Contents - ChaosBook

CHAPTER 1. OVERTURE 18Figure 1.11: Approximation to (a) a smooth dynamicsby (b) the skeleton of periodic points, together withtheir linearized neighborhoods. Indicated are segmentsof two 1-cycles and a 2-cycle that alternates betweenthe neighborhoods of the two 1-cycles, shadowing firstone of the two 1-cycles, and then the other.approximation by shorter cycles {a} and {b}∣)t ab − t a t b = t ab (1 − t a t b /t ab ) = t ab(1 −Λ ab ∣∣∣∣∣ , (1.11)Λ a Λ bwhere a and b are symbol sequences of the two shorter cycles. If all orbits areweighted equally (t p = z n p), such combinations cancel exactly; if orbits of similarsymbolic dynamics have similar weights, the weights in such combinations almostcancel.This can be understood in the context of the pinball game as follows. Considerorbits 0, 1 and 01. The first corresponds to bouncing between any two disks whilethe second corresponds to bouncing successively around all three, tracing out anequilateral triangle. The cycle 01 starts at one disk, say disk 2. It then bouncesfrom disk 3 back to disk 2 then bounces from disk 1 back to disk 2 and so on, so itsitinerary is 2321. In terms of the bounce types shown in figure1.6, the trajectory isalternating between 0 and 1. The incoming and outgoing angles when it executesthese bounces are very close to the corresponding angles for 0 and 1 cycles. Alsothe distances traversed between bounces are similar so that the 2-cycle expandingeigenvalue Λ 01 is close in magnitude to the product of the 1-cycle eigenvaluesΛ 0 Λ 1 .To understand this on a more general level, try to visualize the partition ofa chaotic dynamical system’s state space in terms of cycle neighborhoods as atessellation (a tiling) of the dynamical system, with smooth flow approximated byits periodic orbit skeleton, each ‘tile’ centered on a periodic point, and the scaleof the ‘tile’ determined by the linearization of the flow around the periodic point,figure 1.11.The orbits that follow the same symbolic dynamics, such as {ab} and a ‘pseudoorbit’ {a}{b}, lie close to each other in state space; long shadowing pairs have tostart out exponentially close to beat the exponential growth in separation withtime. If the weights associated with the orbits are multiplicative along the flow(for example, by the chain rule for products of derivatives) and the flow is smooth,the term in parenthesis in (1.11) falls off exponentially with the cycle length, andtherefore the curvature expansions are expected to be highly convergent.[chapter 21]intro - 13jun2008.tex