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.
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)
Some further reading has led me to hyperchaos:
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.