Systems in the Plane

(b) Plot the two stable limit cycles in the bistable region for Exercise 10(a)

on one phase portrait. This shows that the system is multistable. For example, take Ŵ = 0.07. These limit cycles correspond to steady states on the upper and lower branches of the bistable cycle.