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By Simon Rubinstein-Salzedo

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Of course, for k Ͼ 1, G has fewer than 2k points of period-k. ) For example, x ϭ 0 is a period one point and therefore not a period-k point for k Ͼ 1, although it is one of the 2k fixed points of Gk . 8 Let G(x) ϭ 4x(1 Ϫ x). Prove that for each positive integer k, there is an orbit of period-k. 23 ONE-DIMENSIONAL MAPS Number of fixed Period Number of points of G k fixed points due to lower Orbits of period orbits period k 0 2 2 4 .. 2 1 2 3 .. k of G 1 2 3 4 .. 2 4 8 16 .. 3 The periodic table for the logistic map.

The proof that period three implies chaos is given below in outline form. In each part, you are expected to fill in a reason or argument. Step 1 Let d denote the length of the subinterval RR. Denote by J ϭ S1 и и и Sk R any subinterval that ends in R. 15 Schematic itineraries for period-three map. The rules: (1) an interval ending in R splits into two subintervals ending in RR and RL; the order is switched if there is an even number of R’s, (2) an interval ending in L contains a shorter subinterval ending in LR, and a gap on the left (for an odd number of R’s) or the right (for an even number).

In general, the two large stars would travel in ellipses, but Poincar´e made another assumption, that the initial conditions were chosen such that the two moved in circles, at constant speed, circling about their combined center of mass. It is simplest to observe the trajectory of the asteroid in the rotating coordinate system in which the two stars are stationary. Imagine looking down on the plane in which they are moving, rotating yourself with them so that the two appear fixed in position. 1 shows a typical path of the asteroid.

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