Unstable Periodic Orbits as a Unifying Principle in the Presentation of Dynamical Systems in the Undergraduate Physics Curriculum*

Unstable periodic orbits are a ubiquitous feature of a wide variety of dynamical systems that exhibit behavior that may be characterized as chaotic, complex or turbulent. In physics pedagogy, they may be used as a bridge to the understanding of symbolic dynamics, topological entropy and pressure, and the dynamical zeta function formalism. As high-performance scientific computation makes the application of this methodology possible to ever-larger dynamical systems, this methodology emerges as an effective way to introduce students -- from secondary school, to undergraduate education, to graduate education -- to advanced concepts in the theory of dynamical systems.