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 $$\epsilon$$...