Classical Dynamics of Particles and Systems · Problem 4-24

Gravitation — Marion & Thornton

Physics — Mechanics · van-der-pol-perturbation

Problem

Assume that

x(t) = b\cos(\omega_0 t) + u(t)

is a solution of the van der Pol Equation 4.19. Assume that the damping parameter

\mu

is small and keep terms in

u(t)

to first order in

\mu

. Show that

b = 2a

and

u(t) = -(\mu a^3/4\omega_0)\sin(3\omega_0 t)

is a solution. Produce a phase diagram of

\dot{x}

versus

x

and produce plots of

x(t)

and

\dot{x}(t)

for values of

a = 1

\omega_0 = 1

, and

\mu = 0.05