Abstract
The present work is about providing visual experience on the 3D chaotic systems described in the book Elegant Automation - Robotic Analysis of Chaotic Systems by Julien Clinton Sprott (Sprott (2023)). There are 50 chaotic dynamical systems described in the book and each system is visualized in the present work in two ways: pre-computed animation and real-time simulation. These visualizations were made using the opensource tools such as P5.js, Processing language, quarto and C++. This work was made from the inspiration on the fact that the book was mostly writen by a computer program developed by the author Sprott, and all systems are visualizable in the 3D space. All the computer programs created in this work were made available as opensource. This work is made to complement the author Sprott for his book.
1 Introduction
I would like to begin by quoting a sentence from Sprott’s book that states my inspiration: “There is no substitute for the thrill and insight of seeing the solution of a simple equation unfold as the trajectory wanders in real time across your computer screen using a program of your own making”. Almost all of my works (can be found here) were driven by this thrill and insight that I get from them. This work was also made in the same line and added to my list.
The book Elegant Automation - Robotic Analysis of Chaotic Systems is the recent work of Julien Clinton Sprott (Sprott (2023)) that contains 50 dynamical systems in 3 dimensions. These dynamical systems were discovered by Sprott through automation by computer programming. In fact, most portion of the book was writen by a computer, which Sprott himself has stated in the book.
One of my hobbies is the computer programming. The inventions of P5.js and Processing creative programming languages have boosted enthusiasts like me to create and share visualizations related to scientific and non-scientific fields. A beginner with basic programming knowledge can make impressive visualizations in his/her field by taking this book (Shiffman (2024)) by Daniel Shiffman, one of the pioneers in the field creative coding.
Sprott has given visualizations of the solution trajectories of the dynamical systems in his book. But, those visualizations were the images of the solution trajectories presented in classic 4-view geometry style (front, top, side and isometric views). More thrill and insight can be felt by the readers of his book if they have access to the 3D visualizations of the dynamical systems given in the book. The present work is all about providing those 3D visualizations to augment the feel of readers with more thrill and insight.
Two modes of visualizations were presented for each dynamical system given in the book: pre-computed animations and real-time simulations. The pre-computed animations of solution trajectories were made in two steps: computing accurate solution trajectories with C++, and visualizing in 3D space using Processing. This provides an accurate 3D view of the solution trajectory for each dynamical system.
The thrill of watching the trajectory evolution was provided by the real-time simulations. These were made as html embeddings using P5.js for each dynamical system. This visualization is interactable using mouse/touchpad and have option of setting the simulation runtime. Please note that these real-time simulations will be less accurate due to large fixed timestep size, and can cause your web-browser to slow when run for long simulation times. All the computer codes were made available as opensource here, hence interested readers can download and run the codes on their local machine for longer simulation times and experience.
2 Chaotic systems and Lyapunov exponents
All the dynamical systems given in the Sprott’s book are chaotic. Another way of stating chaotic is “sensitive dependence on initial conditions”. The shape and path of the solution trajectory of a given dynamical system will depend on its initial condition.
A significant difference in initial condition can lead to a significantly different solution trajectory. However, there is a class of dynamical systems to which a machine epsilon level change (an infinitesimal change) in the initial condition will lead to a completely different trajectory. Such systems are chaotic in nature, a quick example will be the system modeling weather in the earth’s atmosphere. The interested readers can learn more at Fitzgerald (1991) book by James Gleik on Chaos.
The Lyapunov exponent is a quantity that defines if a given dynamic system is chaotic. The solution trajectories given in Sprott’s book are colored based on local Lyapunov exponents (LLE). In the book, the regions on a trajectory with a red shade describes a positive LLE, indicating that the system will produce a completely different trajectory if an infinitesimally close point parallel to that region is chosen as the initial condition. The regions with blue shade describes a negative LLE, indicating that the system is not chaotic in those regions.
A similar colorscheme was used for the present work. For example, the below animation shows the 360o view of the Diffusionless Lorenz system (5th system in the book).
The left-to-right 360o degree view animation of the Diffusionless Lorenz system.
Here, the colorscheme used is blue-green-red, with green being neutral. Blue shade shows regions with negative LLE and the red shade regions are with positive LLE. For a given dynamical system, the LLEs were computed by first calculating the largest eigen value in magnitude of the jacobian matrix of the system, then computing the natural log of absolute of the computed eigen value. In the present work, the largest eigen value was computed using Power iteration algorithm.
Computing LLE is expensive for real-time simuations. Thus, all the real-time simulations will have a solid color for the solution trajectory.
3 Visualizations of chaotic systems
It will be cumbersome to include all 50 dynamical system visualizations in a single manuscript. Thus, they are given as a separate online book made using quarto. It can be found here
4 Conclusion
In the present work, the 3D visualizations were made for all the 50 chaotic systems described in the book Elegant Automation - Robotic Analysis of Chaotic Systems by Julien Clinton Sprott (Sprott (2023)). Two modes of visualizations were made: pre-computed animation shaded with local Lyapunov exponents, and real-time simulations. This work was made out of inspiration and thrill obtained from the book, with a motive of sharing the same but enhanced thrill to the readers of the book all around the world.
The present work was made as a complement to the author Julien Clinton Sprott for his book. All the computer programs created for this work are made opensource and can be found here.
5 Acknowledgements
I sincerely thank the following people for their actions that led to the present work.
For the mentorship and introduction to quarto
Dr. Devendra Prakash Ghate,
Department of Aerospace Engineering,
Indian Institute of Space Science and Technology,
Trivandrum, Kerala, India.For the introduction to chaotic systems
Dr. Vinoth B.R.,
Department of Aerospace Engineering,
Indian Institute of Space Science and Technology,
Trivandrum, Kerala, India.For bringing Sprott’s book within reach
Dr. Anil Kumar C.V.,
Department of Mathematics,
Indian Institute of Space Science and Technology,
Trivandrum, Kerala, India.For knowledge sharing on P5.js and Processing
Daniel Shiffman,
Creator of the youtube channel: The Coding Train,
https://www.youtube.com/c/TheCodingTrain
Ofcourse!, this wouldn’t have possible without Sprott’s book!