In the spectral stability analysis of localized patterns to singular perturbed evolution problems, one often encounters that the Evans function respects the scale separation. In such cases the... Show moreIn the spectral stability analysis of localized patterns to singular perturbed evolution problems, one often encounters that the Evans function respects the scale separation. In such cases the Evans function of the full linear stability problem can be approximated by a product of a slow and a fast reduced Evans function, which correspond to properly scaled slow and fast singular limit problems. This feature has been used in several spectral stability analyses in order to reduce the complexity of the linear stability problem. In these studies the factorization of the Evans function was established via geometric arguments that need to be customized for the specific equations and solutions under consideration. In this paper we develop an alternative factorization method. In this analytic method we use the Riccati transformation and exponential dichotomies to separate slow from fast dynamics. We employ our factorization procedure to study the spectra associated with spatially periodic pulse solutions to a general class of multi-component, singularly perturbed reaction-diffusion equations. Eventually, we obtain expressions of the slow and fast reduced Evans functions, which describe the spectrum in the singular limit. The spectral stability of localized periodic patterns has so far only been investigated in specific models such as the Gierer-Meinhardt equations. Our spectral analysis significantly extends and formalizes these existing results. Moreover, it leads to explicit instability criteria. Show less
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,... Show moreIn 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. Show less
We consider a two-component system of evolutionary partial differential equations posed on a bounded domain. Our system is pattern forming, with a small stationary pattern bifurcating from the... Show moreWe consider a two-component system of evolutionary partial differential equations posed on a bounded domain. Our system is pattern forming, with a small stationary pattern bifurcating from the background state. It is also equipped with a multiscale structure, manifesting itself through the presence of spectrum close to the origin. Spatial processes are associated with long time scales and affect the nonlinear pattern dynamics strongly. To track these dynamics past the bifurcation, we develop an asymptotics-based method complementing and extending rigorous center manifoldreduction. Using it, we obtain a complete analytic description of the pattern stability problem in terms of the linear stability of the background state. Through this procedure, we portray with precision how slow spatial processes can destabilize small patterns close to onset. We further illustrate our results on a model describing phytoplankton whose growth is co-limited by nutrient and light. Localized colonies forming at intermediate depths are found to be subject to oscillatory destabilization shortly after emergence, whereas boundary-layer type colonies at the bottom persist. These analytic results are in agreement with numerical simulations for the full model, which we also present. Show less
Siero, E.; Doelman, A.; Eppinga, M.B.; Rademacher, J.; Rietkerk, M.; Siteur, K. 2015
