By Salomon Bochner, Kentaro Yano
Booklet by way of Bochner, Salomon, Yano, Kentaro
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Additional info for Curvature and Betti Numbers.
Other initial conditions would lie on 2-toruses nested with the one shown. These nested toruses are successively smaller, ultimately forming an invariant circle at the center of the island. Each trajectory eventually comes arbitrarily close to every point on the surface of its torus, mapping out a two-dimensional surface in the three-dimensional state space. The surface has a cut where z (mod 2π) = 0, but this cut could be removed by plotting the torus in cylindrical coordinates as was done in Fig.
12 State space plot showing an invariant 2-torus for the conservative forced pendulum in Eq. 8, 0, π/2), λ = (0, 0, 0). energy decays to zero only when there is a point attractor. Otherwise, the forcing function or antidamping term supplies energy to replace that lost through dissipation, and the energy fluctuates about some average value as the trajectory moves around on the attractor. 8 Two-toruses and Quasiperiodicity The holes in the Poincar´e section in Fig. 10 represent regions in which the trajectories lie on the surface of a 2-torus, like the surface of a doughnut or inner tube, although the shape is only topologically equivalent to a doughnut and may be greatly distorted.
One way to quantify the elegance of Eq. 10) is thus to count the number of nonzero parameters and then add to that count the total number of digits including the decimal point but excluding leading and trailing zeros for any parameters that are not ±1 so that integer parameters are preferred. The resulting number, which perhaps should be called the inelegance, is the quantity to be minimized. By this criterion, Eq. 9) has an inelegance of 39, whereas Eq. 6), when viewed as a special case of Eq.