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Inmathematics andtheoretical physics,quasiperiodic motion is motion on atorus that never comes back to the same point. This behavior can also be called quasiperiodic evolution, dynamics, orflow. 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]

InHamiltonian mechanics withn position variables and associated rates of change it is sometimes possible to find a set ofn conserved quantities. This is called the fully integrable case. One then has new position variables calledaction-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 ann-manifold – locally having the topology ofn-dimensional space.^[5] The concept is closely connected to the basic facts aboutlinear flow on the torus. These essentially linear systems and their behaviour under perturbation play a significant role in the general theory ofnon-linear dynamic systems.^[6] Quasiperiodic motion does not exhibit thebutterfly effect characteristic ofchaotic 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]

Rectilinear motion along a line in aEuclidean space gives rise to a quasiperiodic motion if the space is turned into a torus (acompact space) by making every point equivalent to any other point situated in the same way with respect to theinteger lattice (the points with integer coordinates), so long as thedirection cosines of the rectilinear motion form irrational ratios. When the dimension is 2, this means the direction cosines areincommensurable. In higher dimensions it means the direction cosines must belinearly independent over thefield of rational numbers.^[5]

If we imagine that thephase space is modelled by atorusT (that is, the variables are periodic, like angles), the trajectory of the quasiperiodic system is modelled by acurve onT that wraps around the torus without ever exactly coming back on itself. Assuming the dimension ofT is at least two, these can be thought of asone-parameter subgroups of the torus givengroup structure (by specifying a certain point as theidentity element).

A quasiperiodic motion can be expressed as a function of time whose value is a vector of "quasiperiodic functions".A quasiperiodic functionf on thereal line is a function obtained from a functionF on a standard torusT (defined byn angles), by means of a trajectory in the torus in which each angle increases at a constant rate.^[7] There aren "internal frequencies", being the rates at which then angles progress, but as mentioned above the set is not uniquely determined. In many cases the function in the torus can be expressed as a multipleFourier series. Forn equal to 2 this is:

${\displaystyle F({\theta }_1,{\theta }_2)=\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{C}_{jk}\exp (ij{\theta }_1)\exp (ik{\theta }_2)}$

If the trajectory is

${\displaystyle {\theta }_1=a_1+{\omega }_{1}t}$

${\displaystyle {\theta }_2=a_2+{\omega }_{2}t}$

then the quasiperiodic function is:

${\displaystyle f(t)=\sum _{j=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{C}_{jk}\exp (ij{a}_1+ik{a}_2+i(j{\omega }_1+k{\omega }_2)t)}$

This shows that there may be an infinite number of frequencies in the expansion, not multiples of a finite number of frequencies. Depending on which coefficients${\displaystyle C_{jk}}$ are non-zero the "internal frequencies"${\displaystyle {\omega }_1}$ and${\displaystyle {\omega }_2}$ themselves may not contribute terms in this expansion, even if one uses an alternative set of internal frequencies such as${\displaystyle {\omega }_1}$ and${\displaystyle {\omega }_1+{\omega }_2.}$^[8] If the${\displaystyle C_{jk}}$ are non-zero only when the ratio${\displaystyle i/j}$ is some specific constant, then the function is actually periodic rather than quasiperiodic.

SeeKronecker's theorem for the geometric and Fourier theory attached to the number of modes. The closure of (the image of) any one-parameter subgroup inT is a subtorus of some dimensiond. In that subtorus the result of Kronecker applies: there ared 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 theergodicity of the motion: for anymeasurable subsetA ofT (for the usual probability measure), the average proportion of time spent by the motion inA is equal to the measure ofA.^[9]

The theory ofalmost periodic functions is, roughly speaking, for the same situation but allowingT to be a torus with an infinite number of dimensions. The early discussion of quasi-periodic functions, byErnest Esclangon following the work ofPiers Bohl, in fact led to a definition of almost-periodic function, the terminology ofHarald Bohr.^[10]Ian Stewart wrote that the default position of classicalcelestial mechanics, at this period, was that motions that could be described as quasiperiodic were the most complex that occurred.^[11] For theSolar 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.^[12] The research direction begun byAndrei Kolmogorov in the 1950s led to the understanding that quasiperiodic flow on phase space tori could survive perturbation.^[13]

NB: The concept ofquasiperiodic function, for example the sense in whichtheta functions and theWeierstrass zeta function incomplex analysis are said to havequasi-periods with respect to aperiod lattice, is something distinct from this topic.

• Quasiperiodicity
• Kolmogorov–Arnold–Moser theorem

• Dynamical systems

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