# Derivation of van der pol oscillator

• Slightly
In summary, the conversation discusses fitting a polynomial to a characteristic equation in order to obtain the van der pol equation. The author is trying to fill in the details on how to do this, particularly for a polynomial of degree three. There is some confusion on how to manipulate the polynomial and whether plugging it into the characteristic equation is the correct approach. The conversation also mentions trying to get rid of the U^3 term, but so far, the attempts have not been successful.

## Homework Statement

Given characteristic equation for a circuit containing a diode, I must figure out how to fit a polynomial to the curve so that the van der pol equation is obtained.

The paper I am reading is here:

http://www.unige.ch/math/hairer60/pres/pres_rentrop.pdf

My doubts are located on page three where the author talks about fitting the polynomial of degree three. I am trying to fill in the details as it isn't quite clear what he is actually doing.

Suggestions?

Should I plug the polynomial into the characteristic equation and manipulate it? Or is there some other technique?

He is just filling it in I think.

$$C\ddot{U} + \left( \frac{RC}{L} + \frac{d}{du}\left(a_1U +a_2U^2 +a_3U^3\right)\right)\dot{U} + \frac{1}{L}\left(R(a_1U +a_2U^2 +a_3U^3) + U - U_{op}\right)$$

It looks like it is going to look as is stated. Have you tried it and doesn't it work?

barefeet said:
He is just filling it in I think.

$$C\ddot{U} + \left( \frac{RC}{L} + \frac{d}{du}\left(a_1U +a_2U^2 +a_3U^3\right)\right)\dot{U} + \frac{1}{L}\left(R(a_1U +a_2U^2 +a_3U^3) + U - U_{op}\right)$$

It looks like it is going to look as is stated. Have you tried it and doesn't it work?
Yea, I tried plugging everything in like you just did, but I could not figure out how to get rid of the U^3 term.

In the bracket it vanishes due to the derivative, but outside it does not.
Setting y=U+d with the right constant d gets rid of the U term in the brackets, but then you still have those U^2 and U^3 outside.

That's exactly what I don't want. I'm trying to figure out how they get to the van der pol equation.

## 1. What is the van der pol equation?

The van der pol equation is a mathematical model that describes the behavior of a non-linear oscillator. It was developed by Dutch physicist Balthasar van der Pol in the early 20th century.

## 2. How is the van der pol equation derived?

The van der pol equation is derived from the equations of motion for a damped harmonic oscillator, with the addition of a non-linear term. This term captures the phenomenon of self-sustained oscillations, which is commonly observed in physical systems.

## 3. What are the applications of the van der pol equation?

The van der pol equation has applications in various fields, including electronics, mechanics, and biology. It is commonly used to model phenomena such as electrical circuits, mechanical systems, and biological systems like neuron firing.

## 4. What is the significance of the van der pol oscillator?

The van der pol oscillator is significant because it exhibits a variety of interesting behaviors, including self-sustained oscillations, limit cycles, and chaos. It has been extensively studied and has provided insights into the dynamics of complex systems.

## 5. Are there any limitations to the van der pol equation?

Like any mathematical model, the van der pol equation has certain limitations. It is a simplified representation of real-world systems and may not accurately capture all aspects of their behavior. Additionally, it can become unstable under certain conditions, making it less suitable for modeling those situations.