# 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

* period 1
* Cauliflower_Julia_set - period 1 - parabolic case
* parabolic case : field lines
* period 2
* period 2 attracting case
* Basilica Julia set - period 2 superattracting case
* Fat Basilica: period 2 parabolic - field lines
* disconnected
* Imploded cauliflower
Full source code is on the commons page ( click on the image)

# key words
## complex map

Map

## Riemann sphere

Point $s$ of the Riemann sphere

## inverse stereographic projection

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

## p-norm

The $p$-norm (also called $\ell _{p}$-norm) of vector $\mathbf {x} =(x_{1},\ldots ,x_{n})$ is
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

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

where:
* N is a fixed number of iterations
* $z_{0}=x_{0}+y_{0}i\$ 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
# 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
## Fragmentarium

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

## c
# Papers

* 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** through arxiv.org, free but not peer-reviewed preprint

# www

* fractalforums.org : unveiling-the-fractal-structure-of-julia-sets-with-lagrangian-descriptors/3376/msg19803

# references

* ↑ C. Mendoza, A. M. Mancho. The hidden geometry of ocean flows. Physical Review Letters 105 (2010), 3, 038501-1-038501-4.
* ↑ A. M. Mancho, S. Wiggins, J. Curbelo, C. Mendoza. Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems. Communications in Nonlinear Science and Numerical Simulation. 18 (2013) 3530-3557
* ↑ C. Lopesino, F. Balibrea-Iniesta, V. J. García-Garrido, S. Wiggins, A. M. Mancho. A theoretical framework for Lagrangian descriptors. International Journal of Bifurcation and Chaos 27, 1730001 (2017).
* ↑ Lagrangian description of the flow field in wikipedia
* ↑ C. Lopesino, F. Balibrea, S. Wiggins, A.M. Mancho. Lagrangian Descriptors for Two Dimensional, Area Preserving Autonomous and Nonautonomous Maps. Communications in Nonlinear Science and Numerical Simulation 27 (1-3) (2015) 40-51.
* ↑ V. J . Garcia Garrido. Unveiling the Fractal Structure of Julia Sets with Lagrangian Descriptors. https://arxiv.org/abs/2001.08937
* ↑ V. J. García-Garrido, F. Balibrea-Iniesta, S. Wiggins, A. M. Mancho, C. Lopesino. Detection of Phase Space Structures of the Cat Map with Lagrangian Descriptors. Regular and Chaotic Dynamics 23, (6) 751-766 (2018).
* ↑ G. G. Carlo and F. Borondo. Lagrangian descriptors for open maps Phys. Rev. E 101, 022208 (2020)
* ↑ Lp space when 0<p <1 in wikipedia
* ↑ fractalforums.org: lagrangian-descriptors-fragment-code
V. J. García-Garrido. Unveiling the fractal structure of Julia sets with Lagrangian descriptors. Communications in Nonlinear Science and Numerical Simulation 91 (2020) 105417.