Abstract: "The Falkner-Skan equation is a reversible three dimensional system of ODEs without fixed points. A novel sequence of bifurcations, each of which creates a large set of periodic and other interesting orbits 'from infinity', occurs for each positive integer value of a parameter. Another sequence of bifurcations destroys these orbits as the parameter increases; topological constraints allow us to understand this sequence of bifurcations in considerable detail. While outlining these results, we can also make a number of possibly illuminating remarks connecting parts of the proof, well-known numerical techniques for locating and continuing perodic orbits, and recent ideas in the control of chaos."