A complex quadratic polynomial is a
quadratic polynomial
In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
whose
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
s and
variable are
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 fo ...
s.
Properties
Quadratic polynomials have the following properties, regardless of the form:
*It is a unicritical polynomial, i.e. it has one
finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: basin of infinity and basin of finite critical point ( if finite critical point do not escapes)
*It can be postcritically finite, i.e. the orbit of the critical point can be finite, because the critical point is periodic or preperiodic.
* It is a
unimodal function,
* It is a
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
,
* It is an
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
.
Forms
When the quadratic polynomial has only one variable (
univariate
In mathematics, a univariate object is an expression, equation, function or polynomial involving only one variable. Objects involving more than one variable are multivariate. In some cases the distinction between the univariate and multivariate ...
), one can distinguish its four main forms:
* The general form:
where
* The factored form used for the
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 popula ...
:
*
which has an indifferent
fixed point with
multiplier at the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
* The
monic and centered form,
The monic and centered form has been studied extensively, and has the following properties:
* It is the simplest form of a
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
function with one coefficient (
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
),
* It is a centered polynomial (the sum of its critical points is zero).
* it is a
binomial
Binomial may refer to:
In mathematics
*Binomial (polynomial), a polynomial with two terms
*Binomial coefficient, numbers appearing in the expansions of powers of binomials
*Binomial QMF, a perfect-reconstruction orthogonal wavelet decomposition
* ...
The lambda form
is:
* the simplest non-trivial perturbation of unperturbated system
* "the first family of
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s in which explicit necessary and sufficient conditions are known for when a small divisor problem is stable"
Conjugation
Between forms
Since
is
affine
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine comb ...
conjugate to the general form of the quadratic polynomial it is often used to study
complex dynamics
Complex dynamics is the study of dynamical systems defined by iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions.
Techniques
*General
**Montel's theorem
** P ...
and to create images of
Mandelbrot,
Julia
Julia is usually a feminine given name. It is a Latinate feminine form of the name Julio and Julius. (For further details on etymology, see the Wiktionary entry "Julius".) The given name ''Julia'' had been in use throughout Late Antiquity (e.g ...
and
Fatou sets.
When one wants change from
to
:
:
When one wants change from
to
, the parameter transformation is
:
and the transformation between the variables in
and
is
:
With doubling map
There is semi-conjugacy between the
dyadic transformation (the doubling map) and the quadratic polynomial case of ''c'' = –2.
Notation
Iteration
Here
denotes the ''n''-th
iterate
Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...
of the function
:
:
so
:
Because of the possible confusion with exponentiation, some authors write
for the ''n''th iterate of
.
Parameter
The monic and centered form
can be marked by:
* the parameter
* the external angle
of the ray that lands:
** at ''c'' in Mandelbrot set on the parameter plane
** on the critical value:''z'' = ''c'' in Julia set on the dynamic plane
so :
:
:
Examples:
* c is the landing point of the 1/6
external ray of the
Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
, and is z->z^2+i (where i^2=-1)
* c is the landing point the 5/14
external ray and is z->z^2+c with c = -1.23922555538957 + 0.412602181602004*i
Paritition of dynamic plane of quadratic polynomial for 1 4.svg, 1/4
Paritition of dynamic plane of quadratic polynomial for 1 6.svg, 1/6
Paritition of dynamic plane of quadratic polynomial for 9 56.svg, 9/56
Paritition of dynamic plane of quadratic polynomial for 129 over 16256.svg, 129/16256
Map
The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials, is typically used with variable
and
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
:
:
When it is used as an
evolution function
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
of the
discrete nonlinear dynamical system
:
it is named the
quadratic map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
:
:
The
Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
is the set of values of the parameter ''c'' for which the initial condition ''z''
0 = 0 does not cause the iterates to diverge to infinity.
Critical items
Critical points
complex plane
A
critical point of
is a point
on
the dynamical plane such that the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
vanishes:
:
Since
:
implies
:
we see that the only (finite) critical point of
is the point
.
is an initial point for
Mandelbrot set
The Mandelbrot set () is the set of complex numbers c for which the function f_c(z)=z^2+c does not diverge to infinity when iterated from z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value.
This ...
iteration.
For the quadratic family
the critical point z = 0 is the
center of symmetry of the
Julia set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values wi ...
Jc, so it is a
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
combination of two points in Jc.
extended complex plane
In the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
polynomial has 2d-2 critical points. Here zero and
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions am ...
are critical points.
Critical value
A
critical value of
is the image of a critical point:
:
Since
:
we have
:
So the parameter
is the critical value of
.
Critical level curves
A critical level curve the level curve which contain critical point. It acts as a sort of skeleton of dynamical plane
Example : level curves cross at
saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the functi ...
, which is a special type of critical point.
Julia set for z^2+0.7i*z.png, attracting
IntLSM_J.jpg, attracting
ILSMJ.png, attracting
Level sets of attraction time to parabolic fixed point in the fat basilica Julia set.png, parabolic
Quadratic Julia set with Internal level sets for internal ray 0.ogv, Video for c along internal ray 0
Critical limit set
Critical limit set is the set of forward orbit of all critical points
Critical orbit
The
forward orbit of a critical point is called a critical orbit. Critical orbits are very important because every attracting
periodic orbit In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given a ...
attracts a critical point, so studying the critical orbits helps us understand the dynamics in the
Fatou set.
:
:
:
:
::
This orbit falls into an
attracting periodic cycle if one exists.
Critical sector
The
critical sector is a sector of the dynamical plane containing the critical point.
Critical set
Critical set is a set of critical points
Critical polynomial
:
so
:
:
:
:
These polynomials are used for:
* finding centers of these Mandelbrot set components of period ''n''. Centers are
roots
A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients.
Root or roots may also refer to:
Art, entertainment, and media
* ''The Root'' (magazine), an online magazine focusing ...
of ''n''-th critical polynomials
::
* finding roots of Mandelbrot set components of period ''n'' (
local minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
of
)
*
Misiurewicz points
::
Critical curves
Diagrams of critical polynomials are called critical curves.
These curves create the skeleton (the dark lines) of a
bifurcation diagram
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the sy ...
.
Spaces, planes
4D space
One can use the Julia-Mandelbrot 4-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
al (4D) space for a global analysis of this dynamical system.
In this space there are two basic types of 2D planes:
* the dynamical (dynamic) plane,
-plane or ''c''-plane
* the parameter plane or ''z''-plane
There is also another plane used to analyze such dynamical systems ''w''-plane:
* the conjugation plane
* model plane
2D Parameter plane
phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
of a quadratic map is called its parameter plane. Here:
is variable.
There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.
The parameter plane consists of:
* The
with
*** root points
** Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set with internal rays
* exterior of Mandelbrot set with
**
s
** equipotential lines
There are many different subtypes of the parameter plane.
which maps exterior of Mandelbrot set to the exterior of unit disc
* multiplier map which maps interior of hyperbolic component of Mandelbrot set to the interior of unit disc
of three-dimensional space of continuous dynamical system.
Dynamical ''z''-planes can be divided into two groups:
*
.
and referred to as the multiplier or the Lyapunov characteristic number. Its
is known as the Lyapunov exponent. Absolute value of multiplier is used to check the
This derivative is used for computing the external distance to the Julia set.