Van Der Pol Dynamics: Understanding Relationships with Infinitesimals

[muzialis]

Hi All,

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

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

i read at http://www.scholarpedia.org/article/Van_der_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

 

[jvicens]

Hello Muzialis,

Thank you for bringing up this interesting topic. The relationship between the infinitesimals in the Van Der Pol oscillator can be understood through the concept of time scales. In this system, there are two main time scales: the fast time scale (represented by x') and the slow time scale (represented by y').

When the system is away from the curve y=x−x3/3, the fast time scale dominates and the system moves quickly in the horizontal direction. This means that the changes in x are much larger than the changes in y, resulting in the relationship |x˙| >> |y˙|=O(1/ϵ). This is because the term 0.3x^3 in the first equation is much larger than the term y, making the changes in x dominant.

However, as the system enters the region where |x−x3/3−y|=O(1/ϵ2), the slow time scale starts to become more significant. This is because the term x^3 is now comparable to the term y, making the changes in both x and y comparable. This results in the relationship x˙ and y˙ being of similar magnitude, as both are now O(1/ϵ).

In summary, the relationship between the infinitesimals in the Van Der Pol oscillator is justified by considering the two time scales in the system and how they affect the dynamics of x and y. I hope this explanation helps. Let me know if you have any further questions.