Quasi Periodic

A bounded orbit that is not Asymptotically Periodic and does not exhibit SDIC. Sits between periodic and chaotic — never repeats, never settles, but nearby orbits don’t diverge either. Lyapunov exponent $h=0$.

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The prototypical example is irrational rotation:

$$f(x)=(x+q)(\mathrm{mod}1)q\in \mathbb{R}\setminus \mathbb{Q}$$

Desmos

$f^{\prime }(x)=1$ everywhere, so $h=0$, no stretching, no contraction. The orbit is dense in $[0,1]$ (visits every subinterval), never repeating, but neighbors travel with it at constant separation.

If $q$ were rational, orbits would be Eventually Periodic. The irrational case keeps the orbit structurally recurrent without ever closing.