Fractals/Iterations in the complex plane/Discrete Lagrangian Descriptors

Lagranian descriptor for continoust-time dynamical systems ( Lagrangian way to describe the flow) is a method for analyzing structure of the phase space. Here this method is extended to discrete dynamical systems: open maps in the complex plane.

=Images=

Full source code is on the commons page ( click on the image)

=key words=

complex number
$$ z = x + y i$$

complex map
Map

$$z_{n+1} = f(z_n ) = f^{(n+1)}(z_0) \quad\forall \ n \ \in \mathbb{N} \cup \left\{0\right\}$$

Riemann sphere
Point $$ s $$ of the Riemann sphere

$$ s = (\xi_1, \xi_2, \xi_3 ) $$

inverse stereographic projection


inverse stereographic projection maps point $$ z $$ of complex plane to point $$ s $$  of Riemann sphere :

$$ S^{-1}(z) = \left(\frac{2 x}{1 + x^2 + y^2}, \frac{2 y}{1 + x^2 + y^2}, \frac{x^2 + y^2 - 1}{1 + x^2 + y^2}\right)$$

so

$$ s = S^{-1}(z) = (\xi_1, \xi_2, \xi_3 )$$ and $$ \xi_1 = \frac{2 x}{1 + x^2 + y^2}$$ $$ \xi_2 = \frac{2 y}{1 + x^2 + y^2}$$ $$ \xi_3 = \frac{x^2 + y^2 - 1}{1 + x^2 + y^2}$$

p-norm
The $$p$$-norm (also called $$\ell_p$$-norm) of vector $$\mathbf{x} = (x_1, \ldots, x_n)$$ is $$ \left\| \mathbf{x} \right\| _p :=  \sum_{i=1}^n \left| x_i \right| ^p $$

where
 * number p is a real number $$p \in (0, 1] $$. It is called a power. It affects the steepness of the gradient near singularities (fractal features like a Julia set)

p-norm is used to measure the distance between sequential iterations of the mapping f on the Riemann sphere

Discrete Lagrangian Descriptor = DLD
The simple idea is to compute the p-norm version of Lagrangian descriptors, not for the points on the complex plane, but for their projections on the Riemann sphere in the extended complex plane. ... in the complex mappings that we consider in this work, the functions that define the dynamics are not invertible, and therefore we will only keep the forward part of the definition. DLD:
 * is a scalar value
 * accumulates the p-norm along the orbit (= has information about the history of the orbit) and therefore unveiles the structure of interior and exterior of Julia set

sum
summing is what the original paper does ( pauldelbrot)

$$ D_p( z_0, N) = \sum_{k= 0}^{N - 1} \left\| s_{k+1} - s_k \right\| _p  = \sum_{j = 1}^{3} \sum_{k= 0}^{N - 1} \left | \xi_j^{(k+1)} - \xi_j^{(k)} \right |^p$$

where:
 * N is a fixed number of iterations
 * $$z_0 = x_0 + y_0i\ $$ is any initial condition selected on a bounded subset D of the complex plane
 * $$s$$ is a point of Riemann sphere: $$ s = (\xi_1, \xi_2, \xi_3 ) $$

Averaging
"averaging keeps the coloring stable if maxiters is changed but can lead to low variation over the image" pauldelbrot

$$ \frac{D_p( z_0, N)}{N}$$

=Steps= For each point z of complex plane
 * compute DLD ( scalar value)
 * color is proportional to DLD

Substeps: To compute DLD of z :
 * iterate point z under the map f = compute zn
 * map each point zn from complex plane to the Riemann sphere ( Inverse Stereographic projection)
 * for each zn compute summand
 * ... ( to do )

=Code=

UltraFractal
" Here's the latest version I've been using in UF. It handles escaping points with no-bail formulae via the isInf/isNaN test (puts the Riemann sphere point at the north pole for those), allows averaging or summing (summing is what the original paper does, whereas averaging keeps the coloring stable if maxiters is changed but can lead to low variation over the image), and can use or not use absolute values on the differences being summed." pauldelbrot

Fragmentarium
Code based on the UF code, modified and optimized for GLSL by 3Dickulus

c
= Papers = = www = = references= V. J. García-Garrido. Unveiling the fractal structure of Julia sets with Lagrangian descriptors. Communications in Nonlinear Science and Numerical Simulation 91 (2020) 105417.
 * Unveiling the Fractal Structure of Julia Sets with Lagrangian Descriptors by Víctor J. García-Garrido
 * in Communications in Nonlinear Science and Numerical Simulation, Volume 91, December 2020, 105417, a single anonymized review but paywalled
 * through arxiv.org, free but not peer-reviewed preprint
 * fractalforums.org : unveiling-the-fractal-structure-of-julia-sets-with-lagrangian-descriptors/3376/msg19803