Homoclinic Loops, Heteroclinic Cycles, and Rank One Dynamics

We prove that genuine nonuniformly hyperbolic dynamics emerge when flows in RN with homoclinic loops or heteroclinic cycles are subjected to certain time-periodic forcing. In particular, we establish the emergence of strange attractors and SRB measures with strong statistical properties (central limit theorem, exponential decay of correlations, et cetera). We identify and study the mechanism responsible for the nonuniform hyperbolicity: saddle point shear. Our results apply to concrete systems of interest in the biological and physical sciences, such as MayLeonard models of Lotka-Volterra dynamics.