# Searching for the proof of a certain claim in Arnold's ODE book from 1992

I was reading today the book of Stephen Wiggins called "Global Bifurcations and Chaos" (the 1988 edition). On pages 12-13 he writes the following:

Consider the following ordinary differential equation $$\dot{\theta}_1=\omega_1 \ \ \ \dot{\theta}_2=\omega_2 \ \ \ \ \theta_i\in (0,2\pi] \ \ \forall i\in\{ 1,2\}$$ where $$\omega_1,\omega_2$$ positive constants. Since $$\theta_1,\theta_2$$ are two angular variables the phase space of the above ode is $$\mathbb{S}^1\times\mathbb{S}^1=\mathbb{T}^2$$. If we draw the torus as the surface of donut in $$\mathbb{R}^3$$ the orbits of this ode spiral around the surface and close (i.e, they are periodic) when $$\omega_1/\omega_2$$ is a rational; alternatively they densely fill the surface when $$\omega_1/\omega_2$$ is irrational, see Arnold  for a detailed proof of these statements.

The 1973 book is the ODE book; I cannot find a suitable copy, but I have the 1992 copy — does someone know where in this book the proof of these statements appears? on which page?

Thanks!

• And this is why all citations should include a page or theorem number. Aug 18 at 11:08