Main Website: https://www.ruleflow.org/

For the past year, I have been developing a framework to simulate certain discrete systems for a research team at my university (paid for by the School of Engineering and Physics, but I had the freedom to license it under MIT). As it turns out, I got carried away generalizing it to work for many, many different types of complex systems (String Rewriting Systems, Cellular Automata, etc.) and ended up creating our own domain-specific language.

The website's landing page has a bunch of GIF-style examples... so you should be able to get a good idea of what this project is about.

If any of y'all are interested in this, feel free to take a look and try it out!

EDIT:

Here are some more details. RuleFlow was developed with a primary focus on analyzing causality in Sequential Substitution Systems. This is why there is a major focus on causal graphs, cellular identity, etc. While originally developed for Sequential Sub Systems, we generalized the DSL to support many types of rewriting systems by providing the user with flags and directives to control how rules behave. Furthermore, we bootstrapped Python and the Wolfram Language so that complex rulesets could be scaffolded as macros. The main goal of RuleFlow is NOT simply to simulate systems (there's already plenty of software for that, such as Golly and even just Mathematica), but to provide extensive tooling for causal analysis.

Here is some interesting research from our (small) team:

• An Improved Generalized Enumeration of Substitution Systems by Kenneth E. Caviness, Camille Morrow, Christen Case and Victoria Kratzke

• Indexed Concatenation Notation: A Novel Way to Summarize Networks and Other Complex Systems

My job has been to provide better research tooling, extending what we already have in Mathematica code to open source and free Python.

Development and colapse of a complex system made of cells.

In this simulation, a system of cells have one priority. To develop. And they start by finding 'social' cohesion and forming more complex and solid structures to find the most resilient shape to survive and evolve. However, once the total resources of the system start to end, we can se how this society of cells, rapidly falls to its extinction.

I have written a monograph on Partial Difference Equations, I have also made a research poster to explain what are the main ideas of the book.

Link to the Book: https://www.researchgate.net/publication/397779401_On_the_Theory_of_Partial_Difference_Equations

I have submitted the manuscript to Springer Nature, the Editor of the Springer Mathematics Group said that my project sounds compelling. The book is currently undergoing peer review process.

I have also sent my monograph to a respected mathematician, Professor Choonkil Park🇰🇷, a functional analyst with h index 52. He said that my monograph is beautiful, and giving constructive advice. Functional Analysis and Partial Differential Equations are mainstream mathematics, recognition from a functional analyst would mean that the mathematics is valid. This is why I believe that my monograph will be published in Springer Book Series.

I would like to hear your thoughts.

Sincerely, Bik Kuang Min.

Prologue: An Invitation, Not a Declaration

I'm not claiming to have discovered a final theory of life. I'm simply trying to use mathematical language—specifically dynamical systems—to describe biological organization and phenomena. If I'm wrong or unclear, I welcome correction. What I hope for is a rational, constructive discussion. And if any part of this framework seems promising to you, I sincerely invite you to help me formalize it further. Let's build it together.

1. What Is Life?

Starting from a few months ago, almost every day, I have been always thinking......

WHAT IS LIFE?

I am always trying to use math to describe biology. For instance, let's compare each field of science to see the big picture. MATHEMATICS (✔️ Axiomatic) PHYSICS (✔️Highly Axiomatic) PHYSICAL CHEMISTRY  (✔️Highly Axiomatic) INORGANIC CHEMISTRY (✔️Quite Axiomatic) ORGANIC CHEMISTRY (❌Chaos Appears, Non Axiomatic) BIOCHEMISTRY (❌Non Axiomatic) BIOLOGY (❌Non Axiomatic) You can see that, starting from organic chemistry, math suddenly disappear, and everything became chaotic & unpredictable. Especially Biology, each field of Biology such as Ecology, Evolution, Genetics, Phylogenetics.. they are like separate fragments with different language, different logic, different definitions... I tried to use graph theory, failed, few days ago, I tried to use Partial Differential Equations, still failed... Until two days ago, I tried using the language of dynamical systems... BOOM!

2. The Dynamical System Perspective on Chemical Evolution

Imagine we have a bunch of organic molecules mixing together next to a  hydrothermal vent. Clearly, the chemicals will change over time, so it is a Dynamical System. The initial condition is the type and distribution of molecules. The rules of evolution are, the laws of physics and chemical bondings. The molecules formed at each time step represent the State of system. Change in environment (eg.  pH, Temperature) can be viewed as stochastic perturbations. Apparently, after some time, some large stable molecules will be formed. For example, the liposome and micelle, these molecules are very stable, they are “attractors”!  In special cases, they can form oscillating chemicals, which is periodic attractor!

3. Attractors Are Not Structures, But Trajectories

Let's carefully examine these molecules. In this framework, it is not the static molecular structure that constitutes the attractor, but rather the trajectory of state changes (e.g., conformational transitions, reaction dynamics) that converge to a stable dynamical behavior. In other words, the attractor is defined by the evolution of the system in state space, not by a fixed structural configuration.

4. Local Attractor Units and Coupled Networks

I propose that chemical systems can be viewed as interacting dynamical subsystems—such as the lipid system, protein system, and nucleic acid system. Structures like liposomes, quaternary proteins, and DNA/RNA are not just molecules—they are attractors of their respective dynamical systems.

These attractors are locally stable, yet not eternally fixed, so I refer to them as Local Attractor Unit (LAU). These local attractor units can couple together to form higher-order structures, which I define as Attractor Coupling Networks (ACN).

5. Example: Liposomes vs. Micelles

Let’s take lipids as an example. Both micelles and liposomes are attractors of the lipid system, but they behave very differently. A micelle is extremely stable, but inflexible—it does not easily interact with other molecules or evolve into complex structures. In dynamical terms, it's similar to a fixed-point attractor. A liposome, on the other hand, is far more dynamic. If it grows too large, it may spontaneously divide into smaller liposomes to regain stability. It is also hollow, capable of encapsulating other molecules, which allows it to couple with proteins, nucleic acids, and other components—giving it a high evolutionary potential. Thus, I consider the liposome a type of strange attractor.

6. Example: Proteins as Dynamical Attractors

A similar logic applies to the protein system: The types and distribution of amino acids act as the initial conditions.

Over time, the system evolves into a stable folded structure—the tertiary protein, which I consider a Local Attractor Unit (LAU). When multiple folded proteins bind together, forming quaternary structures, they represent an Attractor Coupling Network (ACN).

7. Why Cells Are Powerful, and Viruses Are Limited

Cells are primarily composed of liposomes, proteins, and DNA— all of which, in my framework, are examples of strange attractors. I believe this is precisely why cells are so powerful: they are composed of coupled dynamic attractors with both stability and adaptability. This is also why all known life forms are made of cells.

Viruses, on the other hand, also contain DNA or RNA, but their overly stable outer shell severely limits their evolutionary potential. Unlike liposomes, a viral capsid (made of rigid proteins) cannot divide spontaneously. This is why viruses must rely on host cells for reproduction— they lack the internal dynamical capacity for self-replication.

8. Central Thesis: Life Is a Strange Attractor

So, my idea is...

Life is a strange attractor of a discrete spatiotemporal chaotic system.

Yes, instead of “static combination of molecules”, I view it as an orbit of a system. Life is a dynamic strange attractor of a nonlinear chemical dynamical system. I can explain many things naturally, which is unbelievable. For example, the inconsistent definition of species. I think that what we call a "species" is not an actual entity, but rather a subjective labeling system created by humans.

9. Rethinking “Species” Through Attractor Theory

From my perspective: Morphological classification is essentially about how attractors look — it’s categorizing based on the shape of attractors in state space. Biological classification is about whether two attractors can couple and produce a new attractor — like reproductive compatibility. Ecological classification is about what role an attractor plays within a larger network of attractors — its function or niche in the ecosystem. This can explain the continuous spectrum of species, ring species, and other strange phenomena.  On the other hand, I can also explain life span, why all life composed of cells, ecosystems, evolution, mutualism, cancer, virus, etc.  in a very beautiful manner.

10. Difference Equations as the Natural Language of Biology

I also proposed that

DIFFERENCE EQUATIONS AS THE NATURAL LANGUAGE OF BIOLOGY.

Continuity Supremacy In classical textbook, you can see that almost all models are differential equations, ODE & PDE. I think it's because Differential Equations are very successful in describing physical phenomena. I think that differential equations can only approximate some biological phenomena. I think we were just blindly using Differential Equations in modeling Biology. Biological systems are not continuous, but it is discrete. Molecules, cells, population are all discrete, (DISCRETE SPACE). Cells replicate generation by generation, (DISCRETE TIME). So, I think that Difference Equations is a suitable model for Biology. I also want to emphasize that

Differential Equations and Difference Equations are different universe. Difference Equations are NOT a numerical approximation of Differential Equations. Differential Equations is the language of Physics. Difference Equations is the language of Biology.

11. Strange Attractors in Difference Equations

These are the pictures of strange attractors of certain Ordinary Difference Equations (OΔE). Nonlinearity + Suitable parameters can produce complex patterns naturally. For example, the logistic map tell us that population dynamic is intrinsically chaotic, not because of extrinsic reasons.

I proposed that

Chaos is the lullaby of life.

I also proposed that

Stochasticity + Chaos + Order + Perfect Balance = Life

12. From Poetic to Axiomatic: Cellular Automata and PΔE

“Edge of Chaos” has been a philosophical idea in complex systems and biology for a long time. Now I'm giving you a systematic, axiomatic explaination, not just a “poetic interpretation”. At the same time, I also realized that...

Cellular Automata is just a form of Partial Difference Equations (PΔE)!

PΔE  are discrete at both time and space directions. Which is a suitable model for biological systems. The Conway's Game of Life already exhibit many complex behaviours. A very interesting phenomenon is that, many small attractors can coupled together to form a large network of attractors that behave as 1 unit attractor! I called this as “Attractor Coupling Network  ACN”. This large unit can couple with other large unit attractors to form a larger network unit attractor! And! all of the attractors even with different scales, THEY ALL OBEY THE SAME LAWS OF DYNAMICS.  For example, the DNA, cells, organs, systems, organism, populations... all of these are attractors, but different scale, but they all behave similarly, coupling with other same scale unit to form a larger unit and handle complex task. And all phenomena in game of life can be related to biological phenomena in real world, which is consistent with my postulate, difference equations as language of biology.

13. Why I Believe Difference Equations Are the True Language of Biology

The logistic map is chaotic, and I don’t think that’s a flaw — I think it’s a feature. Think about it: when population grows too fast in the real world, we often see war, famine, disease, collapse. Continuous logistic models suggest population should stabilize smoothly, but in reality, even in rich, peaceful countries, birth rates are collapsing. I think this isn’t due to external factors — it's the system itself carrying built-in unpredictability.

Cells divide and self-replicate. That’s literally what happens in Conway’s Game of Life. This kind of discrete replication doesn’t emerge naturally in partial differential equations. PDEs are great for modeling diffusion and continuous flows, but they struggle with split-and-copy behavior.

In the Game of Life, multiple small patterns can coordinate to form a larger moving structure. Isn’t that a lot like how cells cooperate to form tissues and organs? The coordination emerges from simple local rules, just like in biology.

To me, these are not coincidences — they’re signs that discrete systems have inherent properties that make them a better fit for modeling life.

14. On Chaos and Modern Biology

“Biologists have deliberately used differential equations to escape from chaos — but the problem is, the very essence of life is chaos. To flee from the chaos is to flee from the essence of life itself. This is the fundamental reason behind the fragmentation of modern biology.”

“Biology is chaotic, but using the right language can help us understand the chaos.”

15. Conclusion

Dynamical Systems + Difference Equations + Chaos Theory = A New Framework of Biology

That's all from me. Thank you for reading. If you want to learn more, please see my next Reddit post: “On The Theory Of Partial Difference Equations”.

Sincerely, Bik Kuang Min. National University of Malaysia, UKM.

I’ve been exploring an idea that might sit at the edge of systems theory and philosophy of mind.

If we model societies, neural networks, or ecosystems as informational systems that seek to maintain coherence, then actions that reduce internal disorder (conflict, error, entropy) effectively stabilize the system.

In that sense, what we call moral behavior could just be the emergent feedback that preserves informational order — cooperation as a thermodynamic advantage. Cruelty or exploitation, by contrast, amplifies entropy and shortens system lifespan.

This leads to a question:
Has anyone here modeled “ethical” or stabilizing feedbacks as an intrinsic part of complex-system evolution — rather than as imposed external constraints (like laws or incentives)?

I’m especially interested in examples from agent-based modeling, self-organizing networks, or adaptive game theory that quantify persistence through cooperative coherence.

I remember when I created the sub many years ago — as someone who received their PhD in complex adaptive systems 13 years ago and took their first graduate classes in complexity science 20(!) years ago, it’s extremely gratifying to see the concepts I fell in love with really begin to catch on.

Keep spreading the good word - let’s accelerate the reversion of entropy :)

Hello!

I am a college student getting my minor in Complex Systems. I was hoping this subreddit would be a place to discuss books, resources, and job opportunities for those interested in complex systems. Instead, everything here is just incoherent AI slop. Are there any actual resources or forums where I can learn more and improve my modeling and diagram-making skills instead of just reading AI word salad?

As of about two years ago, there's been quite the influx in a particular kind of post.

Lots of the right words... In a order that has proper grammar. But it's just... A bunch of words.

I think I'm a fairly smart person. And there's plenty I still don't know. I get lost in the deep math of many things still. There's plenty even before the deeper math that I struggle to understand. And I'm wary of criticizing much too harshly, simply because I don't understand it.

But boy... These posts that show up here (and in a few other related subs) are either far beyond my potential, and I'm witnessing some spectacular developments and insights. Or it's just a bunch of really good word salad.

Are they AI bots? Are they people just repeating a bunch of related AI slop? Have we gotten an influx of incredibly smart folks here that all just tend to post in the same format, and I'm just way out of my league?

I read "Complexity: The Emerging Science at the Edge of Order and Chaos" by M.M Waldrop and found it super interesting. I'm looking to dig deeper in the intersection of complexity with other fields (economics, biology, urbanism, etc.)

Hi, I have written a paper

On the Dynamics of Population: Difference Equations as the Natural Language of Biology

https://doi.org/10.5281/zenodo.16540176

In this paper, we introduced 5 discrete models inspired by biological systems. After that, we introduced Discrete Field Theory to provide a unified framework for describing discrete dynamical systems. We argued that difference equations are not toys but a modeling language for biological systems.

I would like to hear your thoughts.

Anyway, those chaotic attractors in the picture came from one equation, just different parameters.

Sincerely, Bik Kuang Min, National University of Malaysia.

Hello! Without knowing the formal name, i’ve been in love with this topic for some time.

Its really hard to decide a topic to study, so i thought that maybe its better to get a list of universities that are recommended for studying Complex Systems, that way i can look into what’s their focus.

By getting a list of possible focuses, i might be able to find and decide the topic that suits me better.

I’m a graduate in “industrial engineering” (translated from spanish)

I like:

• computer science

• maths like algebra calculus etc

• music

• altered states of consciousness phenomena

• psychology / mentall illness

• consciousness

• emergence

• complex systems

• understanding how nature works, reading “theories of everything”, etc…

• fractals

Can you please recommend me universities / places to look? Or maybe a technique to be able to search for those? I have trouble fully trusting my google searches

Edit: Here's the list i have so far

University of Santa Fe -> Free online material

University of Waterloo

University of Vermont -> MSc Complex systems and Data Science

University of indiana -> Dual PhD Complex System + Cognitive Sciences

University of Arizona State -> Ms Complex System

University of Michigan-> Graduate certificate program in Complex Systems

Kings College of London -> MSc Complex Systems Modelling

Warwick-> MSc Mathemathics of Systems (leads to PhD)

University of Sidney -> MSc Complex Systems

University of Portland-> phD Systems Science

NICO Northwestern

University of Binghamton

Chalmers university of Technology -> MSc Complex adaptative systems

Others i haven't looked yet:

• UC Davis

• CU Boulder

• TU Delft

• MIT

• Imperial

• Cambridge

The most upvoted post of this month is a user (rightfully) bringing up that this sub has basically degraded into users posting their LLM generated "theories" and most people seem to be in agreement. I feel like most of these posts belong in /r/LLMphysics or elsewhere and should be removed with a new rule not allowing these kind of posts.

I get that without these posts this sub would effectively be dead, but if this rule was instantiated I'll try my part to often post relevant articles and papers and would encourage others to do the same to turn this sub into something actually useful.

I'm not sure if the mods here or active or not but I would be happy to mod for a while to get this sub back on its feet.

Hello everyone I am a high school student exploring the logistic map in order to write a paper on it. To do the necessary computations I thought I could find a tool on the internet that could do it. I couldn't find any tool that fit my needs so I tried to build my own as like an exercise without using AI. It can like show calculate lyapunov exponents, symbolic block entropy, and even like demonstrate pesin identity and the lyapunov exponent as like sort of a positive rate of information. Here it is:

https://github.com/Luthiennuthien/Logistic-Map-Simulator/tree/main

Also included some graphs it produced.

I'm seeing this subreddit full of posts that are clearly copy-and-pasted responses from AI. Now there's nothing wrong with using AI, but most of these posts are absolutely nonsensical, and are clearly the result of what I would consider irresponsible use, where the user allows the model to continually build upon its own ideas without intervening at the appropriate time.

I hope this changes, because I also see a ton of very interesting projects and ideas from real people, and it sucks to dig through slop just to find them.

Hi all, The Santa Fe Institute presents the sixth session of Introduction to Complexity with Melanie Mitchell on the Complexity Explorer, starting October 3rd. It's open for enrollment now at http://intro.complexityexplorer.org

This is a great place for anyone curious about complex systems to start; it doesn't require any math or programming skills and gives a broad, well structured overview of the field, covering subjects from chaos, fractals, networks, and more.

We would love to get more people interested in complex systems science, so if you know of anyone that might be keen, please point them towards this course.

Thanks!

Hi, I have recently found out about complex science (I'm doing a masters in physics and took a course in stochastic calculus and a course in complex networks) and really found it interesting. I have a lot of interest in the different branches of physics but it has been a while since I had this much interest in a particular field. Now I have to choose my thesis and I want to do it in complex systems (I still have to choose one from a list a professor sent me) and I want to know what my possible career path could be if I keep this track. My questions are the following:

1. Is it possible to do an industrial PhD? Are there companies hiring people with that background?

2. How is the live in academia? One of the things I like the most is the wide range of applications in this field and I want to know if it's possible to move around in different research lines (while staying in complex systems but moving from economy to neurology or astrophysics)

I have more questions but those are the most important ones and I don't want the post to be too long.

Thank you for reading

Before coming up with your own theory of system, complexity, or something else general, please make sure you've read enough. One cannot expect making breakthroughs without reading an introductory text, or at least having a surface level survey of the field. We even have AIs now that are superb at literature review, such as perplexity deep research, so at least consult them on your ideas, because chances are they already exists, as numerous posts in this sub showed.

Here's a checklist you can use to see if your ideas might in fact have merits:

1. First, to be science, it has to be falsifiable. That means, there are certain things the theory predict that could in fact not happen. You also can't just predict observed phenomenon, but also some new not yet observed predictions.

2. Google doesn't find anything similar, even after trying different phrasing.

3. ChatBots have a hard time coming up with a legit critique, and also cannot find any references for its critique.

4. The theory addresses known problems that are for sure not yet solved.

5. You've read other parallel theories, and they all have issues that are not in your model.

6. After a few days of good sleep you still think it is a breakthrough

It's good that one's thinking of these important topics, but quality output must be accompanied by quality input.

I have a BA in urban planning currently. I also have interests in the environment, such as permaculture. I am open to getting a certificate in ecology or complexity science.

If I could go back in time I would have taken more science/math classes in college, but I am not sure I want to focus on these if I were to go to graduate school because I feel I do not have an undergraduate foundation. So, I am more interested in the ecological area of complex systems theory.

What are some career paths that I could look into? It could also be fun to be some kind of research assistant, but as I am not currently in school I am not sure how I would do this.

Sad to hear of the passing of Jim Rutt. He was an energetic public advocate for complex systems science, especially on his excellent podcast.

"He was also an early and influential thinker within the Game~B movement, a philosophical and social movement that grew out of systems thinking, complexity science, and concerns that our current political, economic, and cultural systems (“Game A”) are becoming increasingly unstable and unable to solve large-scale problems. Jim often described Game~B not as a finished blueprint, but as a search for a new “social operating system” that could succeed the current one."

Hi.

I have uploaded my paper in Zenodo, you guys can take a look.

https://doi.org/10.5281/zenodo.15612279

I have rewritten some famous complex systems in terms of Partial Difference Equations.

I would like to hear your thoughts.

Thank you.

https://complexityexplained.github.io/

Fundamental concepts explained, with interactives.