(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We're considering the following special case of the Duffing oscillator, in the limit of small nonlinearity:

[itex]\ddot x + x + \epsilon x^3 = 0[/itex].

The problem asks us to find a conserved quantity, and use it to write the period as an integral. Then, to write it as a power series in [itex]\epsilon[/itex] and evaluate it, then check the first two terms agains a power series that was given in our text. Mine didn't match up. I'm pretty sure I got the integrand right, but I really just don't know how select the bounds.

2. Relevant equations

3. The attempt at a solution

First of all, this system resembles a force on a unit mass which doesn't depend on velocity. Therefore, there is a well-defined potential energy as a function of only x, which in this case is

[itex]\frac{1}{2}x^2+\frac{\epsilon}{4}x^4[/itex]

Therefore,

[itex]C=\frac{1}{2}x^2+\frac{\epsilon}{4}x^4+\frac{1}{2}\dot{x}^2[/itex]

is conserved. (We can check that will time derivatives but I won't.)

Therefore, along a particular trajectory, there is some constant C such that

[itex]\frac {dx}{dt}=\dot x=\sqrt{C-x^2-\frac{\epsilon}{2}x^4}[/itex].

Therefore

[itex]dt=\frac{dx}{\sqrt{C-x^2-\frac{\epsilon}{2}x^4}}[/itex].

So the period will be

[itex]T=\int \frac{dx}{\sqrt{C-x^2-\frac{\epsilon}{2}x^4}}[/itex].

The problem is I have no idea what to set the bounds of the integral as. The x-value at the start and end of the period are the same, so the integral will equal zero.

I recognize also that it'd make more sense to cast it as a closed line integral on the set

[itex]y=\sqrt{C-x^2-\frac{\epsilon}{2}x^4}[/itex],

but I'm not sure how to effectively parametrize that set, or else I should be writing it in a very different form.

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# Finding the period of a limit cycle

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