Quasiperiodic motion, in mathematics and theoretical physics, is in rough terms the type of motion executed by a dynamical system containing two incommensurable frequencies.
That is, if we imagine that the phase space is modelled by a torus T, the trajectory of the system is modelled by a curve on T that wraps around without ever exactly coming back on itself.
A quasiperiodic function on the real line is the type of function (continuous, say) obtained from a function on T, by means of a curve
- R → T
which is linear (when lifted from T to its covering Euclidean space), by composition. It is therefore oscillating, with a finite number of underlying frequencies. (NB the sense in which theta functions and the Weierstrass zeta function in complex analysis are said to have quasi-periods with respect to a period lattice is something distinct from this.)
The theory of almost periodic functions is, roughly speaking, for the same situation but allowing T to be a torus with an infinite number of dimensions.