This proposal focuses on dynamical bifurcation phenomena for which it is necessary to use a combination of theoretical and rigorous computational methods in order to study the global attractor structure for complicated systems. The investigators consider specific phenomena which are not robust under approximation, such as the occurrence of an infinite number of bifurcations, or solutions for systems of partial differential equations converging to one point in phase space as an associated parameter increases. Specifically, the proposal addresses the appearance of period-doubling cascades in bifurcation diagrams as well as the occurrence of equilibrium clusters in extended systems, and aims at the development of rigorous computational tools for establishing heteroclinic connections. The developed methodology will be applied to the study of transitional states in Cahn-Morral systems for phase separation in multicomponent alloys, and the study of complicated transition patterns in the Ohta Kawasaki diblock copolymer model. The methods developed for these examples are of general interest. They will serve as prototypes for solving similar intractable computational problems. In addition, the numerical tools developed for rigorous computation are general purpose and widely applicable.
This work is focused at the boundary between theory and computation, such that each method fails individually, but theory and computation together yield successful results. Several of the examples focus on answering specific questions in materials science, including models for soft materials in synthetic chemistry such as block copolymers used as additive in adhesives, and phase separation for binary and multicomponent metal alloys. The research enables the better understanding and development of new materials. One of the studied models is immediately relevant in atmospheric science, as it is used to study the creation of raindrops within a cloud. In the longer term, there will be additional applications of these methods to a variety of pattern formation problems arising in chemistry and biology. Students will be incorporated into this project from within an already existing research group of undergraduate and graduate students. Both investigators are committed to recruiting students who are women and members of underrepresented minorities.
