Fractals/Iterations in the complex plane/Fatou set

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Fatou components[edit]

In case of discrete dynamical system based on complex quadratic polynomial Fatou set can consist of components :

  • attarcting ( basin of attraction of fixed point / cycle )
    • superattracting ( Boettcher coordinate )
    • attracting but not superattracting (
  • parabolic (Leau-Fatou) basin ( Fatou coordinate ) Local dynamics near rationally indifferent fixed point/cycle ;
  • elliptic basin = Siegel disc ( Local dynamics near irrationally indifferent fixed point/cycle )

Local discrete dynamics[edit]

Types of dynamics
  • attracting : hyperbolic dynamics
    • superattracting : the very fast ( = exponential) convergence to periodic cycle ( fixed point )
  • parabolic component = slow ( lazy ) dynamics = slow ( exponential slowdown) convergence to parabolic fixed point ( periodic cycle)
  • Siegel disc component = rotation around fixed point and never reach the fixed point

Tests[edit]

Analysis of local dynamics :

  • drawing critical orbit(s)
  • finding periodic points
  • dividing complex move into simple paths
  • topological graph,[1]
  • drawing grid ( polar or rectangular )
method test description resulting sets true sets
binary escape time bailout abs(zn)>ER escaping and not escaping Escaping set contains fast escaping pixels and is a true exterior.

Not escaping set is treated as a filled Julia set ( interior and boundary) but it contains :

  • slow escaping points from exterior,
  • Julia sets
  • interior points
discrete escape time = Level Set Method = LSM bailout Last iteration or final_n = n : abs(zn)>ER escaping set is divided into subsets with the same n ( last iteration). This subsets are called Level Sets and create bands surrounding and approximating Julia set. Boundaries of level sets are called dwell-bands
continous escape time Example Example Example

Bailout test[edit]

References[edit]

  1. A Topology Simplification Method For 2D Vector Fields by Xavier Tricoche, Gerik Scheuermann and Hans Hagen