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1D dynamical systems can have attractors, 2D systems can have cycles, and 3D systems can have chaos. Is there a distinct behavior that only arises in 4D systems?

a month ago by AbouBenAdhem to c/askscience

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Hapankaali 10 points a month ago

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.

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AbouBenAdhem 3 points a month ago

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.

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Hapankaali 4 points a month ago

"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.

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AbouBenAdhem 1 point a month ago

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.

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Hapankaali 1 point a month ago

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.

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AbouBenAdhem 2 points a month ago

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.

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SubArcticTundra 2 points a month ago

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

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Hairyfishnuts 0 points a month ago

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ptz -4 points a month ago

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Eq0 2 points a month ago

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.

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