The equation u?? ? ? ( 1 ? 1 3 u?2 ) u? + u = 0 is often

The equation u?? ? ? ( 1 ? 1 3 u?2 ) u? + u = 0 is often called the Rayleigh equation. (a) Write the Rayleigh equation as a system of two first order equations. (b) Show that the origin is the only critical point of this system. Determine its type and whether it is asymptotically stable, stable, or unstable. (c) Let ? = 1. Choose initial conditions and compute the corresponding solution of the system on an interval such as 0 ? t ? 20 or longer. Plot u versust and also plot the trajectory in the phase plane. Observe that the trajectory approaches a closed curve (limit cycle). Estimate the amplitude A and the period T of the limit cycle. (d) Repeat part (c) for other values of ?, such as ? = 0.2, 0.5, 2, and 5. In each case, estimate the amplitude A and the period T. (e) Describe how the limit cycle changes as ? increases. For example, make a table of values and/or plot A and T as functions of ?