Chapter 25

The targeting of chaotic trajectories uses the sensitive dependence on initial conditions to produce large (desired) changes to the trajectory using small perturbations. One important application is to directing spacecraft using very small amounts of fuel, relying on the (Hamiltonian) chaos of the three body problem consisting of the spacecraft, earth, and moon or other planet. Another application is to improving the control of chaos, and we will establish the concept in this context. One problem with the methods of controlling chaos described in the previous chapter is that they require the trajectory to approach the control point to within some small tolerance ε before the control method can be applied. The number of iterations for this approach can be estimated as τ0 ∼ 1/μ(ε) with μ(ε) the natural measure of the attractor associated with the ε-ball. The measure of a small ball at a point x scales with the ball size according to the pointwise dimension DP (x) which can be estimated as the information dimensionD1, so that τ0 ∼ ε−D1 . This power law dependence on ε can lead to large capture times, especially for high dimensional chaos. The method of targeting can be used to convert this to a much weaker logarithmic dependence. Suppose we are starting from a point x0 and wish to control the system at a