The wiki link is very general, can you explain the mathematical meaning of causality?

My answer would have been that no, 3D systems already display all known behaviors of dynamical systems.

Chaos can occur in 1+1D systems (1 spatial-like, 1 timelike dimension). A basic example is the logistic map.

Counterintuitively, systems with 4 and higher dimensions can often be simpler than 3D systems in a certain sense, because mean-field theories are often more easily applied to 4D+ systems.

Wait, could you post me a link so I can learn about the things you mentioned?

Hmm... the book I was reading (Complex and Adaptive Dynamical Systems by Claudius Gros) actually said strange attractors only occur in three dimensions or higher, but I changed that to “chaos” in the title to fit the character limit—I thought they were essentially synonymous.

Here’s the quote:

Strange attractors can only occur in dynamical system of dimension three and higher, in one dimension fixpoints are the only possible attracting states and one needs at least two dimensions for limit cycles.

Edit: I see that Gros clarifies a few paragraphs later:

Chaos may arise in one dimensional maps ... but continuous-time dynamical systems need to be at least three dimensional in order to show chaotic behavior.

I guess I was mentally thinking of continuous-time systems, but there was no room to fit that in the title.

"Chaos" describes dynamical systems that exhibit sensitivity to initial conditions. This can be made mathematically precise, but it boils down to when you start with two initial conditions that are "close," they will after some "sufficient" time be "far apart."

For example, turbulent air flow is an example of a chaotic system. This is fundamentally the reason why weather forecasts become less accurate further in the future. The data that goes into the weather model does not have infinite precision, small errors eventually lead to large deviations in the state of the system.

I edited/clarified my comment before I saw your reply—apparently the restriction to three dimensions only apples to continuous-time systems, which is what I had in mind when I posted the question.

I'm not sure what the author had in mind exactly, but it's not accurate that at least 3D is required in continuous-time systems.

A common example in 2D is the chaotic billiard.

I am not familiar with the book and haven't studied strange attractors.

Some further reading has led me to hyperchaos:

A hyperchaotic system is a dynamical system with a bounded attractor set, on which there are at least two positive Lyapunov exponents. Since on an attractor, the sum of Lyapunov exponents is non-positive, there must be at least one negative Lyapunov exponent. If the system has continuous time, then along the trajectory, the Lyapunov exponent is zero, and so the minimal number of dimensions in which continuous-time hyperchaos can occur is 4.

The physical motion of the billiards is two-dimensional, but wouldn’t the phase space be four-dimensional (since it tracks both position and momentum)?

I think the reason chaos needs at least three-dimensional phase space for continuous-time systems is that the orbits in phase space can’t intersect (which the paths of the billiards in real space obviously do).

Sensitivity is only one of three requests, there is also mixing and dense periodic orbits.

You are correct in the identification of the phase space for billiards.

Would that “just” mean that along any given orbit there are two expanding directions? What sort of qualitative difference would that entail?

That’s how I understand it: the volume element surrounding any given point has to be contracting, but it can expand in some direction(s) if it has at least one other direction to contract in for compensation.

As for qualitative differences, I'm not really picturing what the difference would be—but trying to intuit the effect of extra parametric dimensions might be like trying to explain trichromancy to a color-blind person.

I guess one would not have 2D flat strange attractors like the Lorentz butterfly, but the chaotic region must then be at least 3D (two expanding directions + the time direction)

Yeah, that’s my understanding.