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?


  • 3
    $\begingroup$ And this is why all citations should include a page or theorem number. $\endgroup$ Aug 18 at 11:08

See at p. 163 of this PDF link:


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).


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