Kim-Ostlund Torus Map

This figure was made using dstool. It shows the attractor of the Kim-Ostlund torus map:

x' = x + wx - a/(2 pi) sin(2 pi y)
y' = y + wy - a/(2 pi) sin(2 pi x)

   wx = 0.5979714
   wy = 0.8065
   a  = 0.7
See "Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos," by Baesens et al., Physica D49 (1991) pp. 387-475. (I'm not one of the authors.)

The attractor (shown in black) is a single invariant circle that wraps around the torus many times. The dynamics are quasiperiodic, not chaotic. The colors help identify the period-five fixed point that is found where a corner of one color meets a half-plane of another color. In reality, all the part that is not black is connected (on the torus) so the colors are a fiction. When 4 copies of the attractor are shown the colors help convince you that the attractor lies on a torus.

The fifth iterate of the map is near the identity map, and the dynamics are well-approximated by a flow (ODE) on the torus. The flow is a `cherry flow' of type C(12,11) in the notation of the aforementioned paper, meaning that the invariant circle twists around the torus 11 times in the x direction and 12 times in the y direction. This is after we make a reduction to a smaller torus that contains only one fixed point. (See the paper.)

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