What Is the Next Step in Simplifying the Van der Pol Oscillator Equation?

AI Thread Summary
The discussion focuses on simplifying the Van der Pol oscillator equation, particularly when ε = 0, revealing a circular limit cycle in phase space. The Fourier series representation indicates that the limit cycle's behavior is independent of initial conditions, except at the origin. The participant is attempting to simplify the equation of motion by substituting the Fourier series back into it, leading to a simplified form involving ε. They express uncertainty about the next steps, suggesting that trigonometric identities might be necessary for further simplification. The conversation emphasizes the importance of understanding the oscillatory behavior and the role of Fourier series in analyzing the system.
kornelthefirst
Messages
1
Reaction score
0
Homework Statement
We have a Van der Pol oscillator with small ##\epsilon## and after writing up a Fourier-series we have to bring it to a simpler form.
Relevant Equations
Equation of motion for the Van der Pol oscillator$$\ddot{x} + \epsilon(x^{2} - 1)\dot{x} + x = 0$$ Fourier-series for the limit cycle(already given) $$x_\epsilon^p(t) = \frac{a_0}{2} + \sum\limits_{k=1}^{\infty } [a_k \cos(k \omega t) + b_k \sin(k \omega t)]$$ Equation we need to arrive to$$\epsilon (x_p^2 - 1)\dot{x_p} = \epsilon a_1\omega[(1-\frac{a_1^2}{4})\sin(\omega t)-\frac{a_1^2}{4}\sin(3 \omega t)]$$
First i looked at the case of ## \epsilon = 0## and came to the conclusion, that this oscillator has a circular limit cycle in a phase space trajectory, when plotted with the axes x and ##\dot{x}##.
I have found that ##x_0^p (t) = a_1 \cos(t)## which implies that all other Fourier- coefficients have ##\epsilon## of at least power of 1
The limit cycle is independent of the starting conditions unless ##x = 0## and ##\dot{x} = 0##, so we can choose ##a_1## to be > 0 and ##b_1 > 0##.
If we put the equation of the Fourier-series back to the equation of motion we get$$\sum\limits_{k=1}^{\infty } [ - a_k \cos(k \omega t) - b_k sin( k \omega t)] + \epsilon (x^2-1)\dot{x} + \frac{a_0}{2}+\sum\limits_{k=1}^{\infty } [a_k \cos(k \omega t) + b_k sin( k \omega t)]$$ so simplified $$\epsilon (x^2-1)\dot{x} + \frac{a_0}{2} = 0$$ I am currently stuck here and can't find the next step. I can only assume it will include trigonometric identities, because i can see some patterns for some.
 
Physics news on Phys.org
kornelthefirst said:
Homework Statement: We have a Van der Pol oscillator with small ##\epsilon## and after writing up a Fourier-series we have to bring it to a simpler form.

I'm not sure this is the best approach to this problem; could you please post the exact problem statement?

Relevant Equations: Equation of motion for the Van der Pol oscillator$$\ddot{x} + \epsilon(x^{2} - 1)\dot{x} + x = 0$$ Fourier-series for the limit cycle(already given) $$x_\epsilon^p(t) = \frac{a_0}{2} + \sum\limits_{k=1}^{\infty } [a_k \cos(k \omega t) + b_k \sin(k \omega t)]$$ Equation we need to arrive to$$\epsilon (x_p^2 - 1)\dot{x_p} = \epsilon a_1\omega[(1-\frac{a_1^2}{4})\sin(\omega t)-\frac{a_1^2}{4}\sin(3 \omega t)]$$

First i looked at the case of ## \epsilon = 0## and came to the conclusion, that this oscillator has a circular limit cycle in a phase space trajectory, when plotted with the axes x and ##\dot{x}##.
I have found that ##x_0^p (t) = a_1 \cos(t)## which implies that all other Fourier- coefficients have ##\epsilon## of at least power of 1
The limit cycle is independent of the starting conditions unless ##x = 0## and ##\dot{x} = 0##, so we can choose ##a_1## to be > 0 and ##b_1 > 0##.
If we put the equation of the Fourier-series back to the equation of motion we get$$\sum\limits_{k=1}^{\infty } [ - a_k \cos(k \omega t) - b_k sin( k \omega t)] + \epsilon (x^2-1)\dot{x} + \frac{a_0}{2}+\sum\limits_{k=1}^{\infty } [a_k \cos(k \omega t) + b_k sin( k \omega t)]$$

You should have <br /> \ddot x = -\sum_{n=1}^\infty n^2\omega^2 (a_n \cos (n\omega t) + b_n \sin (n\omega t)).

I am currently stuck here and can't find the next step. I can only assume it will include trigonometric identities, because i can see some patterns for some.

I think the idea is that x(t) = a_1 \cos \omega t + \epsilon x_p(t) so that \begin{split}<br /> \ddot x + x &amp;= -\epsilon(x^2 - 1)\dot x \\<br /> (1 - \omega^2) \cos \omega t + \epsilon (\ddot x_p + x_p) &amp;= \epsilon a_1 \omega (a_1^2 \cos^2\omega t - 1)\sin \omega t + O(\epsilon^2)\end{split} subject to \dot x_p(0) = x_p(0) = 0. We do not need a sine term in the leading order solution since that just amounts to a shift in the origin of time, which merely moves us to a different point on the limit cycle. It is not necessary to expand x_p as a fourier series in order to solve this, although expressing the right hand side as a series in \sin n\omega t and knowing \ddot y + y where y = \sin n\omega t or t\cos n\omega t will assist.
 
Thread 'Help with Time-Independent Perturbation Theory "Good" States Proof'
(Disclaimer: this is not a HW question. I am self-studying, and this felt like the type of question I've seen in this forum. If there is somewhere better for me to share this doubt, please let me know and I'll transfer it right away.) I am currently reviewing Chapter 7 of Introduction to QM by Griffiths. I have been stuck for an hour or so trying to understand the last paragraph of this proof (pls check the attached file). It claims that we can express Ψ_{γ}(0) as a linear combination of...
Back
Top