# Fractals/Iterations of real numbers/r iterations

Dynamics:

# Diagram types
## 2D diagrams

* parameter is a variable on horizontal axis
** bifurcation diagram : P-curves ( = periodic points) versus parameter
** orbit diagram : points of critical orbit versus parameter
** skeleton diagram ( critical curves = q-curves)
** Lyapunow diagram : Lyapunov exponent versus parameter
** multiplier diagrams : multiplier of periodic orbit versus parameter
* constant parameter diagrams
** cobweb diagram or a Verhulst diagram = Graphical iteration
** Iterates versus time diagram = connected scatter graph = time series
** invariant density diagram , histogram, the distribution of the orbit, frequency distribution = power spectrum
** Poincare plot
* orbit diagram
* critical curves
* Bifurcation diagram for real quadratic map. Periodic points for periods 1,2,4,and 8 are shown
* Orbit and Lyapunov diagrams
* Multiplier diagram
* Logistic map cobweb and time evolution a=3.2

### Transformation

Exponential transformation of the parameter axis
* normal horizontal axis
* transformed horizontal axis to show period doubling cascade
* 3 windows with non-linear, i.e. logarithmic x-axis scale:

## 3D diagrams

* An animated cobweb diagram
* An animated connected scatter diagram

# Maps

* Due to numerical errors different implementations of the same equation can give different trajectories. For example: r*x*(1-x) and r*x - r*x*x
* Sensitivity to initial conditions: "small difference in the initial condition will produce large differences in the long-term behaviour of the system. This property is sometimes called the 'butterfly effect'."
Map types
* doubling map

## Tent map

Orbits of tent map:
* double precision computation and numerical error
* increased precision with no error

## Logistic map

names :
* logistic map: $f(x)=rx(1-x),$
* logistic equation $x_{n+1}=f(x_{n}),$
* logistic difference equation $x_{n+1}=rx_{n}(1-x_{n}),$
* discrete dynamical system
The logistic map is defined by a recurrence relation ( difference equation) :
    $x_{n+1}=rx_{n}(1-x_{n}),$
where :
* $r$ is a given constant parameter
* $x_{0}$ is given the initial term
* $x_{n}$ is subsequent term determined by this relation
Do not confuse it with :
* differential equation ( which gives continous version )
Bash code Javascript code from Khan Academy MATLAB code:
See also Lasin
Maxima CAS code

### explicit solutions

The system with r=4 has the explicit solution for the nth iteration :

### Precision

Numerical Precision in the Chaotic Regime : " the number of digits of precision which must be specified is about 0.6 of the number of iterations. Hence, to determine is x10 000, we need about 6000 digits."
"Hence it is not possible to predict the value of xn for very large n in the chaotic regime."

### Better image

To show more detaile use tips by User:PAR:
"The horizontal axis is the r parameter, the vertical axis is the x variable. The image was created by forming a 1601 x 1001 array representing increments of 0.001 in r and x. A starting value of x=0.25 was used, and the map was iterated 1000 times in order to stabilize the values of x. 100,000 x -values were then calculated for each value of r and for each x value, the corresponding (x,r) pixel in the image was incremented by one. All values in a column (corresponding to a particular value of r) were then multiplied by the number of non-zero pixels in that column, in order to even out the intensities. Values above 250,000 were set to 250,000, and then the entire image was normalized to 0-255. Finally, pixels for values of r below 3.57 were darkened to increase visibility."
See also :
* tips from learner.org

### Lyapunov exponent

* code in Matlab
* G. Pastor, M. Romera and F. Montoya, "A revision of the Lyapunov exponent in 1D quadratic maps", Physica D, 107 (1997), 17-22. Preprint

### Invariant Measure

An invariant measure or probability density in state space
Video on Youtube

## Real quadratic map

Great images by Chip Ross
For $x_{n+1}=x_{n}^{2}-c$, the code in MATLAB can be written as:
Maxima CAS code for drawing real quadratic map : $f_{c}(z)=z^{2}+c$ :

### Lyapunov exponent
# Zoom

* fractal forums: rendering-bifurcation-diagrams

# Cycles

* n-cycles of polynomial maps by Cheng Zhang and paper: Cycles of the logistic map

# Points

* Misiurewicz Points
** of the Logistic Map by the J. C. Sprott
** M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535
** G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Misiurewicz point patterns generation in one-dimensional quadratic maps", Physica A, 292 (2001), 207-230
** G. Pastor, M. Romera and F. Montoya, "On the calculation of Misiurewicz patterns in one-dimensional quadratic maps" Physica A, 232 (1996), 536-553. Preprint
* Accumulation points
* bifurcation points

# Constants

* Feigenbaum constants in wikipedia

# Properities
## self-similarity, scaling and renormalization

* Feigenbaums Scaling Law For TheLogistic Map

# See also

* Geogebra: The Feigenbaum loci Author:a.zampa
* oeis search: logistic map
* Logistic Map by B. L. Badger
* Analytical properties of horizontal visibility graphs in the Feigenbaum scenario Bartolo Luque, Lucas Lacasa, Fernando J. Ballesteros, Alberto Robledo
* The Quadratic Map is Topologically Conjugate to the Shift Map - Gareth Roberts
* Numerical Errors in Logistic Map Calculation J. C. Sprott
* Numbers and Computers by Ronald T. Kneusel, Reviewed by David S. Mazel , on 01/27/2016
* Chaotic Modelling and Simulation Analysis of Chaotic Models, Attractors and Forms Christos H. Skiadas Charilaos Skiadas
* delayed_logistic
* The logistic map revisited Jerzy Ombach, Cracow, Poland
* Structure in the Parameter Dependence of Order and Chaos for the Quadratic Map Brian R. Hunt and Edward Ott
* Windows of periodicity scaling by Evgeny Demidov
* in Webgl by Ricky Reusser
* high-precision-feigenbaum-alpha-calc-using-julia by Stuart Brorson
* Feigenbaum Tree by 3DXM Consortium
* Pascal Yang Hui triangles and power laws_in_the_logistic_map by Carlos Velarde Alberto Robledo
* demonstrations.wolfram : EstimatingTheFeigenbaumConstantFromAOneParameterScalingLaw
* Egwald Mathematics: Nonlinear Dynamics: The Logistic Map and Chaos by Elmer G. Wiens

# software

* pynamical - Python package for modeling, simulating, visualizing, and animating discrete nonlinear dynamical systems and chaos
* Chaos. Visualizations connecting chaos theory, fractals, and the logistic map! Written by Jonny Hyman, 2020

# Videos:

* Illustration of Chaotic Systems Using Logistic Map by Zylasable
* Bifurcation Diagram for the Logistic Map Using R by Oxlajuj N'oj

# References

* ↑ FORTY YEARS OF UNIMODAL DYNAMICS: ON THE OCCASION OF ARTUR AVILA WINNING THE BRIN PRIZE by MIKHAIL LYUBICH
* ↑ Bifurcation and Orbit Diagrams by Chip Ross
* ↑ math.stackexchange question: what-are-the-lines-on-a-bifurcation-diagram ?
* ↑ A revision of the Lyapunov exponent in 1D quadratic maps by Gerardo Pastor, Miguel Romera, Fausto Montoya Vitini. PHYSICA D NONLINEAR PHENOMENA 107(1):17 · AUGUST 1997
* ↑ wiki : Cobweb plot
* ↑ One-Dimensional Dynamical Systems Part 3: Iteration by Hinke Osinga
* ↑ wikipedia : histogram
* ↑ CHAOS THEORY: DEPENDENCE ON PARAMETER R
* ↑ bat-country by xian
* ↑ Power Spectrum of the Logistic Map from wolframmathematica
* ↑ Poincare plot in wikipedia
* ↑ physics.stackexchange question : Poincaré plane and Logistic Map
* ↑ The logistic equation by Didier Gonze
* ↑ May, Robert M. (1976). "Simple mathematical models with very complicated dynamics". Nature. 261 (5560): 459–467. Bibcode:1976Natur.261..459M. doi:10.1038/261459a0. hdl:10338.dmlcz/104555. PMID 934280.
* ↑ Simple mathematical models with very complicated dynamics by R May
* ↑ The Logistic Map and Chaos by Elmer G. Wiens
* ↑ Ausloos, Marcel, Dirickx, Michel (Eds.) : The Logistic Map and the Route to Chaos
* ↑ Logistic map by M.R. Titchener
* ↑ khanacademy ; logistic-map
* ↑ Lasin - Matlab code
* ↑ Logistic diagram by Mario Rodriguez Riotorto using Maxima CAS draw package
* ↑ Double precision errors in the logistic map: Statistical study and dynamical interpretation J. A. Oteo J. Ros
* ↑ The Logistic Map by A. Peter Young
* ↑ earner.org textbook
* ↑ fractalforums.org : what-could-these-artifacts-be-on-my-lyapunov-fractal-renders
* ↑ calculate lyapunov of the logistic map -LASIN
* ↑ Invariant Measure Exercise by James P. Sethna, Christopher R. Myers.
* ↑ Invariant measure of logistic map by todo314
* ↑ Chip Ross : Bifurcation and Orbit diagrams
* ↑ demonstrations from wolfram : FeigenbaumsScalingLawForTheLogisticMap