A Few Notions of Stability and Bifurcation Theory

Abstract

While numerical approaches are a very important step in investigating the patterns exhibited by the hyperbolic and kinetic models discussed in the previous chapters, they could be slow and might not offer a full understanding of the models’ dynamics due to the very large parameter space associated with some models. In contrast, stability theory could identify the parameter conditions under which a pattern could form, and eventually could become unstable giving rise to a different pattern. A deeper understanding of the formation of various spatial and spatio-temporal patterns is offered by bifurcation theory, which can distil the mathematical and biological mechanisms not only behind the formation of patterns, but also behind the transitions between different spatial and spatio-temporal patterns. In this Chapter, we review some basic notions of linear stability analysis for pattern formation in ordinary differential equations and partial differential equations, as well as basic notions of symmetry theory and bifurcation theory. We also present in more details the weakly nonlinear analysis approach for pattern investigation and classification. Finally, we discuss some drawbacks of bifurcation theory (e.g. centre manifold reduction) for infinite-dimensional dynamical systems.