Understanding the Van Der Pol System: Muzialis' Struggles

[muzialis]

Hello All,

I am struggling to understand the following passage (all equations can be found in a much better format at http://www.scholarpedia.org/article/Van_der_Pol_oscillator, eq(8) especially).

Starting from (C being a constant)

u' = C * u [4 - (u^2*v^2)]
v' = C * v [4 - (u^2*v^2)]

introducing

r^2 = u^2 * v^2

the equation

r' = C * r [4 - (u^2*v^2)]

I observe that squaring each equation above and summing one obtains

(u')^2 = C^2 * u^2 [4 - (u^2*v^2)^2]
(v')^2 = C^2 * v^2 [4 - (u^2*v^2)^2]

Summing top and below

(u')^2 + (v')^2 = C^2 * [4-(u^2*v^2)^2] * [u^2 + v^2]

Is this not a contradiction with the eqaution in bold i would like to understand? I am sure of course I made a trivial mistake somewhere. Thank you for any help

Muzialis

 

[gtoguy97]

No, there is no contradiction. The equation you obtained is the same as the one in bold, just written in a different form. The equation in bold states that the derivative of r^2 is equal to C*r[4 - (u^2*v^2)]. Squaring both sides of the equation and expanding gives the result that (u')^2 + (v')^2 = C^2 * [4-(u^2*v^2)^2] * [u^2 + v^2].