TY - JOUR

T1 - Multi-dimensioned intertwined basin boundaries

T2 - Basin structure of the kicked double rotor

AU - Grebogi, Celso

AU - Kostelich, Eric

AU - Ott, Edward

AU - Yorke, James A.

N1 - Funding Information:
This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences (Applied Mathematics Program), DARPA under NIMMP, and the Office of Naval Research.

PY - 1987

Y1 - 1987

N2 - 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,..., dN. 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,..., dN. For example, in our double rotor map it appears that N = 2, and d1 = 3.0 and d2 = 3.9.

AB - 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,..., dN. 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,..., dN. For example, in our double rotor map it appears that N = 2, and d1 = 3.0 and d2 = 3.9.

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U2 - 10.1016/0167-2789(87)90108-4

DO - 10.1016/0167-2789(87)90108-4

M3 - Article

AN - SCOPUS:0023310831

SN - 0167-2789

VL - 25

SP - 347

EP - 360

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 1-3

ER -