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 [1973] 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!



Searching for the proof of a certain claim in Arnold's ODE book from 1992
See at p. 163 of this PDF link:


[https://eclass.uoa.gr/modules/document/file.php/PHYS289/Βιβλία/Arnold%2C%20V.I.%20-%20Ordinary%20differential%20equations\_Red.pdf](https://eclass.uoa.gr/modules/document/file.php/PHYS289/%CE%92%CE%B9%CE%B2%CE%BB%CE%AF%CE%B1/Arnold%2C%20V.I.%20-%20Ordinary%20differential%20equations_Red.pdf)


Maybe you got confused because between the statement and the proof of the proposition you are interested in, there is a technical Lemma with its "proof". Moreover, this "proof" is willingly flawed (plain Arnold's style!), and just after the flaw is explained (but not corrected; that is left as a problem for the reader). Then the proof of the proposition you want starts (p.164).


