Quasiperiodic motion
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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.[3] 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.[4]
Torus model
[edit]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.[5] 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.[6]
See Kronecker's theorem for the geometric and Fourier theory attached to the number of modes. The closure 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.
Terminology
[edit]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.
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.
References
[edit]- ^ Vasilevich, Sidorov Sergey; Alexandrovich, Magnitskii Nikolai. New Methods For Chaotic Dynamics. World Scientific. pp. 23–24. ISBN 9789814477918.
- ^ Weisstein, Eric W. (12 December 2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 2447. ISBN 978-1-4200-3522-3.
- ^ "Quasi-periodic motion", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- ^ Broer, Hendrik W.; Huitema, George B.; Sevryuk, Mikhail B. (25 January 2009). Quasi-Periodic Motions in Families of Dynamical Systems: Order amidst Chaos. Springer. pp. 1–4. ISBN 978-3-540-49613-7.
- ^ Komlenko, Yu. V.; Tonkov, E. L. (2001) [1994], "Quasi-periodic function", Encyclopedia of Mathematics, EMS Press
- ^ Ruelle, David (7 September 1989). Chaotic Evolution and Strange Attractors. Cambridge University Press. p. 4. ISBN 978-0-521-36830-8.