In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies.[1] [2]
In Hamiltonian mechanics, the action-angle coordinates allow such motions to be defined on level sets. The concept is closely connected to the basic facts about linear flow on the torus. These essentially linear systems and their behaviour under perturbation play a significant role in the general theory of non-linear dynamic systems.[3] Quasiperiodic motion does not exhibit the butterfly effect characteristic of chaotic systems.[4]
Rectilinear motion along a line in a Euclidean space gives rise to a quasiperiodic motion when the integer lattice is used to compactify the space, under a typical condition on the direction cosines. When the dimension is 2, the condition is that the direction cosines are incommensurable. In higher dimensions the condition is that the direction cosines are linearly independent over the field of rational numbers.
If we imagine that the phase space is modelled by a torus T (that is, the variables are periodic, like angles), the trajectory of the quasiperiodic system is modelled by a curve on T that wraps around the torus without ever exactly coming back on itself. Assuming the dimension of T is at least two, there are one-parameter subgroups of that kind.
A quasiperiodic function f on the real line is the type of function (continuous, say) obtained from a function on T, by means of a curve that is such a one-parameter subgroup. Therefore f is oscillating, with a finite number of underlying frequencies. David Ruelle comments that it makes no sense to ask which are those frequencies, as specific numbers: there are as many as the modes of the system, but there is no unique basis to choose, just any independent set such that the frequencies are rational linear combinations of those.[5]
See Kronecker's theorem for the geometric and Fourier theory attached to the number of modes. The closure of (the image of) any one-parameter subgroup in T is a subtorus of some dimension d. In that subtorus the result of Kronecker applies: there are d real numbers, linearly independent over the rational numbers, that are the corresponding frequencies.
In the quasiperiodic case, where the image is dense, a result can be proved on the ergodicity of the motion: for any measurable subset A of T (for the usual probability measure), the average proportion of time spent by the motion in A is equal to the measure of A.[6]
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. The early discussion of quasi-periodic functions, by Ernest Esclangon following the work of Piers Bohl, in fact led to a definition of almost-periodic function, the terminology of Harald Bohr.[7] Ian Stewart wrote that the default position of classical celestial mechanics, at this period, was that motions that could be described as quasiperiodic were the most complex that occurred.[8] For the solar system, that would apparently be the case if the gravitational attractions of the planets to each other could be neglected: but that assumption turned out to be the starting point of complex mathematics.[9] The research direction begun by Andrei Kolmogorov in the 1950s led to the understanding that quasiperiodic flow on phase space tori could survive perturbation.[10]
NB: The concept of quasiperiodic function, for example 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 topic.