Parameterized function crosses own path

1. Feb 27, 2013

mariush

Hi!

Given a function $$r:\mathbb{R} \rightarrow \mathbb{R}^2, r(t) = (f_1(t), f_2(t))$$, is there a way to analytically determine if there are points (x1, x2) where r(t) = (x1, x2) for multiple t-values?

Lets say i was to find such points for the function $$r(t) = (t^3-t, 3t^2 + 1)$$

How should i go about finding the points without having to plot the graf?

Thanks!

2. Feb 27, 2013

CompuChip

In theory, you could assume that given some time t, the graph crosses the same point r(t) after time a again, leading to the equation
r(t) = r(t - a).

This leads to two equations in t and a.
The question would then be if there is a non-trivial solution (i.e. $a \neq 0$).

3. Feb 27, 2013

mariush

Exactly!

Thanks alot :)