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

x' = x + wSee "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.)_{x}- a/(2 pi) sin(2 pi y) y' = y + w_{y}- a/(2 pi) sin(2 pi x) w_{x}= 0.5979714 w_{y}= 0.8065 a_{ }= 0.7

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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