Pattern Formation via Braids


J. B. van den Berg, M. Kramar  and R. C. Vandervorst,   The order of bifurcation points in fourth order conservative systems via braids , SIAM Journal on Applied Dynamical Systems, Volume 10, (2011).

The mathematical study of pattern formation phenomena covers a plethora of methods, results and equations. One of the most successful attempts in this area concerns fourth order conservative systems. These equations exhibit complicated dynamic behavior and the depenedence on parameters is intricate. Our goal is to uncover the topological structure that underlies many individual patterns generated by fourth order systems. We use the theory of braid invariants to study the bifurcation branches of the Swift-Hohenberg equation. These invariants also provide a framework for forcing solutions based on the existence of other previously known solutions.