Transient Chaos in Higher Dimensions

This chapter is devoted to transient chaos in higher-dimensional dynamical systems. The defining characteristic of high-dimensional transient chaos is that the underlying chaotic set has unstable dimension more than one, in contrast to most situations discussed in previous chapters, where chaotic sets have one unstable dimension. We shall call nonattracting chaotic sets with one unstable dimension low-dimensional, while those having unstable dimension greater than one high-dimensional. The increase in the unstable dimension from one represents a highly nontrivial extension in terms of what has been discussed so far about transient chaos. For instance, the PIM-triple algorithm, which is effective for finding an approximate continuous trajectory on a low-dimensional chaotic saddle, is generally not applicable to high-dimensional chaotic saddles. In a scattering experiment in high-dimensional phase space, the presence of a chaotic saddle cannot guarantee that chaos can be physically observed. In particular, if the box-counting dimension of the chaotic saddle is low, its stable manifold may not intersect a set of initial conditions prepared in the corresponding physical space; only when the dimension is high enough can chaotic scattering be observed.