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Phys601/F11/Problem Set 6
Due 10/17/11
(upgraded)
GPS:
Do Ch8 #1, #27
6.1H
Van der Pol Oscillator
Consider the Van der Pol Oscillator for x(t)
d
2
x/dt
2
+ x
=
 2
ν
(dx/dt) (1x
2
).
Let x ~ 1 (“of order unity”), i.e., make no assumptions on whether x is small or large.
This means that the sign of the antifriction term on the RHS could change.
a.
Suppose
ν
<< 1.
Solve for x perturbatively by expanding in a series x = x
0
+ x
1
+ x
2
….. assuming that each term is smaller than the previous term in the series.
Show that
the regular perturbation series exhibits secular behaviour.
b.
Solve by assuming a multiple time scale solution, i.e., assuming that x = x(t,
τ
), where
τ
is over a longer time scale, in particular d/dt
=
∂
/
∂
t +
∂
/
∂τ
, where 
∂
/
∂τ
 << 
∂
/
∂
t.
Obtain solutions to lowest significant order, i.e., make sure you get to the interesting
stuff.
Make a sketch of x(t) for the two cases 0 < x(0) < 1 and 1 < x(0).
You will have to solve an equation using the “energy method”.
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This note was uploaded on 12/29/2011 for the course PHYSICS 601 taught by Professor Hassam during the Fall '11 term at Maryland.
 Fall '11
 Hassam
 mechanics

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