Continuation methods and numerical bifurcation analysis

Te-Sheng Lin

Department of Applied Mathematics

National Chiao Tung University

    A numerical continuation method is developed to follow time-periodic travelling-wave solutions of both local and non-local evolution partial differential equations (PDEs). It is found that the equation for the speed of the moving coordinate can be derived naturally from the governing equations together with a condition that breaks the translational symmetry. The derived system of equations allows one to follow the branch of travelling-wave solutions as well as solutions that are time-periodic in a frame of reference travelling at a constant speed. We then show examples of the bifurcation and stability analysis of long-wave models of electrified falling films as well as films on a rotating cylinder. Finally, we present our recent work on spontaneous autophoretic motion of colloidal particles in two-dimensional space.

Keyword: numerical continuation method, time-periodic travelling-wave solution, numerical bifurcation analysis