Rigorous homoclinicity proofs for the Lorenz equations and other differentiable flows

[Truncated] Mathematicians (or chaoticians) frequently find that they need to establish that a given system, or a model of that system, exhibits chaotic dynamics. Until the last decade, however, few systems had been rigorously proven to possess chaotic dynamics. With recent computational and topological advances, it has been verified that certain dynamical systems exhibit horseshoe-type dynamics for a given point in parameter space. A horseshoe is a smooth folding-type map that has been shown to imply chaos. The identification of compact invariant sets on which a dynamical system acts chaotically or bifurcates to chaos is one important but sometimes analytically difficult approach to proving chaos. Establishing horseshoe-type dynamics, for example, often relies on computer-aided proofs and possibly on index theory. On the other hand, proving that nonhyperbolic compact invariant sets such as homoclinic orbits exist for smooth flows is a much more difficult procedure, both in geometrical theory and in our ability to adapt numerical techniques to handle nonhyperbolicity. Consequently, the presence of nonhyperbolic compact invariant sets for flows such as homoclinic orbits is usually contained as an assumption in texts and papers.