Complex quadratic polynomial

<h2 id="properties">Properties</h2>
<p>Quadratic polynomials have the following properties, regardless of the form:
</p>
<ul><li>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)</li>
<li>It can be postcritically finite, i.e. the orbit of the critical point can be finite, because the critical point is periodic or preperiodic.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li>
<li>It is a <a href="/facts/Unimodality/Xo3bhgeE">unimodal function</a>,</li>
<li>It is a <a href="/facts/Rational_function/2nJGu0Ko">rational function</a>,</li>
<li>It is an <a href="/facts/Entire_function/XNluKzu7">entire function</a>.</li></ul>
<h2 id="forms">Forms</h2>
<p>When the quadratic polynomial has only one variable (<a href="/facts/Univariate/PYwC7u5H">univariate</a>), one can distinguish its four main forms:
</p>
<ul><li>The general form: 
  
    
      
        f
        (
        x
        )
        =
        
          a
          
            2
          
        
        
          x
          
            2
          
        
        +
        
          a
          
            1
          
        
        x
        +
        
          a
          
            0
          
        
      
    
    {\displaystyle f(x)=a_{2}x^{2}+a_{1}x+a_{0}}
  
 where 
  
    
      
        
          a
          
            2
          
        
        ≠
        0
      
    
    {\displaystyle a_{2}\neq 0}
  
</li>
<li>The factored form used for the <a href="/facts/Logistic_map/wskv7mRQ">logistic map</a>: 
  
    
      
        
          f
          
            r
          
        
        (
        x
        )
        =
        r
        x
        (
        1
        −
        x
        )
      
    
    {\displaystyle f_{r}(x)=rx(1-x)}
  
</li>
<li>
  
    
      
        
          f
          
            θ
          
        
        (
        x
        )
        =
        
          x
          
            2
          
        
        +
        λ
        x
      
    
    {\displaystyle f_{\theta }(x)=x^{2}+\lambda x}
  
 which has an indifferent <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed point</a> with <a href="/facts/Periodic_points_of_complex_quadratic_mappings/IlRwhEdR">multiplier</a> 
  
    
      
        λ
        =
        
          e
          
            2
            π
            θ
            i
          
        
      
    
    {\displaystyle \lambda =e^{2\pi \theta i}}
  
 at the <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li>
<li>The <a href="/facts/Monic_polynomial/4JzuIJ5R">monic</a> and centered form, 
  
    
      
        
          f
          
            c
          
        
        (
        x
        )
        =
        
          x
          
            2
          
        
        +
        c
      
    
    {\displaystyle f_{c}(x)=x^{2}+c}
  
</li></ul>
<p>The monic and centered form has been studied extensively, and has the following properties:
</p>
<ul><li>It is the simplest form of a <a href="/facts/Nonlinearity_(disambiguation)/XXFrzHka">nonlinear</a> <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> with one coefficient (<a href="/facts/Parameter/zFvVFMv9">parameter</a>),</li>
<li>It is a centered polynomial (the sum of its critical points is zero).<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li>
<li>it is a <a href="/facts/Binomial_(polynomial)/cfFcsR8h">binomial</a></li></ul>
<p>The lambda form 
  
    
      
        
          f
          
            λ
          
        
        (
        z
        )
        =
        
          z
          
            2
          
        
        +
        λ
        z
      
    
    {\displaystyle f_{\lambda }(z)=z^{2}+\lambda z}
  
 is:
</p>
<ul><li>the simplest non-trivial perturbation of unperturbated system 
  
    
      
        z
        ↦
        λ
        z
      
    
    {\displaystyle z\mapsto \lambda z}
  
</li>
<li>"the first family of <a href="/facts/Dynamical_system/782gREl5">dynamical systems</a> in which explicit necessary and sufficient conditions are known for when a small divisor problem is stable"<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li></ul>
<h2 id="conjugation">Conjugation</h2>
<h3>Between forms</h3>
<p>Since 
  
    
      
        
          f
          
            c
          
        
        (
        x
        )
      
    
    {\displaystyle f_{c}(x)}
  
 is <a href="/facts/Affine_transformation/BD7yDr10">affine</a> <a href="/facts/Topological_conjugation/vQvYmcIA">conjugate</a> to the general form of the quadratic polynomial it is often used to study <a href="/facts/Complex_dynamics/NXHJfR3s">complex dynamics</a> and to create images of <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot</a>, <a href="/facts/Julia_set/dmn8xIcG">Julia</a> and <a href="/facts/Fatou_set/dmn8xIcG">Fatou sets</a>.
</p><p>When one wants change from 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
  to 
  
    
      
        c
      
    
    {\displaystyle c}
  
:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>

c
        =
        c
        (
        θ
        )
        =
        
          
            
              e
              
                2
                π
                θ
                i
              
            
            2
          
        
        
          (
          
            1
            −
            
              
                
                  e
                  
                    2
                    π
                    θ
                    i
                  
                
                2
              
            
          
          )
        
        .
      
    
    {\displaystyle c=c(\theta )={\frac {e^{2\pi \theta i}}{2}}\left(1-{\frac {e^{2\pi \theta i}}{2}}\right).}

<p>When one wants change from 
  
    
      
        r
      
    
    {\displaystyle r}
  
 to 
  
    
      
        c
      
    
    {\displaystyle c}
  
, the parameter transformation is<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>

c
        =
        c
        (
        r
        )
        =
        
          
            
              1
              −
              (
              r
              −
              1
              
                )
                
                  2
                
              
            
            4
          
        
        =
        −
        
          
            r
            2
          
        
        
          (
          
            
              
                r
                −
                2
              
              2
            
          
          )
        
      
    
    {\displaystyle c=c(r)={\frac {1-(r-1)^{2}}{4}}=-{\frac {r}{2}}\left({\frac {r-2}{2}}\right)}

<p>and the transformation between the variables in 
  
    
      
        
          z
          
            t
            +
            1
          
        
        =
        
          z
          
            t
          
          
            2
          
        
        +
        c
      
    
    {\displaystyle z_{t+1}=z_{t}^{2}+c}
  
 and 
  
    
      
        
          x
          
            t
            +
            1
          
        
        =
        r
        
          x
          
            t
          
        
        (
        1
        −
        
          x
          
            t
          
        
        )
      
    
    {\displaystyle x_{t+1}=rx_{t}(1-x_{t})}
  
 is
</p>

z
        =
        r
        
          (
          
            
              
                1
                2
              
            
            −
            x
          
          )
        
        .
      
    
    {\displaystyle z=r\left({\frac {1}{2}}-x\right).}

<h3>With doubling map</h3>
<p>There is semi-conjugacy between the <a href="/facts/Dyadic_transformation/Hopsjd88">dyadic transformation</a> (the doubling map) and the quadratic polynomial case of <i>c</i> = –2.
</p>
<h2 id="notation">Notation</h2>
<h3>Iteration</h3>
<p>Here 
  
    
      
        
          f
          
            n
          
        
      
    
    {\displaystyle f^{n}}
  
 denotes the <i>n</i>-th <a href="/facts/Iterated_function/Vzsx64An">iterate</a> of the function 
  
    
      
        f
      
    
    {\displaystyle f}
  
:
</p>

f
          
            c
          
          
            n
          
        
        (
        z
        )
        =
        
          f
          
            c
          
          
            1
          
        
        (
        
          f
          
            c
          
          
            n
            −
            1
          
        
        (
        z
        )
        )
      
    
    {\displaystyle f_{c}^{n}(z)=f_{c}^{1}(f_{c}^{n-1}(z))}

z
          
            n
          
        
        =
        
          f
          
            c
          
          
            n
          
        
        (
        
          z
          
            0
          
        
        )
        .
      
    
    {\displaystyle z_{n}=f_{c}^{n}(z_{0}).}

<p>Because of the possible confusion with exponentiation, some authors write 
  
    
      
        
          f
          
            ∘
            n
          
        
      
    
    {\displaystyle f^{\circ n}}
  
 for the <i>n</i>th iterate of 
  
    
      
        f
      
    
    {\displaystyle f}
  
.
</p>
<h3>Parameter</h3>
<p>The monic and centered form 
  
    
      
        
          f
          
            c
          
        
        (
        x
        )
        =
        
          x
          
            2
          
        
        +
        c
      
    
    {\displaystyle f_{c}(x)=x^{2}+c}
  
 can be marked by:
</p>
<ul><li>the parameter 
  
    
      
        c
      
    
    {\displaystyle c}
  
</li>
<li>the external angle 
  
    
      
        θ
      
    
    {\displaystyle \theta }
  
 of the ray that lands:
<ul><li>at <i>c</i> in Mandelbrot set on the parameter plane</li>
<li>on the critical value:<i>z</i> = <i>c</i> in Julia set on the dynamic plane</li></ul></li></ul>
<p>so :
</p>

f
          
            c
          
        
        =
        
          f
          
            θ
          
        
      
    
    {\displaystyle f_{c}=f_{\theta }}

c
        =
        c
        (
        
          θ
        
        )
      
    
    {\displaystyle c=c({\theta })}

<p>Examples:
</p>
<ul><li>c is the landing point of the 1/6 <a href="/facts/External_ray/bR9yUYuq">external ray</a> of the <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a>, and is 
  
    
      
        z
        →
        
          z
          
            2
          
        
        +
        i
      
    
    {\displaystyle z\to z^{2}+i}
  
 (where i^2=-1)</li>
<li>c is the landing point the 5/14 <a href="/facts/External_ray/bR9yUYuq">external ray</a> and is 
  
    
      
        z
        →
        
          z
          
            2
          
        
        +
        c
      
    
    {\displaystyle z\to z^{2}+c}
  
 with 
  
    
      
        c
        =
        −
        1.23922555538957
        +
        0.412602181602004
        ∗
        i
      
    
    {\displaystyle c=-1.23922555538957+0.412602181602004*i}
  
</li></ul>

<h3>Map</h3>
<p>The monic and centered form, sometimes called the Douady-Hubbard family of quadratic polynomials,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> is typically used with variable 
  
    
      
        z
      
    
    {\displaystyle z}
  
 and <a href="/facts/Parameter/zFvVFMv9">parameter</a> 
  
    
      
        c
      
    
    {\displaystyle c}
  
:
</p>

f
          
            c
          
        
        (
        z
        )
        =
        
          z
          
            2
          
        
        +
        c
        .
      
    
    {\displaystyle f_{c}(z)=z^{2}+c.}

<p>When it is used as an <a href="/facts/Dynamical_system_(definition)/hcaspdq4">evolution function</a> of the <a href="/facts/Dynamical_system/782gREl5">discrete nonlinear dynamical system</a>
</p>

z
          
            n
            +
            1
          
        
        =
        
          f
          
            c
          
        
        (
        
          z
          
            n
          
        
        )
      
    
    {\displaystyle z_{n+1}=f_{c}(z_{n})}

<p>it is named the <a href="/facts/Quadratic_polynomial/mKxHIC3U">quadratic</a> <a href="/facts/Map_(mathematics)/FABF1l2f">map</a>:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p>

f
          
            c
          
        
        :
        z
        →
        
          z
          
            2
          
        
        +
        c
        .
      
    
    {\displaystyle f_{c}:z\to z^{2}+c.}

<p>The <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a> is the set of values of the parameter <i>c</i> for which the initial condition <i>z</i>0 = 0 does not cause the iterates to diverge to infinity.
</p>
<h2 id="critical-items">Critical items</h2>
<h3>Critical points</h3>
<h4>complex plane</h4>
<p>A <a href="/facts/Critical_point_(mathematics)/MrEk9PnY">critical point</a> of 
  
    
      
        
          f
          
            c
          
        
      
    
    {\displaystyle f_{c}}
  
 is a point 
  
    
      
        
          z
          
            c
            r
          
        
      
    
    {\displaystyle z_{cr}}
  
 on the dynamical plane such that the <a href="/facts/Derivative/7Ito70Dh">derivative</a> vanishes:
</p>

f
          
            c
          
          ′
        
        (
        
          z
          
            c
            r
          
        
        )
        =
        0.
      
    
    {\displaystyle f_{c}'(z_{cr})=0.}

<p>Since
</p>

f
          
            c
          
          ′
        
        (
        z
        )
        =
        
          
            d
            
              d
              z
            
          
        
        
          f
          
            c
          
        
        (
        z
        )
        =
        2
        z
      
    
    {\displaystyle f_{c}'(z)={\frac {d}{dz}}f_{c}(z)=2z}

<p>implies
</p>

z
          
            c
            r
          
        
        =
        0
        ,
      
    
    {\displaystyle z_{cr}=0,}

<p>we see that the only (finite) critical point of 
  
    
      
        
          f
          
            c
          
        
      
    
    {\displaystyle f_{c}}
  
 is the point 
  
    
      
        
          z
          
            c
            r
          
        
        =
        0
      
    
    {\displaystyle z_{cr}=0}
  
.
</p><p>
  
    
      
        
          z
          
            0
          
        
      
    
    {\displaystyle z_{0}}
  
 is an initial point for <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a> iteration.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
</p><p>For the quadratic family 
  
    
      
        
          f
          
            c
          
        
        (
        z
        )
        =
        
          z
          
            2
          
        
        +
        c
      
    
    {\displaystyle f_{c}(z)=z^{2}+c}
  
 the critical point z = 0 is the <a href="/facts/Center_of_symmetry/NLmmEsue">center of symmetry</a> of the <a href="/facts/Julia_set/dmn8xIcG">Julia set</a> Jc, so it is a <a href="/facts/Convex_hull/D4Id3JDS">convex</a> combination of two points in Jc.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>
</p>
<h4>Extended complex plane</h4>
<p>In the <a href="/facts/Riemann_sphere/4XgBzplC">Riemann sphere</a> polynomial has 2d-2 critical points. Here zero and <a href="/facts/Point_at_infinity/GAuS0wHQ">infinity</a> are critical points.
</p>
<h3>Critical value</h3>
<p>A critical value 
  
    
      
        
          z
          
            c
            v
          
        
      
    
    {\displaystyle z_{cv}}
  
 of 
  
    
      
        
          f
          
            c
          
        
      
    
    {\displaystyle f_{c}}
  
 is the image of a critical point:
</p>

z
          
            c
            v
          
        
        =
        
          f
          
            c
          
        
        (
        
          z
          
            c
            r
          
        
        )
      
    
    {\displaystyle z_{cv}=f_{c}(z_{cr})}

<p>Since
</p>

z
          
            c
            r
          
        
        =
        0
      
    
    {\displaystyle z_{cr}=0}

z
          
            c
            v
          
        
        =
        c
      
    
    {\displaystyle z_{cv}=c}

<p>So the parameter 
  
    
      
        c
      
    
    {\displaystyle c}
  
 is the critical value of 
  
    
      
        
          f
          
            c
          
        
        (
        z
        )
      
    
    {\displaystyle f_{c}(z)}
  
.
</p>
<h3>Critical level curves</h3>
<p>A critical level curve the level curve which contain critical point. It acts as a sort of skeleton<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> of dynamical plane
</p><p>Example : level curves cross at <a href="/facts/Saddle_point/v5MT6YbM">saddle point</a>, which is a special type of critical point.
</p>

<h3>Critical limit set</h3>
<p>Critical limit set is the set of forward orbit of all critical points
</p>
<h3>Critical orbit</h3>

<p>The <a href="/facts/Orbit_(dynamics)/WBpc3bUH">forward orbit</a> of a critical point is called a critical orbit. Critical orbits are very important because every attracting <a href="/facts/Periodic_points_of_complex_quadratic_mappings/IlRwhEdR">periodic orbit</a> attracts a critical point, so studying the critical orbits helps us understand the dynamics in the <a href="/facts/Fatou_set/dmn8xIcG">Fatou set</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a>
</p>

z
          
            0
          
        
        =
        
          z
          
            c
            r
          
        
        =
        0
      
    
    {\displaystyle z_{0}=z_{cr}=0}

z
          
            1
          
        
        =
        
          f
          
            c
          
        
        (
        
          z
          
            0
          
        
        )
        =
        c
      
    
    {\displaystyle z_{1}=f_{c}(z_{0})=c}

z
          
            2
          
        
        =
        
          f
          
            c
          
        
        (
        
          z
          
            1
          
        
        )
        =
        
          c
          
            2
          
        
        +
        c
      
    
    {\displaystyle z_{2}=f_{c}(z_{1})=c^{2}+c}

z
          
            3
          
        
        =
        
          f
          
            c
          
        
        (
        
          z
          
            2
          
        
        )
        =
        (
        
          c
          
            2
          
        
        +
        c
        
          )
          
            2
          
        
        +
        c
      
    
    {\displaystyle z_{3}=f_{c}(z_{2})=(c^{2}+c)^{2}+c}

⋮
      
    
    {\displaystyle \ \vdots }

<p>This orbit falls into an <a href="/facts/Periodic_points_of_complex_quadratic_mappings/IlRwhEdR">attracting periodic cycle</a> if one exists.
</p>
<h3>Critical sector</h3>
<p>The <a href="/facts/Orbit_portrait/tMxWIa5Q">critical sector</a> is a sector of the dynamical plane containing the critical point.
</p>
<h3>Critical set</h3>
<p>Critical set is a set of critical points
</p>
<h3>Critical polynomial</h3>

P
          
            n
          
        
        (
        c
        )
        =
        
          f
          
            c
          
          
            n
          
        
        (
        
          z
          
            c
            r
          
        
        )
        =
        
          f
          
            c
          
          
            n
          
        
        (
        0
        )
      
    
    {\displaystyle P_{n}(c)=f_{c}^{n}(z_{cr})=f_{c}^{n}(0)}

P
          
            0
          
        
        (
        c
        )
        =
        0
      
    
    {\displaystyle P_{0}(c)=0}

P
          
            1
          
        
        (
        c
        )
        =
        c
      
    
    {\displaystyle P_{1}(c)=c}

P
          
            2
          
        
        (
        c
        )
        =
        
          c
          
            2
          
        
        +
        c
      
    
    {\displaystyle P_{2}(c)=c^{2}+c}

P
          
            3
          
        
        (
        c
        )
        =
        (
        
          c
          
            2
          
        
        +
        c
        
          )
          
            2
          
        
        +
        c
      
    
    {\displaystyle P_{3}(c)=(c^{2}+c)^{2}+c}

<p>These polynomials are used for:
</p>
<ul><li>finding centers of these Mandelbrot set components of period <i>n</i>. Centers are <a href="/facts/Root_of_a_polynomial/mlK3dhoT">roots</a> of <i>n</i>-th critical polynomials</li></ul>

centers
        
        =
        {
        c
        :
        
          P
          
            n
          
        
        (
        c
        )
        =
        0
        }
      
    
    {\displaystyle {\text{centers}}=\{c:P_{n}(c)=0\}}

<ul><li>finding roots of Mandelbrot set components of period <i>n</i> (<a href="/facts/Local_minimum/ADlgnyzV">local minimum</a> of 
  
    
      
        
          P
          
            n
          
        
        (
        c
        )
      
    
    {\displaystyle P_{n}(c)}
  
)</li>
<li><a href="/facts/Misiurewicz_point/qP4DDoxw">Misiurewicz points</a></li></ul>

M
          
            n
            ,
            k
          
        
        =
        {
        c
        :
        
          P
          
            k
          
        
        (
        c
        )
        =
        
          P
          
            k
            +
            n
          
        
        (
        c
        )
        }
      
    
    {\displaystyle M_{n,k}=\{c:P_{k}(c)=P_{k+n}(c)\}}

<h3>Critical curves</h3>

<p>Diagrams of critical polynomials are called critical curves.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a>
</p><p>These curves create the skeleton (the dark lines) of a <a href="/facts/Bifurcation_diagram/NIAKhdDr">bifurcation diagram</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a>
</p>
<h2 id="spaces-planes">Spaces, planes</h2>
<h3>4D space</h3>
<p>One can use the Julia-Mandelbrot 4-<a href="/facts/Dimension/0ktmUyac">dimensional</a> (4D) space for a global analysis of this dynamical system.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a>
</p>

<p>In this space there are two basic types of 2D planes:
</p>
<ul><li>the dynamical (dynamic) plane, 
  
    
      
        
          f
          
            c
          
        
      
    
    {\displaystyle f_{c}}
  
-plane or  <i>c</i>-plane</li>
<li>the parameter plane or <i>z</i>-plane</li></ul>
<p>There is also another plane used to analyze such dynamical systems  <i>w</i>-plane: 
</p>
<ul><li>the conjugation plane<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li>
<li>model plane<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a></li></ul>
<h4>2D Parameter plane</h4>

<p>The <a href="/facts/Phase_space/qPe4Fq57">phase space</a> of a quadratic map is called its parameter plane. Here:
</p><p>
  
    
      
        
          z
          
            0
          
        
        =
        
          z
          
            c
            r
          
        
      
    
    {\displaystyle z_{0}=z_{cr}}
  
 is constant and 
  
    
      
        c
      
    
    {\displaystyle c}
  
 is variable.
</p><p>There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.
</p><p>The parameter plane consists of:
</p>
<ul><li>The <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a>
<ul><li>The <a href="/facts/Bifurcation_locus/LjJF7CsS">bifurcation locus</a> = boundary of <a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a> with
<ul><li>root points</li></ul></li>
<li>Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> with internal rays</li></ul></li>
<li>exterior of Mandelbrot set with
<ul><li><a href="/facts/External_ray/bR9yUYuq">external rays</a></li>
<li>equipotential lines</li></ul></li></ul>
<p>There are many different subtypes of the parameter plane.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a><a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a>
</p>

<p>See also :
</p>
<ul><li><a href="/facts/External_ray/bR9yUYuq">Boettcher map</a> which maps exterior of Mandelbrot set to the exterior of unit disc</li>
<li>multiplier map which maps interior of hyperbolic component of Mandelbrot set to the interior of unit disc</li></ul>
<h4>2D Dynamical plane</h4>
<blockquote><p>"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 Kauko<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a></p></blockquote><p>On the dynamical plane one can find:
</p><ul><li>The <a href="/facts/Julia_set/dmn8xIcG">Julia set</a></li>
<li>The <a href="/facts/Filled_Julia_set/zGGSFDas">Filled Julia set</a></li>
<li>The <a href="/facts/Fatou_set/dmn8xIcG">Fatou set</a></li>
<li>Orbits</li></ul>
<p>The dynamical plane consists of:
</p>
<ul><li><a href="/facts/Fatou_set/dmn8xIcG">Fatou set</a></li>
<li><a href="/facts/Julia_set/dmn8xIcG">Julia set</a></li></ul>
<p>Here, 
  
    
      
        c
      
    
    {\displaystyle c}
  
 is a constant and 
  
    
      
        z
      
    
    {\displaystyle z}
  
 is a variable.
</p><p>The two-dimensional dynamical plane can be treated as a <a href="/facts/Poincar%25C3%25A9_map/pd15LII2">Poincaré cross-section</a> of three-dimensional space of continuous dynamical system.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a><a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a>
</p><p>Dynamical <i>z</i>-planes can be divided into two groups:
</p>
<ul><li>
  
    
      
        
          f
          
            0
          
        
      
    
    {\displaystyle f_{0}}
  
 plane for 
  
    
      
        c
        =
        0
      
    
    {\displaystyle c=0}
  
 (see <a href="/facts/Complex_squaring_map/TxKqC4tp">complex squaring map</a>)</li>
<li>
  
    
      
        
          f
          
            c
          
        
      
    
    {\displaystyle f_{c}}
  
 planes (all other planes for 
  
    
      
        c
        ≠
        0
      
    
    {\displaystyle c\neq 0}
  
)</li></ul>
<h3>Riemann sphere</h3>
<p>The extended complex plane plus a <a href="/facts/Point_at_infinity/GAuS0wHQ">point at infinity</a>
</p>
<ul><li><a href="/facts/Riemann_sphere/4XgBzplC">the Riemann sphere</a></li></ul>
<h2 id="derivatives">Derivatives</h2>
<h3>First derivative with respect to <i>c</i></h3>
<p>On the parameter plane:
</p>
<ul><li>
  
    
      
        c
      
    
    {\displaystyle c}
  
 is a variable</li>
<li>
  
    
      
        
          z
          
            0
          
        
        =
        0
      
    
    {\displaystyle z_{0}=0}
  
 is constant</li></ul>
<p>The first <a href="/facts/Derivative/7Ito70Dh">derivative</a> of 
  
    
      
        
          f
          
            c
          
          
            n
          
        
        (
        
          z
          
            0
          
        
        )
      
    
    {\displaystyle f_{c}^{n}(z_{0})}
  
 with respect to <i>c</i> is
</p>

z
          
            n
          
          ′
        
        =
        
          
            d
            
              d
              c
            
          
        
        
          f
          
            c
          
          
            n
          
        
        (
        
          z
          
            0
          
        
        )
        .
      
    
    {\displaystyle z_{n}'={\frac {d}{dc}}f_{c}^{n}(z_{0}).}

<p>This derivative can be found by <a href="/facts/Iterated_function/Vzsx64An">iteration</a> starting with
</p>

z
          
            0
          
          ′
        
        =
        
          
            d
            
              d
              c
            
          
        
        
          f
          
            c
          
          
            0
          
        
        (
        
          z
          
            0
          
        
        )
        =
        1
      
    
    {\displaystyle z_{0}'={\frac {d}{dc}}f_{c}^{0}(z_{0})=1}

<p>and then replacing at every consecutive step
</p>

z
          
            n
            +
            1
          
          ′
        
        =
        
          
            d
            
              d
              c
            
          
        
        
          f
          
            c
          
          
            n
            +
            1
          
        
        (
        
          z
          
            0
          
        
        )
        =
        2
        ⋅

f
          
            c
          
          
            n
          
        
        (
        z
        )
        ⋅
        
          
            d
            
              d
              c
            
          
        
        
          f
          
            c
          
          
            n
          
        
        (
        
          z
          
            0
          
        
        )
        +
        1
        =
        2
        ⋅
        
          z
          
            n
          
        
        ⋅
        
          z
          
            n
          
          ′
        
        +
        1.
      
    
    {\displaystyle z_{n+1}'={\frac {d}{dc}}f_{c}^{n+1}(z_{0})=2\cdot {}f_{c}^{n}(z)\cdot {\frac {d}{dc}}f_{c}^{n}(z_{0})+1=2\cdot z_{n}\cdot z_{n}'+1.}

<p>This can easily be verified by using the <a href="/facts/Chain_rule/aMLYxP0x">chain rule</a> for the derivative.
</p><p>This derivative is used in the <a href="/facts/Mandelbrot_set/2NVUgrVG">distance estimation method for drawing a Mandelbrot set</a>.
</p>
<h3>First derivative with respect to <i>z</i></h3>
<p>On the dynamical plane:
</p>
<ul><li>
  
    
      
        z
      
    
    {\displaystyle z}
  
 is a variable;</li>
<li>
  
    
      
        c
      
    
    {\displaystyle c}
  
 is a constant.</li></ul>
<p>At a fixed point 
  
    
      
        
          z
          
            0
          
        
      
    
    {\displaystyle z_{0}}
  
,
</p>

f
          
            c
          
          ′
        
        (
        
          z
          
            0
          
        
        )
        =
        
          
            d
            
              d
              z
            
          
        
        
          f
          
            c
          
        
        (
        
          z
          
            0
          
        
        )
        =
        2
        
          z
          
            0
          
        
        .
      
    
    {\displaystyle f_{c}'(z_{0})={\frac {d}{dz}}f_{c}(z_{0})=2z_{0}.}

<p>At a <a href="/facts/Periodic_points_of_complex_quadratic_mappings/IlRwhEdR">periodic point</a> <i>z</i>0 of period <i>p</i> the first derivative of a function
</p>

(
        
          f
          
            c
          
          
            p
          
        
        
          )
          ′
        
        (
        
          z
          
            0
          
        
        )
        =
        
          
            d
            
              d
              z
            
          
        
        
          f
          
            c
          
          
            p
          
        
        (
        
          z
          
            0
          
        
        )
        =
        
          ∏
          
            i
            =
            0
          
          
            p
            −
            1
          
        
        
          f
          
            c
          
          ′
        
        (
        
          z
          
            i
          
        
        )
        =
        
          2
          
            p
          
        
        
          ∏
          
            i
            =
            0
          
          
            p
            −
            1
          
        
        
          z
          
            i
          
        
        =
        λ
      
    
    {\displaystyle (f_{c}^{p})'(z_{0})={\frac {d}{dz}}f_{c}^{p}(z_{0})=\prod _{i=0}^{p-1}f_{c}'(z_{i})=2^{p}\prod _{i=0}^{p-1}z_{i}=\lambda }

<p>is often represented by 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 and referred to as the multiplier or the Lyapunov characteristic number. Its <a href="/facts/Logarithm/QeXaV97q">logarithm</a> is known as the Lyapunov exponent. Absolute value of multiplier is used to check the <a href="/facts/Stability_theory/Su5sHnxs">stability</a> of <a href="/facts/Periodic_points_of_complex_quadratic_mappings/IlRwhEdR">periodic (also fixed) points</a>.
</p><p>At a nonperiodic point, the derivative, denoted by 
  
    
      
        
          z
          
            n
          
          ′
        
      
    
    {\displaystyle z'_{n}}
  
, can be found by <a href="/facts/Iterated_function/Vzsx64An">iteration</a> starting with
</p>

z
          
            0
          
          ′
        
        =
        1
        ,
      
    
    {\displaystyle z'_{0}=1,}

<p>and then using
</p>

z
          
            n
          
          ′
        
        =
        2
        ∗
        
          z
          
            n
            −
            1
          
        
        ∗
        
          z
          
            n
            −
            1
          
          ′
        
        .
      
    
    {\displaystyle z'_{n}=2*z_{n-1}*z'_{n-1}.}

<p>This derivative is used for computing the external distance to the Julia set.
</p>
<h3>Schwarzian derivative</h3>
<p>The <a href="/facts/Schwarzian_derivative/acRAssYR">Schwarzian derivative</a> (SD for short) of <i>f</i> is:<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a>
</p>

(
        S
        f
        )
        (
        z
        )
        =
        
          
            
              
                f
                ‴
              
              (
              z
              )
            
            
              
                f
                ′
              
              (
              z
              )
            
          
        
        −
        
          
            3
            2
          
        
        
          
            (
            
              
                
                  
                    f
                    ″
                  
                  (
                  z
                  )
                
                
                  
                    f
                    ′
                  
                  (
                  z
                  )
                
              
            
            )
          
          
            2
          
        
        .
      
    
    {\displaystyle (Sf)(z)={\frac {f'''(z)}{f'(z)}}-{\frac {3}{2}}\left({\frac {f''(z)}{f'(z)}}\right)^{2}.}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Misiurewicz_point/qP4DDoxw">Misiurewicz point</a></li>
<li><a href="/facts/Periodic_points_of_complex_quadratic_mappings/IlRwhEdR">Periodic points of complex quadratic mappings</a></li>
<li><a href="/facts/Mandelbrot_set/2NVUgrVG">Mandelbrot set</a></li>
<li><a href="/facts/Julia_set/dmn8xIcG">Julia set</a></li>
<li><a href="/facts/Milnor%25E2%2580%2593Thurston_kneading_theory/TgCDheNk">Milnor–Thurston kneading theory</a></li>
<li><a href="/facts/Tent_map/SuKt6Hzz">Tent map</a></li>
<li><a href="/facts/Logistic_map/wskv7mRQ">Logistic map</a></li></ul>

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Complex quadratic polynomials.

<ul><li><a href="/facts/Monica_Nevins/aAa0PI1G">Monica Nevins</a> and Thomas D. Rogers, "<a href="http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/nevins.pdf">Quadratic maps as dynamical systems on the p-adic numbers</a>[<i>permanent dead link</i>]"</li>
<li><a href="https://web.archive.org/web/20040828174339/http://www.math.sunysb.edu/cgi-bin/thesis.pl?thesis02-3">Wolf Jung : Homeomorphisms on Edges of the Mandelbrot Set. Ph.D. thesis of 2002</a></li>
<li>More about Quadratic Maps : <a href="https://mathworld.wolfram.com/QuadraticMap.html">Quadratic Map</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Poirier, Alfredo (1993). "On postcritically finite polynomials, part 1: Critical portraits". arXiv:math/9305207. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials" (PDF). <a href="https://www.ams.org/jams/2001-14-01/S0894-0347-00-00348-9/S0894-0347-00-00348-9.pdf" target="_blank">https://www.ams.org/jams/2001-14-01/S0894-0347-00-00348-9/S0894-0347-00-00348-9.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Bodil Branner: Holomorphic dynamical systems in the complex plane. Mat-Report No 1996-42. Technical University of Denmark <a href="/wiki/Bodil_Branner" target="_blank">/wiki/Bodil_Branner</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Dynamical Systems and Small Divisors, Editors: Stefano Marmi, Jean-Christophe Yoccoz, page 46 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials" (PDF). <a href="https://www.ams.org/jams/2001-14-01/S0894-0347-00-00348-9/S0894-0347-00-00348-9.pdf" target="_blank">https://www.ams.org/jams/2001-14-01/S0894-0347-00-00348-9/S0894-0347-00-00348-9.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>"Show that the familiar logistic map $x_{n+1} = sx_n(1 - x_n)$, can be recoded into the form $x_{n+1} = x_n^2 + c$". Mathematics Stack Exchange. <a href="https://math.stackexchange.com/q/290564" target="_blank">https://math.stackexchange.com/q/290564</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Yunping Jing : Local connectivity of the Mandelbrot set at certain infinitely renormalizable points  Complex Dynamics and Related Topics, New Studies in Advanced Mathematics, 2004, The International Press, 236-264 <a href="http://qcpages.qc.cuny.edu/~yjiang/HomePageYJ/Download/2004MandLocConn.pdf" target="_blank">http://qcpages.qc.cuny.edu/~yjiang/HomePageYJ/Download/2004MandLocConn.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Weisstein, Eric W. "Quadratic Map". mathworld.wolfram.com. <a href="https://mathworld.wolfram.com/" target="_blank">https://mathworld.wolfram.com/</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations Archived 26 April 2012 at the Wayback Machine <a href="http://mathesim.degruyter.de/jws_en/show_simulation.php?id=1052&type=RoessMa&lang=en" target="_blank">http://mathesim.degruyter.de/jws_en/show_simulation.php?id=1052&type=RoessMa&lang=en</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>"Convex Julia sets". MathOverflow. <a href="https://mathoverflow.net/questions/356342/convex-julia-sets" target="_blank">https://mathoverflow.net/questions/356342/convex-julia-sets</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Richards, Trevor (11 May 2015). "Conformal equivalence of analytic functions on compact sets". arXiv:1505.02671v1 [math.CV]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>M. Romera Archived 22 June 2008 at the Wayback Machine, G. Pastor Archived 1 May 2008 at the Wayback Machine, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Archived 11 December 2009 at the Wayback Machine  Fractalia Archived 19 September 2008 at the Wayback Machine 6, No. 21, 10-12 (1997) <a href="http://www.iec.csic.es/~miguel/miguel.html" target="_blank">http://www.iec.csic.es/~miguel/miguel.html</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Burns A M : Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104–116 <a href="https://web.archive.org/web/20041031082406/http://myweb.cwpost.liu.edu/aburns/index.html" target="_blank">https://web.archive.org/web/20041031082406/http://myweb.cwpost.liu.edu/aburns/index.html</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>"Khan Academy". Khan Academy. <a href="https://www.khanacademy.org/computer-programming/mandelbrot-spirals-2/1030775610" target="_blank">https://www.khanacademy.org/computer-programming/mandelbrot-spirals-2/1030775610</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
<li id="fn:15"><p>The Road to Chaos is Filled with Polynomial Curves
by Richard D. Neidinger and R. John Annen III. American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640–653 <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li>
<li id="fn:16"><p>Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific. ISBN 9971-5-0682-3. Archived from the original on 5 December 2009. Retrieved 2 December 2009. <a href="9971-5-0682-3" target="_blank">9971-5-0682-3</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li>
<li id="fn:17"><p>"M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint" (PDF). Archived from the original (PDF) on 2 October 2006. <a href="https://web.archive.org/web/20061002202249/http://www.iec.csic.es/~gerardo/publica/Romera96b.pdf" target="_blank">https://web.archive.org/web/20061002202249/http://www.iec.csic.es/~gerardo/publica/Romera96b.pdf</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li>
<li id="fn:18"><p>"Julia-Mandelbrot Space, Mu-Ency at MROB". www.mrob.com. <a href="http://www.mrob.com/pub/muency/juliamandelbrotspace.html" target="_blank">http://www.mrob.com/pub/muency/juliamandelbrotspace.html</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li>
<li id="fn:19"><p>Carleson, Lennart, Gamelin, Theodore W.: Complex Dynamics Series: Universitext, Subseries: Universitext: Tracts in Mathematics, 1st ed. 1993. Corr. 2nd printing, 1996, IX, 192 p. 28 illus., ISBN 978-0-387-97942-7 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li>
<li id="fn:20"><p>Holomorphic motions and puzzels by P Roesch <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li>
<li id="fn:21"><p>Rempe, Lasse; Schleicher, Dierk (12 May 2008). "Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity". arXiv:0805.1658 [math.DS]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li>
<li id="fn:22"><p>"Julia and Mandelbrot sets, alternate planes". aleph0.clarku.edu. <a href="http://aleph0.clarku.edu/~djoyce/julia/altplane.html" target="_blank">http://aleph0.clarku.edu/~djoyce/julia/altplane.html</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li>
<li id="fn:23"><p>"Exponential Map, Mu-Ency at MROB". mrob.com. <a href="http://mrob.com/pub/muency/exponentialmap.html" target="_blank">http://mrob.com/pub/muency/exponentialmap.html</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li>
<li id="fn:24"><p>Trees of visible components in the Mandelbrot set by Virpi K a u k o , FUNDAM E N TA MATHEMATICAE 164 (2000) <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm164/fm16413.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/fm/fm164/fm16413.pdf</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li>
<li id="fn:25"><p>"The Mandelbrot Set is named after mathematician Benoit B". www.sgtnd.narod.ru. <a href="http://www.sgtnd.narod.ru/science/complex/eng/main.htm" target="_blank">http://www.sgtnd.narod.ru/science/complex/eng/main.htm</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li>
<li id="fn:26"><p>Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, <a href="http://www.scholarpedia.org/article/Periodic_orbit#Stability_of_a_Periodic_Orbit%7CJeff" target="_blank">http://www.scholarpedia.org/article/Periodic_orbit#Stability_of_a_Periodic_Orbit%7CJeff</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li>
<li id="fn:27"><p>"Lecture Notes | Mathematical Exposition | Mathematics". MIT OpenCourseWare. <a href="https://ocw.mit.edu/courses/18-091-mathematical-exposition-spring-2005/pages/lecture-notes/" target="_blank">https://ocw.mit.edu/courses/18-091-mathematical-exposition-spring-2005/pages/lecture-notes/</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li>
</ol>

Complex quadratic polynomial open-in-new

Complex quadratic polynomial