Quasiperiodic motion
Encyclopedia
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.
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
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 function
s is, roughly speaking, for the same situation but allowing T to be a torus with an infinite number of dimensions.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
containing a finite number (two or more) of incommensurable
Commensurability (mathematics)
In mathematics, two non-zero real numbers a and b are said to be commensurable if a/b is a rational number.-History of the concept:...
frequencies.
That is, if we imagine that the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
is modelled by a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
T, the trajectory of the system is modelled by a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
on T that wraps around without ever exactly coming back on itself.
A quasiperiodic function on the real line
Real line
In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one...
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
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
), 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
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
are said to have quasi-periods with respect to a period lattice is something distinct from this.)
The theory of almost periodic function
Almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov,...
s is, roughly speaking, for the same situation but allowing T to be a torus with an infinite number of dimensions.