Tracking pattern evolution through extended center manifold reduction and singular perturbations

In this paper we develop an extended center manifold reduction method: a methodology to analyze the formation and bifurcations of small-amplitude patterns in certain classes of multi-component, singularly perturbed systems of partial differential equations. We specifically consider systems with a spatially homogeneous state whose stability spectrum partitions into eigenvalue groups with distinct asymptotic properties. One group of successive eigenvalues in the bifurcating group are widely interspaced, while the eigenvalues in the other are stable and cluster asymptotically close to the origin along the stable semi-axis. The classical center manifold reduction provides a rigorous framework to analyze destabilizations of the trivial state, as long as there is a spectral gap of sufficient width. When the bifurcating eigenvalue becomes commensurate to the stable eigenvalues clustering close to the origin, the center manifold reduction breaks down. Moreover, it cannot capture subsequent bifurcations of the bifurcating pattern. Through our methodology, we formally derive expressions for low-dimensional manifolds exponentially attracting the full flow for parameter combinations that go beyond those allowed for the (classical) center manifold reduction, i.e. to cases in which the spectral gap condition no longer can be satisfied. Our method also provides an explicit description of the flow on these manifolds and thus provides an analytical tool to study subsequent bifurcations. Our analysis centers around primary bifurcations of transcritical type–that can be either of co-dimension 1 or 2–in two- and three-component PDE systems. We employ our method to study bifurcation scenarios of small-amplitude patterns and the possible appearance of low-dimensional spatio-temporal chaos. We also exemplify our analysis by a number of characteristic reaction–diffusion systems with disparate diffusivities.