Contents - ChaosBook

CHAPTER 1. OVERTURE 14x(t)δ x(t) = J t δ x(0)Figure 1.10: The fundamental matrix J t maps an infinitesimaldisplacement δx at x 0 into a displacementJ t (x 0 )δx finite time t later.x(0)δx(0)As a trajectory evolves, it carries along and distorts its infinitesimal neighborhood.Letx(t) = f t (x 0 )denote the trajectory of an initial point x 0 = x(0). Expanding f t (x 0 + δx 0 )tolinear order, the evolution of the distance to a neighboring trajectory x i (t) + δx i (t)is given by the fundamental matrix J:δx i (t) =d∑J t (x 0 ) ij δx 0 j ,j=1J t (x 0 ) ij = ∂x i(t)∂x 0 j.A trajectory of a pinball moving on a flat surface is specified by two positioncoordinates and the direction of motion, so in this case d = 3. Evaluation of acycle fundamental matrix is a long exercise - here we just state the result. The [section 8.2]fundamental matrix describes the deformation of an infinitesimal neighborhoodof x(t) along the flow; its eigenvectors and eigenvalues give the directions andthe corresponding rates of expansion or contraction, figure1.10. The trajectoriesthat start out in an infinitesimal neighborhood separate along the unstable directions(those whose eigenvalues are greater than unity in magnitude), approacheach other along the stable directions (those whose eigenvalues are less than unityin magnitude), and maintain their distance along the marginal directions (thosewhose eigenvalues equal unity in magnitude).In our game of pinball the beam of neighboring trajectories is defocused alongthe unstable eigendirection of the fundamental matrix M.As the heights of the strips in figure 1.9 are effectively constant, we can concentrateon their thickness. If the height is ≈ L, then the area of the ith strip isM i ≈ Ll i for a strip of width l i .Each strip i in figure 1.9 contains a periodic point x i . The finer the intervals,the smaller the variation in flow across them, so the contribution from the stripof width l i is well-approximated by the contraction around the periodic point x iwithin the interval,l i = a i /|Λ i | , (1.4)intro - 13jun2008.tex