Chaos and crises in more than two dimensions

Noisy chaotic trajectories, with finite-time Lyapunov exponents that fluctuate about zero, are basically unshadowable [S. Dawson, C. Grebogi, T. Sauer, and J. A. Yorke, Phys. Rev. Lett 73, 1927 (1994)]. This can occur when periodic orbits, with different numbers of unstable directions, coexist inside the attractor. The presence of a H\'enon-type chaotic saddle (i.e., a nonattracting chaotic set with a structure similar to that of the H\'enon attractor) guarantees such coexistence in a persistent manner [S. P. Dawson, Phys. Rev. Lett. 76, 4348 (1996)]. In this paper, we describe how these sets appear naturally in maps of more than two dimensions, how they can be found, and what crises they produce.