RANK ONE DYNAMICS NEAR HETEROCLINIC CYCLES AND CONDITIONAL MEMORY LOSS FOR NONEQUILIBRIUM DYNAMICAL SYSTEMS

There are two parts in this dissertation. In the first part we prove that genuine nonuniformly hyperbolic dynamics emerge when flows in $\backslash mbbRN$ 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, etc. 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 May-Leonard models of Lotka-Volterra dynamics.

In the second part we introduce a notion of conditional memory loss for nonequilibrium open dynamical systems.
We prove that this type of memory loss occurs at an exponential rate for nonequilibrium open systems generated by one-dimensional piecewise-differentiable expanding Lasota-Yorke maps.