Since is affineconjugate to the general form of the quadratic polynomial it is often used to study complex dynamics and to create images of Mandelbrot, Julia 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
There is semi-conjugacy between the dyadic transformation and the quadratic polynomial case of c = –2.
Notation
Iteration
Here denotes the n-th iteration of the function : so Because of the possible confusion with exponentiation, some authors write for the nth iterate of the function
Parameter
The monic and centered form can be marked by:
the parameter
the external angle of the ray that lands:
* at c in M on the parameter plane
* at z = c in J on the dynamic plane
so :
Map
The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials, is typically used with variable and parameter : When it is used as an evolution function of the discrete nonlinear dynamical system it is named the quadratic map: The Mandelbrot set is the set of values of the parameter c for which the initial conditionz0 = 0 does not cause the iterates to diverge to infinity.
Critical items
Critical point
A critical point of is a point in the dynamical plane such that the derivative vanishes: Since implies we see that the only critical point of is the point. is an initial point for Mandelbrot set iteration.
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 orbit
The forward orbit of a critical point is called a critical orbit. Critical orbits are very important because every attracting periodic orbit 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 polynomial
so These polynomials are used for:
finding centers of these Mandelbrot set components of period n. Centers are roots of n-th critical polynomials
finding roots of Mandelbrot set components of period n
Misiurewicz points
Critical curves
Diagrams of critical polynomials are called critical curves. These curves create the skeleton of a bifurcation diagram.
There is also another plane used to analyze such dynamical systems w-plane:
the conjugation plane
model plane
2D Parameter plane
The phase space of a quadratic map is called its parameter plane. Here: is constant and 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:
multiplier map which maps interior of hyperbolic component of Mandelbrot set to the interior of unit disc
2D Dynamical plane
"The polynomial Pc maps each dynamical ray to another ray doubling the angle, and the dynamical rays of any polynomial “look like straight rays” near infinity. This allows us to study the Mandelbrot and Julia sets combinatorially, replacing the dynamical plane by the unit circle, rays by angles, and the quadratic polynomial by the doubling modulo one map." Virpi K a u k o On the dynamical plane one can find:
Here, is a constant and is a variable. The two-dimensional dynamical plane can be treated as a Poincaré cross-section of three-dimensional space of continuous dynamical system. Dynamical z-planes can be divided in two groups :
The first derivative of with respect to c is This derivative can be found by iteration starting with and then replacing at every consecutive step This can easily be verified by using the chain rule for the derivative. This derivative is used in the distance estimation method for drawing a Mandelbrot set.
First derivative with respect to ''z''
On the dynamical plane:
is a variable;
is a constant.
At a fixed point At a periodic pointz0 of period p the first derivative of a function is often represented by and referred to as the multiplier or the Lyapunov characteristic number. Its logarithm is known as the Lyapunov exponent. It used to check the stability of periodic points. At a nonperiodic point, the derivative, denoted by can be found by iteration starting with and then using This derivative is used for computing the external distance to the Julia set.
