Quasiperiodicity
Encyclopedia
Quasiperiodicity is the property of a system
that displays irregular periodicity
. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpredictability that does not lend itself to precise measurement. It is different from the mathematical concept of an almost periodic function
, which has increasing regularity over multiple periods. It should also be noted that the precise mathematical definition of quasiperiodic function
is a completely different concept; the two should not be confused.
, quasiperiodic is a term used to denote oscillations that appear to follow a regular pattern but which do not have a fixed period.
Within a dynamical system
such as the ocean-atmosphere oscillations may occur regularly, when they are forced by a regular external forcing: for example, the familiar winter-summer cycle is forced by variations in sunlight from the (very close to perfectly) periodic motion of the earth around the sun. Or, like the recent ice age
cycles, they may be less regular but still locked by external forcing. However, when the system contains the potential for an oscillation, but there is no strong external forcing it to be phase-locked to it, the "period" is likely to be irregular.
The canonical example of quasiperiodicity in climatology is El Niño-Southern Oscillation
. In the modern era, it has a "period" somewhere between four to twelve years and a peak spectral density
around five years.
System
System is a set of interacting or interdependent components forming an integrated whole....
that displays irregular periodicity
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...
. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is a pattern of recurrence with a component of unpredictability that does not lend itself to precise measurement. It is different from the mathematical concept of an 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,...
, which has increasing regularity over multiple periods. It should also be noted that the precise mathematical definition of quasiperiodic function
Quasiperiodic function
In mathematics, a function is said to be quasiperiodic when it has some similarity to a periodic function but does not meet the strict definition.A simple case is if the function obeys the equation:...
is a completely different concept; the two should not be confused.
Climatology
In climatologyClimatology
Climatology is the study of climate, scientifically defined as weather conditions averaged over a period of time, and is a branch of the atmospheric sciences...
, quasiperiodic is a term used to denote oscillations that appear to follow a regular pattern but which do not have a fixed period.
Within 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...
such as the ocean-atmosphere oscillations may occur regularly, when they are forced by a regular external forcing: for example, the familiar winter-summer cycle is forced by variations in sunlight from the (very close to perfectly) periodic motion of the earth around the sun. Or, like the recent ice age
Ice age
An ice age or, more precisely, glacial age, is a generic geological period of long-term reduction in the temperature of the Earth's surface and atmosphere, resulting in the presence or expansion of continental ice sheets, polar ice sheets and alpine glaciers...
cycles, they may be less regular but still locked by external forcing. However, when the system contains the potential for an oscillation, but there is no strong external forcing it to be phase-locked to it, the "period" is likely to be irregular.
The canonical example of quasiperiodicity in climatology is El Niño-Southern Oscillation
El Niño-Southern Oscillation
El Niño/La Niña-Southern Oscillation, or ENSO, is a quasiperiodic climate pattern that occurs across the tropical Pacific Ocean roughly every five years...
. In the modern era, it has a "period" somewhere between four to twelve years and a peak spectral density
Spectral density
In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...
around five years.
See also
- QuasicrystalQuasicrystalA quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry...
- Quasiperiodic functionQuasiperiodic functionIn mathematics, a function is said to be quasiperiodic when it has some similarity to a periodic function but does not meet the strict definition.A simple case is if the function obeys the equation:...
- Quasiperiodic motionQuasiperiodic motionIn mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number of incommensurable frequencies....
- Quasi-periodic oscillationsQuasi-periodic oscillationsIn X-ray astronomy, quasi-periodic oscillation is the manner in which the X-ray light from an astronomical object flickers about certain frequencies...
- Quasiperiodic tilingQuasiperiodic tilingA quasiperiodic tiling is a tiling of the plane that exhibits local periodicity under some transformations; we can slide or rotate it such that a finite number of tiles overlap perfectly, yet the entire tiling will not.See...