15.The equation u′′−μ(1−13u'2)u′+u=0u″−μ1−13u′2u′+u=0 is often called the Rayleigh14 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 μ = 1. Choose initial conditions, and graph the corresponding solution of the system on an interval such as 0 ≤ t ≤ 20 or longer. Plot u versus t, 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 μ, such as μ = 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 μ increases. For example, make a table of values and/or plot A and T as functions of μ.
15.The equation
is often called the Rayleigh14 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 μ = 1. Choose initial conditions, and graph the corresponding solution of the system on an interval such as 0 ≤ t ≤ 20 or longer. Plot u versus t, 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 μ, such as μ = 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 μ increases. For example, make a table of values and/or plot A and T as functions of μ.
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