Van Der Pol dynamics

muzialis

Hi All,

for a Van Der Pol dyanmical sysetm , governed by the eqautions

x' = a * (x - 0.3 x ^3 ) - y
y' = x

i read at http://www.scholarpedia.org/article/...Pol_oscillator that when the system is away from the curve y=x−x3/3 , "a relation |x˙| >> |y˙|=O(1/ϵ) is obtained from equations (2) and (3). Therefore, the system moves quickly in the horizontal direction. When the system enters the region where |x−x3/3−y|=O(1/ϵ2) , x˙ and y˙ are comparable because both of them are O(1/ϵ)".

I really do not get this entirely.`I am unsure of how the relationship involving the infinitesimals are derived. How are these realtionship justified?

Can anybody please try to give us a hint? Thank you so much

Muzialis