# Irrational Flow yields dense orbits.

1. Mar 19, 2012

### JuanYsimura

I have the folloring problem:

Given the following flow on the torus (θ_1)' = ω_1 and (θ_2)' = ω_2, where ω_1 /ω_2 is irrational then I am asked to show that each trajectory is DENSE. So I need to prove that Given any point p on the torus, any initial condition q, and any ε > 0, then there exists t finite such that the trajectory starting at q passes within a distance epsilon of p, that is to say find a t such that |q - p| < ε.

My problem is how can I find such a t? Can I prove this by contradiction?

Thank you very much for your help,

Juan

2. Mar 28, 2012