Quasiperiodic motion

In mathematics and theoretical physics, quasiperiodic motion is motion on a torus that never comes back to the same point. This behavior can also be called quasiperiodic evolution, dynamics, or flow. The torus may be a generalized torus so that the neighborhood of any point is more than two-dimensional. At each point of the torus there is a direction of motion that remains on the torus. Once a flow on a torus is defined or fixed, it determines trajectories. If the trajectories come back to a given point after a certain time then the motion is periodic with that period, otherwise it is quasiperiodic.

The quasiperiodic motion is characterized by a finite set of frequencies which can be thought of as the frequencies at which the motion goes around the torus in different directions. For instance, if the torus is the surface of a doughnut, then there is the frequency at which the motion goes around the doughnut and the frequency at which it goes inside and out. But the set of frequencies is not unique – by redefining the way position on the torus is parametrized another set of the same size can be generated. These frequencies will be integer combinations of the former frequencies (in such a way that the backward transformation is also an integer combination). To be quasiperiodic, the ratios of the frequencies must be irrational numbers.[1][2][3][4]

In Hamiltonian mechanics with n position variables and associated rates of change it is sometimes possible to find a set of n conserved quantities. This is called the fully integrable case. One then has new position variables called action-angle coordinates, one for each conserved quantity, and these action angles simply increase linearly with time. This gives motion on "level sets" of the conserved quantities, resulting in a torus that is an n-manifold – locally having the topology of n-dimensional space.[5] 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.[6] Quasiperiodic motion does not exhibit the butterfly effect characteristic of chaotic systems. In other words, starting from a slightly different initial point on the torus results in a trajectory that is always just slightly different from the original trajectory, rather than the deviation becoming large.[4]

  1. ^ Sergey Vasilevich Sidorov; Nikolai Alexandrovich Magnitskii. New Methods For Chaotic Dynamics. World Scientific. pp. 23–24. ISBN 9789814477918.
  2. ^ Weisstein, Eric W. (12 December 2002). CRC Concise Encyclopedia of Mathematics. CRC Press. p. 2447. ISBN 978-1-4200-3522-3.
  3. ^ Ruelle, David (7 September 1989). Chaotic Evolution and Strange Attractors. Cambridge University Press. p. 4. ISBN 978-0-521-36830-8.
  4. ^ a b Broer, Hendrik W.; Huitema, George B.; Sevryuk, Mikhail B. (25 January 2009). Quasi-Periodic Motions in Families of Dynamical Systems: Order amidst Chaos. Springer. p. 2. ISBN 978-3-540-49613-7.
  5. ^ "Quasi-periodic motion", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  6. ^ 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.