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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the complex squaring map, a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
mapping of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
two 2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultur ...
, is a simple and accessible demonstration of
chaos Chaos or CHAOS may refer to: Arts, entertainment and media Fictional elements * Chaos (''Kinnikuman'') * Chaos (''Sailor Moon'') * Chaos (''Sesame Park'') * Chaos (''Warhammer'') * Chaos, in ''Fabula Nova Crystallis Final Fantasy'' * Cha ...
in
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s. It can be constructed by performing the following steps: # Choose any
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
on the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
whose
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
(angle) is not a
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
multiple of π, # Repeatedly square that number. This repetition (iteration) produces a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
of complex numbers that can be described alone by their arguments. Any choice of starting angle that satisfies (1) above will produce an extremely complicated sequence of angles, that belies the simplicity of the steps. It can be shown that the sequence will be
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
, i.e. it is sensitive to the detailed choice of starting angle.


Chaos and the complex squaring map

The informal reason why the iteration is chaotic is that the angle doubles on every iteration and doubling grows very quickly as the angle becomes ever larger, but angles which differ by multiples of 2π (
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s) are identical. Thus, when the angle exceeds 2π, it must ''wrap'' to the remainder on division by 2π. Therefore, the angle is transformed according to the
dyadic transformation The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) : T: , 1) \to forward orbit of ''z''''n'' cannot repeat itself and become periodic. More formally, the iteration can be written as :z_ = z_n^2 where z_n is the resulting sequence of complex numbers obtained by iterating the steps above, and z_0 represents the initial starting number. We can solve this iteration exactly: :z_n = z_0^ Starting with angle ''θ'', we can write the initial term as z_0 = \exp(i\theta) so that z_n = \exp(i2^n\theta). This makes the successive doubling of the angle clear. (This is equivalent to the relation z_n = \cos(2^n\theta)+i \sin(2^n\theta) by Euler's formula.)


Generalisations

This map is a special case of the complex quadratic map, which has exact solutions for many special cases.M. Little, D. Heesch (2004)
Chaotic root-finding for a small class of polynomials
''Journal of Difference Equations and Applications'', 10(11):949–953. The complex map obtained by raising the previous number to any
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
power z_ = z_n^p is also exactly solvable as z_n = z_0^. In the case ''p'' = 2, the dynamics can be mapped to the dyadic transformation, as described above, but for ''p'' > 2, we obtain a shift map in the
number base In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
 ''p''. For example, ''p'' = 10 is a decimal shift map.


See also

*
Logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popular ...
*
Dyadic transformation The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) : T: , 1) \to , 1)^\infty : x \mapsto (x_0, x_1, x_2, ...


References

{{Chaos theory Chaotic maps