Denis Blackmore

Some interesting variants of walking droplet based discrete dynamical bifurcations arising from diffeomorphisms are analyzed in detail. A notable feature of these new bifurcations is that, like Smale horseshoes, they can be represented by simple geometric paradigms, which markedly simplify their analysis. The two-dimensional diffeomorphisms that produce these bifurcations are called sigma maps or double sigma maps for reasons that are made manifest in this investigation. Several examples are presented along with their dynamical simulations.

New type of strange chaotic "attractor" models for discrete dynamical systems of dimension greater than one are constructed geometrically. These model, unlike most of the standard examples of chaotic attractors, have very complicated dynamics that are not generated by transverse (homoclinic) intersections of the stable and unstable manifolds of fixed points, and may include transverse heteroclinic orbits. Moreover, the dynamics of these model are not generally structurally stable (nor $\Omega$-stable) for dimensions greater than two, although the topology and geometry of the nonwandering set $\Omega$ are invariant under small continuously differentiable perturbations. It is shown how these strange chaotic models can be analyzed using symbolic dynamics, and examples of analytically defined diffeomorphisms are adduced that generate the models locally. Possible applications of the exotic dynamical regimes exhibited by these models are also briefly discussed.