Topological characterization of deterministic chaos: enforcing orientation preservation.

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.