Braids, PDEs, and Lagrangian Dynamics

Robert W. Ghrist, Department of Mathematics, University of Illinois

The archetypical forcing theorem in dynamical systems is the well-known "period three implies chaos" for interval maps (aka Sharkovski's theorem). This talk will describe an analogous forcing theory for stationary solutions to scalar uniformly parabolic PDEs, such as are used in (1-d) pattern-formation. The theory also works for second-order Lagrangian dynamics, twist maps, and several other settings. This forcing theory, like the original "period three" theorem, is at heart topological, and doesn't require knowing any analytical details of the PDE. The key is thinking about sets of solutions to parabolic PDE's as topological braids.