Controlling Chaos Theory in VFX
What is Chaos Theory
Chaos theory is a branch of mathematics that focuses on understanding the chaos of the universe. These chaotic systems are controlled by dynamic ones that contain states of disorder and irregularities. Once observed on a smaller level, these states of disorder can be seen as patterns that are controlled by initial conditions in space time. The majority of these systems are not predictable.
So according to chaos theory, all random behavior in the universe is controlled by a pattern. These patterns can be represented as interconnectedness, constant feedback loops, repetition, selfsimilarity, fractals, and selforganization.
Chaotic behavior exists in many different places in nature. Such as fluid flow, heartbeats in animals, weather and climate conditions, and in more man made conditions. As road traffic and computer systems. Because of this it is used in many different fields of study, such as meteorology, anthropology, sociology, environmental science, computer science, engineering, economics, ecology, and philosophy. Chaos theory can also be used to predict future events.
All of these systems can be studied through the Chaotic Mathematical Model, or Recurrence Plots, and Poincaré maps.
Chaos is also very important when it comes to understanding the underlying noise in the universe. As well as it’s Supersymmetry. Supersymmetry is an overarching theory that describes the symmetrical correspondence between two different particles. These particles are Bosons and Fermions. Supersymmetry states that one of these Bosons must have a matching Fermion somewhere in the Universe. And vice versa. Together these two particles are called Superparteners. Each of these Superparteners share the same mass, spic, and quantum numbers. These particles help lay the foundations that chaotic movement in the Universe isn’t really chaotic. Rather, just a noisy pattern.
Complexities of Chaos Systems
As described above, the smaller these systems get. The more complicated they are. So let’s dive into some of the more complex elements. As well as breaking down the larger, and more simpler elements.
Chaotic behavior is controlled and is centered around it’s initial conditions. One of the easiest ways to describe this is by looking at weather systems. For example, one small change in the atmosphere can lead to completely different weather patterns the next day. Such as a forest fire emitting heat and debris into the sky. That can cause water particles and ash to rise into the atmosphere, and then descend quickly to form storm clouds. Another way to describe chaotic behavior is The butterfly effect.
A system described in a mathematical sense, is a name given to an object that might be abstract or concrete, elementary or composite; linear or nonlinear; simple or complicated; complex or chaotic. Right now, we are only focusing on the chaotic ones, but since they can also evolve into a mixture of complex chaotic ones, let’s take a look at those too.
Complex systems are systems made up of large numbers of other systems. Or composites as you will. These systems repeat over each other, and their behavior feeds back into the behavior of other parts of the system.
Chaotic systems are very similar, as they have some composite systems within them, but not as many as a complex system. When it’s systems interact with each other, it produces intricate results, which can be visualized as fractallike patterns. The way it’s system changes is by numerous interactions of the same rule.
However, chaotic systems evolve when they become dynamic. A dynamic system is a system whose state and variables evolve over time, according to the rules in it’s system. How it evolves is dependent on its rules and it’s initial conditions.
Complex and Chaotic systems are both considered to be nonlinear dynamic systems. The difference between a linear and non linear system is as follows. A linear system is a system that solves the superposition principle. The principle states that both A and B are solutions(ways it could evolve, or outcomes) for a system, as well as their sum when they are added together. Basically, what this means is that linear systems can be deconstructed, and each part of it’s system can be solved separately.
Nonlinear systems are a bit different. Their systems cannot be solved separately. It’s inputs are not equal to its outputs. Usually, it’s solution will be of a greater value than its parts.
Now, this is where the process of chaotic systems gets a bit more confusing. Enter…...Time. Systems change over time, but time also interacts with different systems in other ways than age, or when a system starts or finishes. But rather determine the system’s past and future, and current states all at the same time. This can only be done if a system is deterministic. A system that is semideterministic is a system that has its future determined, but not it’s past. Finally, an indeterministic system is a system that cannot have it’s future outcome determined because it’s evolution is random.
It is often easy to discover which systems fit into these categories by looking at how it plots itself on a graph. (More on this later). These plot points a system generates is called a time series.
This time series is used to determine if a system is chaotic or complex. A chaotic systems time series would resemble fractallike lines. It’s pattern of points would also look the same at different scales, and would emit self symmetry.
However, in order to chart the evolution of these systems, the initial conditions of the system must be understood perfectly. As the initial condition pretty much controls how an entire system will change over time. This is not always possible, or done with complete accuracy.
Jumping back to complex systems, there are some important characteristics that help define them. This is also important to point out, as this will help you recognize the difference between a complex and chaotic system.
Complex systems show signs of self organization. Self organization is when a system automatically orders itself without any external interference. Chaotic systems have the opposite behavior of this. Most complex systems are also considered open systems. Meaning that they dissipate energy, and are not closed off to their surrounding environment. Chaotic systems are also considered open systems as they need interaction from their environment to be chaotic.
Complex systems also contain feedback. This means that the output of the system is recycled back into the system to generate another iteration of the system. This feedback can be both positive and negative. A positive feedback will increase the rate of change in variable in a certain direction. A negative feedback will reverse the direction of the system. This feedback loop occurs in between a complex system’s self organization process. Because of this process, the complex system can evolve with its surrounding environment.
Logistical Maps and Chaos Theory
As previously mentioned, systems interact with graphs pretty well. As they are needed to determine what type of system they are, and how they evolve over time. So let’s talk about how they behave on logistical maps.
A logistical map is a “graph” that is used to map how chaotic polynomial systems grow from nonlinear dynamics. The most basic of them use logistic (log) functions to show how populations grow. Log functions are great for mapping growth as they use a differential equation that treats time as continuous. But when viewed on a logistic map, you are able to see the discrete time steps between each iteration of a pattern/system.
When mapping chaotic systems on these maps, depending on if the system is fractal like or is an attractor it will move differently.
Attractors
Attractors are a type of chaotic geometry. They are driven by a set of mathematical dynamic systems. An attractor is a set of states which evolves the current system it resides in. It can be represented as a set of points, curves, manifold geometry, and complex fractal structures.
Attractors don’t really have a set trajectory when they grow. Rather they flow back over themselves. Attractors can be represented in two or three dimensions. They don’t really have any particular constraints when it comes to evolving their system, other than overlapping it’s previous iteration.
They are different from repellers in mathematics. Both repellers and attractors have an initial input and process, but depending how the process evolves makes them different. A repeller's process will move away from the object or itself, and an attractor’s process will stay in the same area.
Currently, most attractors can only be visualized on computers. As they are dynamic systems that create thermodynamic losses, and will dissipate. So they are best represented by 3D geometry.
It’s hard to explain in writing what an attractor really is, but this site here has some great visuals of it: https://www.dynamicmath.xyz/strangeattractors/
This leads us into the different types of attractors that exist. Some are not chaotic, and some are simple in chaos theory terms. When an attractor is chaotic it contains two different points and has two exponentially diverging paths. It also cannot be described or visualized as a set of geometric objects. Such as lines, surfaces, spheres, etc. However, when you take into account other complex forms of attractors, they are harder to visualize as geometric objects. Simply because of how wild their topology is.
The two most simple types of attractors are fixed point and limit cycle attractors.
Fixed Point Attractors:

These are attractors that have an input that never changes. A fixed point for an attractor is a number that when you square it, it never changes. This means the value of the point has to be a 1 or a 0. The closer to the value zero the point is, the more the object is considered an attractor. The closer to 1, the more the system is considered a repeller. However, as soon as this system contains nonnegative numbers, this process reverses. Fixed point attractors can contain more than one fixed point.

One interesting example of a fixed point attracting system is terrain. Any terrain that contains multiple peaks, valleys, saddle points, ridges, ravines, and plains, can be considered a fixed point attractor.
Limit Cycle Attractors:

These attractors are nonlinear systems that follow something called a limit cycle. You can best visualize a limit cycle as a spiral that goes on for infinity. These are oscillating systems that can be found in nature or in mechanical objects. Such as your heartbeat and the swing of a pendulum clock.
Finite Number of Points Attractors:

These are attractors that are modeled over something called a discrete time system. A discrete time system is different from how we would regularly view time on a graph. For example, I’m pretty sure most of us have used the function $T in Houdini. In the Houdini sense, this means time. When we take $T*3 , that is the same equivalent as $T*x. X being the variable we multiply the time by. This in a mathematical sense would be considered continuous time. As the values of time are changed over time by a fixed variable. On a graph this would look like a slope. For a discrete system, the graph already has a bunch of points with set values placed at different time points. So in this scenario, time suddenly becomes the variable. You can read up more on this system here: https://en.wikipedia.org/wiki/Discrete_time_and_continuous_time

So this attractor type takes these points and visualizes the sequence.
Limit Torus Attractor:

This attractor type is similar to a limit cycle attractor. However, it’s trajectories are bound within a torus shape. It has most attributes of the limit cycle attractors, but this limitation forces it to be more contained than other attractors. When visualized on a spectrum, it’s frequency looks like sharp lines.
Strange Attractor:

For a more indepth dive into strange attractors. See this video here: https://www.youtube.com/watch?v=aAJkLh76QnM

An attractor is only strange if it has a fractallike structure. This often happens when the dynamics of it are chaotic. If an attractor is chaotic, it will be extremely sensitive to changes in its initial conditions. Attractors can be strange and not chaotic.

The term strange attractor was created by David Ruelle and Floris Takens. They first came up with the term while describing how attractors interacted with fluid flow.

Some examples of strange attractors are the doublescroll attractor, Hénon attractor, Rössler attractor, and Lorenz attractor. Some of these attractors are chaotic, some are not.
Applications of Chaos Theory
So how exactly do we use chaos theory in our everyday life?
Let’s start with the medical field.
There is a good amount of research out there to say that your heartbeat and the science surrounding it can be mapped to nonlinear chaotic systems. To start, the HisPurkinje system inside your heart has a fractal like vien system that pumps blood through it. However, your pulse is something else. Depending if you are at rest, exercising , or are in a constant state of stress, your pulse will change. But you cannot predict how your heartbeat will react to these environmental changes, and how blood will disperse through your body.
Because of this, scientists have started looking at nonlinear dynamics as a way to understand heart rate control, and cardiac arrest. As well as making these theories accessible to studying physicians. The more we can predict how our bodies will react to certain situations the better we can treat people when something goes wrong.
Doctors have even plotted the heartbeats of health patients and patients with heart failure together on plot graphs to see their iteration structures and time series. By doing this, they have proven that both subjects' heartbeats had the exact same means and variances. However, the healthy patient’s heartbeat generated a complex system of data, and the patient who had heart failure generated a series of periodic oscillations.
In nature, chaos can be found everywhere. Fractals can be found in trees, plants, wrinkles, and other self similar geometry. So it is quite likely we are also made up of these systems. Scientists have already found some of these natural shapes in certain cardiac muscle bundles and the tracheobronchial tree in the lungs.
So as you can see, looking into nonlinear systems can be a huge advantage if we would like to combat cardiac arrest, epilepsy, and even Alzheimer's.
Even our DNA contains fractallike patterns. By doing fractal analysis on DNA, scientists have found that there are series of correlations between nucleuses. Depending on how these patterns are present in your DNA, they might control how susceptible you are to certain health conditions, and what your quality of life may be.
Overall, nonlinear dynamics is being used in epidemiology and infectious disease processes, healthcare organisation, biomedicine, health care social sciences, and health geography.
3D Simulations, Scientific Visualization, and VFX
There are lots of very interesting simulations of chaos theory in VFX and in science. For this chapter, let’s focus on both.
Lab experiments regarding chaos theory are mostly designed for scientists to better understand how to control chaos. They are based around observed system behaviors, the orbits of attractors, and other nonlinear dynamic information.
When scientists attempt to control chaos, they are more or less looking to stabilize the system they are looking into. The end result is to turn the chaotic motion into a more stable and predictable state.
A few techniques have been developed. Some being the OGY (Ott, Grebogi and Yorke) method, and Pyragas continuous control.These algorithms require scientists to determine how unstable the system is beforehand.
After scientists obtain the information from a chaotic system they want, they then take a slice of the system to analyze. Sometimes a Poincaré section. Then after they have studied it, they then determine what type of orbit it has. Studying just a slice of the system is easier than studying an entire chaotic system. As most of them go on for infinity.
Another big thing in visualizing chaos is fractals. Fractals are hugely important in our world, and there are already many great examples of them online. If you ever google fractals on YouTube, you’ll probably find some really nice 4k zooms of fractals going on for infinity.
A fractal is a never ending pattern. They are a simple process that is essentially a never ending feedback loop. They are essentially images of dynamic systems at a micro level. Fractals also appear the same at different scales. Trees, rivers, coastlines, and clouds, all contain some sort of fractal like motion. Because of this fractals are very important to visualize. As they can help us better understand environmental solutions in nature.
Abstract fractals, such as Mandelbrots can be generated by a variety of computer generation software. Such as Houdini! You just need to calculate the same equation over and over.
Fun fact about fractals. There is overall a disagreement in mathematics on how they should be described. Benoit Mandelbrot, who Mandelbrots are named after. Described them as: "Beautiful, damn hard, increasingly useful. That's fractals."
But he later described them in 1982 as: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." So who knows…:)
Fractals can also be visualized as the Koch snowflakes, Cantor set, Haferman carpet,Sierpinski carpet, Sierpinski gasket, Peano curve, HarterHeighway dragon curve, Tsquare, Menger sponge and as Strange attractors. Some of these terms you might recognize, such as koch snowflakes. You might have seen this term under the Lsystem menu functions in Houdini. This is because LSystems also obey the laws of fractals.
Fractals are also implemented in turtle graphics patterns, and are used in medical sciences. You yourself are a fractal. Your biological cells ,neurons, immune system cells, blood vessels, and pulmonary structure all follow a fractal like structure.
There are different types of fractals. They are as follows: The Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal.
Rendering fractals goes well beyond VFX software. Yes they are used for artistic rendering, and medical imaging. But they also are used for developing technology, fractal analysis, and investigation tools in nature. Modeled fractals can also include sounds. Such as electrochemical patterns, and circadian rhythms. Simply because their frequency and waves follow a fractal pattern.
Finally, if you’d like to take a stab at modeling fractals and chaos in Houdini. Feel free to check out the links below:

Mandelbulb Formula: https://www.youtube.com/watch?v=DLjZXYi6smc

Fractal Arabesque: https://www.youtube.com/watch?v=H5gEPeHRtm0

Strange Attractor: https://www.youtube.com/watch?v=saA6edbOE

IFS Fractals: https://www.youtube.com/watch?v=kne16jQszU

Fractal Ornament: https://www.youtube.com/watch?v=tXYO_893cs

Mandelbrot and Mandelbulb: https://www.youtube.com/watch?v=_mwJ7mlYRWg

Chillout Tutorial: Fractal Formulas: https://www.youtube.com/watch?v=LM9czGYQmPc

Fractals in VFX: https://www.youtube.com/watch?v=xwFMha5WKoE
References
The control of chaos: theory and applications
The control of chaos: theory and applications
https://www.sciencedirect.com/science/article/abs/pii/S0370157399000964
Chaos theory
https://www.britannica.com/science/chaostheory
The 17 equations that changed the course of history
https://www.businessinsider.com/17equationsthatchangedtheworld20143
Chaos theory
https://en.wikipedia.org/wiki/Chaos_theory
Controlling chaos
http://www.scholarpedia.org/article/Controlling_chaos
Control of chaos
https://en.wikipedia.org/wiki/Control_of_chaos
Introduction to anticontrol of discrete chaos: theory and applications
https://royalsocietypublishing.org/doi/10.1098/rsta.2006.1833
Control of Chaos: Methods and Applications. I. Methods
Cardiac chaos
http://groups.bme.gatech.edu/groups/pcedl/publications/controlling.pdf
The Control Of Chaos: Theory And Applications
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.32.932
A simple guide to chaos and complexity
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2465602/
The reality of chaos theory – will you ignore, control or embrace?
https://sidewaysthoughts.com/blog/2010/07/realityofchaostheoryignorecontrolembrace/
Nonlinear Dynamics, Fractals, and Chaos Theory for Clinicians
https://archive.physionet.org/tutorials/ndc/
Chaos Theory and the Social Control Thesis: A PostFoucauldian Analysis of Mental Illness and Involuntary Civil Confinement
https://www.jstor.org/stable/29767118?seq=1
Nonlinear Dynamics: Chaos of Control
https://www.youtube.com/watch?v=3G93PeHTEr4
Controlling cardiac chaos
https://science.sciencemag.org/content/257/5074/1230
Chaos and chaos control in biology.
CONTROL OF CHAOS AND BIFURCATIONS
https://www.eolss.net/SampleChapters/C18/E64322.pdf
Chaos Control and its Application in Power Systems
https://www.scientific.net/AMM.316317.60
CHAOS THEORY
https://www.referenceforbusiness.com/management/BunComp/ChaosTheory.html
Chaotic systems
https://www.clausewitz.com/mobile/chaosdemos.htm
Chaos Theory, The Butterfly Effect, And The Computer Glitch That Started It All
Chaos Theory: Modeling, Simulation and Applications
https://www.amazon.com/ChaosTheorySimulationApplicationsInternational/dp/9814350338
Determinism and Chaos
http://pi.math.cornell.edu/~lipa/mec/lesson1.html
An Unpredictable Universe: A Deep Dive Into Chaos Theory
https://www.space.com/chaostheoryexplainerunpredictablesystems.html
Splinter Cell: Chaos Theory HD  Stereoscopic 3D Gameplay
https://www.youtube.com/watch?v=_P5I_kzsz6Q
Fractal
https://en.wikipedia.org/wiki/Fractal
What are Fractals?
https://fractalfoundation.org/resources/whatarefractals/
How Fractals Work
https://science.howstuffworks.com/mathconcepts/fractals.htm
Introduction
https://mathigon.org/course/fractals/introduction
Fractal
https://www.cs.mcgill.ca/~rwest/wikispeedia/wpcd/wp/f/Fractal.htm
Chaos Theory and the Logistic Map: https://geoffboeing.com/2015/03/chaostheorylogisticmap/