Path Equation for 2D weakly-anisotropic harmonic oscillator

[lightbearer88]

Homework Statement

\omega_{x} = \omega

\omega_{y} = \omega + \epsilon

where 0 < \epsilon<<\omega

Question: Find the path equation.

Homework Equations

I started with the 2D equations:

x(t) = A_{x}cos(\omega_{x}t + \phi_{x})
y(t) = A_{y}cos(\omega_{y}t + \phi_{y})

The Attempt at a Solution

by inverting x(t) to get t(x), I then substituted the result into y(t). The result is as below:

y(x) = A_{y}cos[(\omega_{x}/\omega_{y})cos^{-1}(x/A_{x}) - (\omega_{x}/\omega_{y})\phi_{x} + \phi_{y}]

I guess it becomes more of a mathematical problem. How do I simplify this equation, hopefully to find something familiar? An idea I had was to use double-angle trig formulas but I am not sure how it would help.

I personally think this is quite challenging. I have been thinking about it for days now. I tried all the trig identities I know off to manipulate the equation but I can't seem to get it. Unless I am missing something.

Thanks in advance.

PS: A^x is actually A(subscript)x and similarly, A^y is actually A(subscript)y. I don't know what is wrong with the formatting.

 

[lightbearer88]

maybe I should define what the symbols are:

\omega_{x} is the angular frequency in the x-axis
\omega_{y} is the angular frequency in the y-axis

 

[dingo_d]

I'd try to substitute the \omega you have and then try to expand it by small parameter \epsilon...