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!


Actually, exponential divergence by itself is not sufficient for a chaotic behavior. The other major ingredient, curiously many times overlooked, is nontrivial recurrence. Roughly speaking this means that most trajectories get arbitrarily close to their departure point. For example, a single hyperbolic point has exponential divergence but by itself it should be described as a chaotic behavior.

Clearly, quasiperiodic functions have the former nontrivial recurrence property, but they have no exponential behavior. So they are at the other extreme of a single hyperbolic point among the dynamics that are not chaotic.

Something similar happens in higher dimensions, although it depends on your definition of chaos (there are too many definitions, so we really should speak of some specific properties in each case). For example, you can have nonzero Lyapunov exponents in some directions and still call it chaos, say since this lead to positive topological entropy. On the other hand, a quasiperiodic dynamics has zero entropy because all the Lyapunov exponents are then zero.

So, a first necessary criterion for quasiperiodicity is zero entropy. A second necessary criterion for quasiperiodicity is zero Lyapunov exponents (and the second criterion implies the first). But none is a sufficient criterion.

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  • $\begingroup$ Thank you very much for your answer! Do you define a hyperbolic point as a point whose tangent space is a direct sum of stable, unstable and neutral sub spaces? If so, I don't rigorously understand what its opposite would be. Also, are flows with zero Lyapunov exponents which are aperiodic necessarily quasiperiodic? Thanks again for your time! $\endgroup$ – rivendell Aug 4 '17 at 18:35
  • $\begingroup$ Yes, it is as you say. Well, as you can see, my answer on purpose has no formulas, there is no rigorous meaning for "its opposite". :) Your question is really huge (I really appreciated it). As for your last question, it depends on hard number theoretical details, and some partial results do exist, but as far as I known one cannot be so specific as you ask.. $\endgroup$ – John B Aug 4 '17 at 19:20
  • $\begingroup$ @rivendell Hyperbolic equilibrium or fixed point doesn't have the neutral subspace, that's their definition. $\endgroup$ – Evgeny Aug 4 '17 at 20:40
  • $\begingroup$ @Evgeny Didn't notice the "neutral" there... :( $\endgroup$ – John B Aug 4 '17 at 23:12

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