Shadowing Chaotic Motion

Rob Knapp, Wolfram Research, Inc., Champaign, Illinois

Stan Wagon, Macalester College, St. Paul, Minnesota

Abstract

Many dynamical systems are known to admit shadowing. This means that a noisy trajectory computed using machine precision is in fact the exact trajectory from a slightly different set of initial conditions. We will show how, in three very different examples, Mathematica's FindMinimum command can be used to minimize the least squares error and so find the exact values of the shadowing set of initial conditions.

The first case is the classic quadratic map, (Subscript[f, r])=r x (1-x), where r is a value for which iterations are sensitively dependent on initial conditions, such as 4 or 38/10. In the latter case, one can compute a noisy trajectory starting from (Subscript[x, 0])=0.1. Then if one uses high precision and starts with
(Subscript[x, 0])= 0.100000000000000021043239304570322654494662756467086434194016030 the resulting trajectory is within (10^(-15)) of the noisy one.

A second example arises from a billiard relection problem. Again we compute a machine precision path for the particle, and then find a high-precision value (Subscript[y, 0]) so that the noisy path is the exact path for a particle starting at ((1/2),(1/10)+(Subscript[y, 0])).

In these two cases, getting derivative information for the iterated function is a key step, where the derivative is respect to change in the initial conditions, (Subscript[x, 0]) in the first case and (Subscript[y, 0]) in the second. A third example arises from the Duffing differential equation (x''-a x + b (x^3)+c x'=d cos t), where one can use the equation to discover the derivative information that one needs for shadowing.