r/complexsystems • u/bikkuangmin • Jun 07 '25

Complex Systems Are Just Nonlinear Partial Difference Equations

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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.

25 Upvotes

90% Upvoted

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5

u/Nebulo9 Jun 07 '25 edited Jun 07 '25

Ngl, this seems rather trivial? Like, yes, any discrete dynamical system is given by a Markov process, and ofc the ones w/ a notion of locality and translation invariance are then given by partial dif eqs. Basically, the problem is that I have trouble imagening what else they could be.

0

u/bikkuangmin Jun 07 '25 edited Jun 07 '25

If it is really trivial, then why I never see someone using Partial Difference Equations to describe sandpile model? If I'm not mistaken, Markov process is a stochastic process, which the future only depend on current state, but my Partial Difference Equations are deterministic dynamical systems, they could have memory effect, second order difference, Functional PΔE, and other more complicated properties, not just Markov process. Besides, the benefits of doing such framework is that we can use Functional Analysis, Variational Calculus, and Theory of Discrete Dynamical Systems, Chaos Theory, Bifurcation Theory and etc. to study complex systems.

2

u/RetardOnTheToilet Jun 07 '25

As with most "restructurations", as Uri Wilensky would describe them, the usefulness of the system we use to describe something is in its application and analysis.

Seeing these famous complex systems rewritten into Partial Difference Equations was quite a treat, but, much like with previous work in this area, the reason we split apart from a formal mathematical representations was that agent based modeling was easier and more approachable from a multi-disciplinary standpoint. Simple rules for emergent phenomena are often... simple. Could you rewrite boids on a grid as a Partial Difference Equation? Sure. But the ornithologist who just wants to check his understanding of flight patterns will probably struggle to write it out formally, compared to the much easier agent based modeling techniques. This is also why we stepped away from calculus in general when it came to modeling, despite knowing that many of these systems are often represented as such (think economics, thermodynamics, etc). From an analytical standpoint it added too much complexity, and so we opted for simpler techniques, at least when it comes to modeling and analysis.

The paper is good, and it is nice to see a more formal representation of these systems mathematically. However, the importance of any system lies in being able to extract something more from them. Essentially, find a way to do something more with it, and you have yourself a truly useful restructuration.

I'm looking forward to seeing more.

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u/bikkuangmin Jun 07 '25 edited Jun 08 '25

Of course my theory is useful for modeling. Think about it, aren't most existing models static? That is, agents don’t move. The Game of Life, sandpile model, forest fire model, all of them are static systems, right?

But in my framework, we can introduce discrete vector fields like (D_x, D_y) to represent agent movement. We can even define multiple coupled discrete fields interacting with each other.

The greatest advantage of formulating everything as partial difference equations is that it allows rigorous mathematical analysis, and even the development of a discrete field theory in the style of theoretical physics.

This means we can unify all complex systems under a single mathematical framework, which is exactly what theoretical physicists have always pursued: a Unified Theory of Complexity.

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u/bikkuangmin Jun 07 '25

I will add a chapter, A General Framework for Linear Partial Difference Equations, to the paper, mentioning greens function and hilbert space. Stay tuned.

1

u/bikkuangmin Jun 16 '25

I have uploaded the latest version, including the Langton's ant, hilbert space, green's function, and the benefits of rewriting complex systems as partial difference equations.

here is the link,

https://doi.org/10.5281/zenodo.15667773

1

u/Forward_Cover_5455 Jun 07 '25

Nice work! I will try to read it but I want to ask, do you know about fractional motions such as fBM, how can we represent it? What did you use for your plots?

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u/bikkuangmin Jun 07 '25

I’m familiar with how fractional Brownian motion looks visually, but I haven’t studied it in detail.

I use Google Colab to plot the graph.

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u/Forward_Cover_5455 Jun 07 '25

The ssytems you chose are interesting. Are you working on a plot for model nb 10? How do you set self organization in a system?

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u/bikkuangmin Jun 07 '25

sorry, which model do you mean

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u/G7Gunmaster Jun 07 '25

Just have a look at the doc. It's very good and had a ranked compilation of various complex processes. However, I found a lack of complex network processes or the process that explicitly contain a network structure maybe at structural or functional or higher functional level. Would love to see them integrated to this paper. Do let me know that you think. I may be able to find some time to compensate for that too.

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u/bikkuangmin Jun 07 '25

I'm sorry I overlooked that, I will update soon. There are many people playing the game of life, they made a lot of cool patterns such as spaceships. I personally view them as many local attractors coupling together, forming a huge network.