What are properties of dynamical systems in non-integer dimension spaces?

Question

A 1D dynamical system (R1) exhibits convergence to a fixed point, or escapes to infinity. A 2D dynamical system (R3) can produce oscillations, spiral-shaped trajectories, etc.

A 3D dynamical system (R3) exhibits chaos, and strange attractors that occupy a phase space with non-integer dimension d such that $ 2 < d < 3$. Increasing the number of dimensions exhibits similar results ("hyperchaos").

Suppose we were to define a function $$M : x \rightarrow y; x \in R^d, y \in R^d$$ over the non-integer-dimension phase space that the strange attractor lives on. Let M represent the velocity field over this weird phase space.

What kind of dynamics are possible in non-integer-dimension phase spaces less than 3? What kind of dynamics are possible in non-integer-dimension spaces greater than 3?

Answer 1

Let $A$ be our strange attractor and $D$ be the dynamics whose attractor $A$ is.

What will certainly be possible

First of all, you can have the dynamics $D$, since it can be reudced to $A$ without problems.

Second, for every point on $A$, $A$ has to contain 1-manifold which contains the point, since $A$ contains trajectories (or the trajectory, if you so wish) of $D$. Therefore you can have a system with infinetly many fix points:

• Make some Poincaré section of $A$ that is chosen such that there is an upper bound to the time between two of your original dynamics’ visits of that Poincaré section.
• The intersections of your $D$’s trajectory and that Poincaré section are your fix points.
• Take the velocity field of $D$ and rescale it locally such that every starting point on the strange attractor is taken to the nearest fixed point, where distance is measured along the trajectories of your original dynamics.

My educated guess is that it will not be generally possible to have finitely many fixed points though, as certain starting points would take forever to get there.

What depends

Whatever else you can do depends on the topological dimension of $A$:

• If the topological dimension is two, every point of $A$ must be contained in a 2-manifold, which itself is contained in $A$ and you could have a dynamics with infinitely many limit cycles (oscillations) or spirals. You can construct them analogously to the system with infinitely many fixed points above.
• Something analogous is possible, if the topological dimension is three or higher.