Multi-dimensioned intertwined basin boundaries: Basin structure of the kicked double rotor

Using numerical computations on a map which describes the time evolution of a particular mechanical system in a four-dimensional phase space (The kicked double rotor), we have found that the boundaries separating basins of attraction can have different properties in different regions and that these different regions can be intertwined on arbitrarily fine scale. In particular, for the double rotor map, if one chooses a restricted region of the phase space and examines the basin boundary in that region, then either one observes that the boundary is a smooth three-dimensional surface or one observes that the boundary is fractal with dimension d ≊ 3.9, and which of these two possibilities applies depends on the particular phase space region chosen for examination. Furthermore, for any region (no matter how small) for which d ≊ 3.9, one can choose subregions within it for which d = 3. (Hence d ≊ 3.9 region and d = 3 region are intertwined on arbitrarily fine scale.) Other examples will also be presented and analyzed to show how this situation can arise. These include one-dimensional map cases, a map of the plane, and the Lorenz equations. In one of our one-dimensional map cases the boundary will be fractal everywhere, but the dimension can take on either of two different values both of which lie between 0 and 1. These examples lead us to conjecture that basin boundaries typically can have at most a finite number of possible dimension values. More specifically, let these values be denoted d₁, d₂,..., d_N. Choose a volume region of phase space whose interior contains some part of the basin boundary and evaluate the dimension of the boundary in that region. Then our conjecture is that for all typical volume choices, the evaluated dimension within the chosen volume will be one of the values d₁, d₂,..., d_N. For example, in our double rotor map it appears that N = 2, and d₁ = 3.0 and d₂ = 3.9.