# Motivation for quasiperiodic functions

According to the Scholarpedia defintion:

a function $F$ of real variable $t$ such that $F(t) = f(\omega_1t,\cdots, \omega_m t)$ for some continuous function $f(\phi_1,\cdots,\phi_m)$ of $m$ variables $(m≥2)$, periodic on $\phi_1,\cdots,\phi_m$ with the period $2\pi$, and some set of positive frequencies $\omega_1, \cdots, \omega_m$ , rationally linearly independent, which is equivalent to the condition $(k,\omega)=k_1 \omega_1 + \cdots +k_m ω_m \neq 0$ for any non-zero integer-valued vector $k=(k_1,…,k_m)$. The frequency vector $\omega=(\omega_1,…,\omega_m)$ is often called the frequency basis of a quasiperiodic function.

Strogatz helped me understand the intuition behind this definition for $m = 2$. Consider 2 joggers on the circumference of a circular field at speeds $\omega_1$ and $\omega_2$. Assume they both start at the same point and define their positions using $\theta$, their polar coordinate. That is, they are two uncoupled oscillators: $\dot{\theta_1} = \omega_1$ and $\dot{\theta_2} = \omega_2$. If their speeds are relatively prime, $\omega_1/\omega_2$ is a rational number, any pair $(\theta_1(t),\theta_2(t))$ repeats itself and hence this system is periodic. If we represented $(\theta_1(t),\theta_2(t))$ as the latitude and longitude pairs on a torus, the trajectories would trace out knots for rational speed ratios. If the speed ratios were irrational, there would be no closed orbits on the torus and this case is considered to be quasiperiodic since the trajectories are dense on the torus but not quite periodic.

Now, I understand that quasiperiodicity is different from chaotic behavior (at least in the case $m=2$, we don't have chaos due to Poincaré-Bendixson theorem) since we need exponential divergence of close trajectories to qualify as being chaotic. Are these two behaviors different in n-dimensional systems as well, for n>2? If yes, how do we detect quasiperiodicity? If these are two distinct behaviors, how does the definition above, considered on the n-torus, ensure that Lyapunov exponents are all $<= 0$?

Thank you very much for your time!