Complex quadratic polynomial
A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers. Properties. Quadratic polynomials have the following properties, regardless of the form: Forms. When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms: The monic and centered form has been studied extensively, and has the following properties: The lambda form formula_7 is: Conjugation. Between forms. Since formula_9 is affine conjugate 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 formula_10 to formula_11: When one wants change from formula_13 to formula_11, the parameter transformation is and the transformation between the variables in formula_16 and formula_17 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 formula_19 denotes the "n"-th iterate of the function formula_20: so Because of the possible confusion with exponentiation, some authors write formula_23 for the "n"th iterate of formula_20. Parameter. The monic and centered form formula_6 can be marked by: so : Map. The monic and centered  form, sometimes called the Douady-Hubbard family of quadratic polynomials, is typically used with variable formula_30 and parameter formula_11: 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 condition "z" = 0 does not cause the iterates to diverge to infinity. Critical items. Critical points. complex plane. A critical point of formula_35 is a point formula_36 on the dynamical plane such that the derivative vanishes: Since implies we see that the only (finite ) critical point of formula_35 is the point formula_41. formula_42 is an initial point for Mandelbrot set iteration. For the quadratic family formula_43 the critical point z = 0 is the center of symmetry of the Julia set Jc, so it is a convex combination of two points in Jc. extended complex plane. In the Riemann sphere polynomial has 2d-2 critical points. Here zero and infinity are critical points. Critical value. A critical value formula_44 of formula_35 is the image of a critical point: Since we have So the parameter formula_11 is the critical value of formula_50. 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, which is a special type of critical point. 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 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: Critical curves. Diagrams of critical polynomials are called critical curves. These curves create the skeleton (the dark lines) of a bifurcation diagram. Spaces, planes. 4D space. One can use the Julia-Mandelbrot 4-dimensional (4D) space for a global analysis of this dynamical system. In this space there are two basic types of 2D planes: There is also another plane used to  analyze such dynamical systems "w"-plane: 2D Parameter plane. The phase space of a quadratic map is called its parameter plane. Here: formula_65 is constant and formula_11 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: There are many different subtypes of the parameter plane. See also : 2D Dynamical plane. "The polynomial Pc maps each dynamical ray to another ray doubling the angle (which we measure in full turns, i.e. 0 = 1 = 2π rad = 360°), 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 KaukoOn the dynamical plane one can find: The dynamical plane consists of: Here, formula_11 is a constant and formula_30 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 into two groups: Riemann sphere. The extended complex plane  plus a point at infinity Derivatives. First derivative with respect to "c". On the parameter plane: The first derivative of formula_75 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: At a fixed point formula_42, At a periodic point "z" of period "p" the first derivative of a function is often  represented by formula_84 and referred to as the multiplier or the Lyapunov characteristic number. Its logarithm is known as the Lyapunov exponent. Absolute value of multiplier is used to check the stability of periodic (also fixed) points. At a nonperiodic point, the derivative, denoted by formula_85, can be found by iteration starting with and then using This derivative is used for computing the external distance to the Julia set. Schwarzian derivative. The Schwarzian derivative (SD for short) of "f" is: