Since Dmitri Anosov has proven that manifolds of negative curvature exhibit hyperbolicity, it is natural to ask if we get similar results for manifolds of positive curvature. Consider a Riemannian surface with non-negative Gaussian curvature, composed of a cylinder and two hemispheres at each end. If our surface is flattened in one direction, and we apply geodesic flow, we want to show numerically that this system exhibits characteristics of hyperbolicity. More precisely, we will show that one Lyapunov exponent is positive. In this talk, I will go over how the geodesic flow was numerically approximated and how to compute the Lyapnuov exponent using an ensemble of initial conditions.

Pizza will be served.

Geodesics on a flattened surface

Chaotic billiard

Colloquium sponsored by the NSM Dean's office DUSEM grant