Contents - ChaosBook

CHAPTER 1. OVERTURE 12Figure 1.9: The 3-disk game of pinball Poincarésection, trajectories emanating from the disk 1with x 0 = (s 0 , p 0 ) . (a) Strips of initial points M 12 ,M 13 which reach disks 2, 3 in one bounce, respectively.(b) Strips of initial points M 121 , M 131 M 132and M 123 which reach disks 1, 2, 3 in two bounces,respectively. The Poincaré sections for trajectoriesoriginating on the other two disks are obtained bythe appropriate relabeling of the strips. Disk radius: center separation ratio a:R = 1:2.5. (Y. Lan)(a)sinØ10−1−2.500000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111100000000000000011111111111111112 130000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110000000000000001111111111111110 2.5S(b)sinØ10−1−2.5000000000000000111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111123000000000000000011111111111111111310000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110100000000000000001111111111111111000000000000000012101132000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110000000000000000111111111111111100000000000000001111111111111111000000000000000011111111111111110s2.5the M i th strip consists of all points with itinerary i = s 1 s 2 s 3 ...s n , s = {1, 2, 3}.The unstable cycles as a skeleton of chaos are almost visible here: each such patchcontains a periodic point s 1 s 2 s 3 ...s n with the basic block infinitely repeated. Periodicpoints are skeletal in the sense that as we look further and further, the stripsshrink but the periodic points stay put forever.We see now why it pays to utilize a symbolic dynamics; it provides a navigationchart through chaotic state space. There exists a unique trajectory for everyadmissible infinite length itinerary, and a unique itinerary labels every trappedtrajectory. For example, the only trajectory labeled by 12 is the 2-cycle bouncingalong the line connecting the centers of disks 1 and 2; any other trajectory startingout as 12 ...either eventually escapes or hits the 3rd disk.1.4.3 Escape rateWhat is a good physical quantity to compute for the game of pinball? Such a system,for which almost any trajectory eventually leaves a finite region (the pinballtable) never to return, is said to be open, or a repeller. The repeller escape rateis an eminently measurable quantity. An example of such a measurement wouldbe an unstable molecular or nuclear state which can be well approximated by aclassical potential with the possibility of escape in certain directions. In an experimentmany projectiles are injected into a macroscopic ‘black box’ enclosinga microscopic non-confining short-range potential, and their mean escape rate ismeasured, as in figure 1.1. The numerical experiment might consist of injectingthe pinball between the disks in some random direction and asking how manytimes the pinball bounces on the average before it escapes the region between thedisks.For a theorist, a good game of pinball consists in predicting accurately theasymptotic lifetime (or the escape rate) of the pinball. We now show how periodicorbit theory accomplishes this for us. Each step will be so simple that you canfollow even at the cursory pace of this overview, and still the result is surprisinglyelegant.[example 15.2][exercise 1.2]Consider figure 1.9 again. In each bounce the initial conditions get thinnedout, yielding twice as many thin strips as at the previous bounce. The total areathat remains at a given time is the sum of the areas of the strips, so that the fractionintro - 13jun2008.tex