# Lorenz Attractor 3D

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## WHAT IS IT?

This is a model of the phase-space of a system consisting of three ordinary differential equations, known as Lorenz system.

It has chaotic solutions for certain parameter values. A particular set of chaotic solutions of the Lorenz system when plotted, resembles a butterfly or figure eight and is presented as Lorenz attractor.

The model is intended to visualize Lorenz attractor in 3D view with the possibility of changing one of the key parameters (i.e. rho-parameter)

## HOW IT WORKS

It is a system of three ordinary differential equations:

dX/dt = σ (Y-X)

dY/dt = X (ρ - Z) - Y

dZ/dt = XY - βZ

where σ, ρ and β are positive system parameters. Lorenz used the values ρ=28, σ=10, β=8/3

In the actual model σ=10, β=8/3 and ρ can take a value between 10 and 50.

Governed by these equations and calculations a phase-space is created: with each calculation a turtle is generated and plotted in space with respective coordinates (X, Y, Z).

## HOW TO USE IT

(1) Setup: creates basic conditions for the model to run (i.e. erases data from previous runs, generates X, Y and Z axes, etc.).

(2) Go: starts running the model with generation of new points (turtles) in accordance with numeric values as a result of calculations, performed every time-step.

(3) 'Rho-slider' is used to change ρ-value (before a new run)

Buttons (3) and (4) Zoom, (5) and (6) Orbit L&R, (7) and (8) Orbit Up&Down can be used for adjusting the 3D view

(9) The plot shows X, Y, Z values on each time-step/tick.

## THINGS TO NOTICE

The system was developed by Edward Lorenz as a simplified mathematical model for atmospheric convection. It is a toy-model, but a very interesting one in a mathematical sense and can help to understand some very complex behavior.

After the model starts it looks like points are generated in a chaotic manner but after some time it becomes evident that their trajectories are placed around two attractors.

## THINGS TO TRY

You can change rho-parameter value and perform new runs. How this influences the 'butterfly' shape?

## EXTENDING THE MODEL

An optimization of the code for a faster model run would be an option.

## NETLOGO FEATURES

Initially the model was built with the use of NetLogo System Dynamics Modeler, then it was converted from ‘System Dynamics’ version to a ‘regular’ one by recompiling the code and adding new pieces of the code with respective changes/additions in the model interface (buttons, sliders, etc.). Finally it was wrapped using NetLogo 3D-application.

## RELATED MODELS

* Turtle and Observer Motion Example 3D

* Turtle Perspective Example 3D

* Rossler Attractor Model

First two models can be seen in NetLogo library. The last one is part of a suit of models created to visualize some key concepts of Chaos Theory and Dynamical Systems. Most of the models are available on http://modelingcommons.org/account/models/2495

## CREDITS AND REFERENCES

This simple abstract model was developed by Victor Iapascurta, MD. At time of development he was in the Department of Anesthesia and Intensive Care at University of Medicine and Pharmacy in Chisinau, Moldova / ICU at City Emergency Hospital in Chisinau. Please email any questions or comments to viapascurta@yahoo.com

The model was created in NetLogo 6.0.1, Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

This model was inspired by Introduction to Dynamical Systems and Chaos (Fall, 2017) MOOC by David Feldman @ Complexity Explorer (https://www.complexityexplorer.org/courses).

## Comments and Questions

globals [ mylist-x mylist-y mylist-z g X Y Z dt ] to setup clear-all draw-axes set mylist-x list 0 (X) set mylist-y list 0 (Y) set mylist-z list 0 (Z) zoom 20 orbit-up 5 orbit-right 10 system-dynamics-setup system-dynamics-do-plot end to draw-axes create-turtles 1 [ set shape "line" set heading 90 set color red set size world-width stamp die ] create-turtles 1 [ set shape "line" set color yellow set heading 0 set size world-height stamp die ] create-turtles 1 [ set shape "line" set pitch 90 set color blue set size world-depth stamp die ] ask patch max-pxcor 0 0 [ set plabel "x-axis" ] ask patch 0 max-pycor 0 [ set plabel "y-axis" ] ask patch 0 0 max-pzcor [ set plabel "z-axis" ] end to system-dynamics-setup reset-ticks set dt 0.01 set g 10 set X 1 set Y 0 set z 0 end to go system-dynamics-go system-dynamics-do-plot set mylist-x lput result-x mylist-x set mylist-y lput result-y mylist-y set mylist-z lput result-z mylist-z crt 1 [ set color green set xcor (last mylist-x ) * 0.2 set ycor (last mylist-y ) * 0.2 set zcor (last mylist-z ) * 0.2 set size 0.2 set shape "circle" ] end to system-dynamics-go let local-b b let local-r r let local-inflow inflow let local-inflow1 inflow1 let local-inflow2 inflow2 let new-X ( X + local-inflow1 ) let new-Y ( Y + local-inflow ) let new-Z ( Z + local-inflow2 ) set X new-X set Y new-Y set Z new-Z tick-advance dt end to-report result-x report X end to-report result-y report Y end to-report result-z report Z end to-report inflow report ( X * ( r - Z ) - Y ) * dt end to-report inflow1 report ( g * ( Y - X ) ) * dt end to-report inflow2 report ( X * Y - b * Z ) * dt end to-report b report 8 / 3 end to-report r report rho-slider end to system-dynamics-do-plot if plot-pen-exists? "X" [ set-current-plot-pen "X" plotxy ticks X ] if plot-pen-exists? "Y" [ set-current-plot-pen "Y" plotxy ticks Y ] if plot-pen-exists? "z" [ set-current-plot-pen "z" plotxy ticks z ] end

There is only one version of this model, created about 5 years ago by Victor Iapascurta.

## Attached files

File | Type | Description | Last updated | |
---|---|---|---|---|

Lorenz Attractor 3D.png | preview | Preview for 'Lorenz Attractor 3D' | about 5 years ago, by Victor Iapascurta | Download |

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