Finding the period of a limit cycle

[alexfloo]

Homework Statement

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

$\ddot x + x + \epsilon x^3 = 0$.

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 $\epsilon$ 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.

Homework Equations

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

$\frac{1}{2}x^2+\frac{\epsilon}{4}x^4$

Therefore,

$C=\frac{1}{2}x^2+\frac{\epsilon}{4}x^4+\frac{1}{2}\dot{x}^2$

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

$\frac {dx}{dt}=\dot x=\sqrt{C-x^2-\frac{\epsilon}{2}x^4}$.

Therefore

$dt=\frac{dx}{\sqrt{C-x^2-\frac{\epsilon}{2}x^4}}$.

So the period will be

$T=\int \frac{dx}{\sqrt{C-x^2-\frac{\epsilon}{2}x^4}}$.

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

$y=\sqrt{C-x^2-\frac{\epsilon}{2}x^4}$,

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

 

[Mute]

Homework Statement

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

$\ddot x + x + \epsilon x^3 = 0$.

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 $\epsilon$ 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.

Homework Equations

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

$\frac{1}{2}x^2+\frac{\epsilon}{4}x^4$

Therefore,

$C=\frac{1}{2}x^2+\frac{\epsilon}{4}x^4+\frac{1}{2}\dot{x}^2$

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

$\frac {dx}{dt}=\dot x=\sqrt{C-x^2-\frac{\epsilon}{2}x^4}$.

Therefore

$dt=\frac{dx}{\sqrt{C-x^2-\frac{\epsilon}{2}x^4}}$.

So the period will be

$T=\int \frac{dx}{\sqrt{C-x^2-\frac{\epsilon}{2}x^4}}$.

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

$y=\sqrt{C-x^2-\frac{\epsilon}{2}x^4}$,

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

Remember that when you take the square root there are two solutions to consider. The integral isn't zero because you have to choose the plus sign along one half cycle and the minus sign on the other half cycle. The period is thus more like

$$T = 2\int_0^{X} \frac{dx}{\sqrt{C - x^2 - \epsilon x^4/2}},$$

where X is the maximum value x attains.