- #1

kornelthefirst

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- 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.

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.