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[–] [S] 2 points 3 months ago* (last edited 3 months ago) (4 children)

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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  • [–] 1 point 2 months ago (3 children)

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

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  • [–] [S] 1 point 2 months ago (2 children)

    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.

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