Geertje Hek

Geometric singular perturbation theory in biological practice

Geometric singular perturbation theory is a useful tool in the analysis of problems with a clear separation in time scales. It uses invariant manifolds in phase space in order to understand the global structure of the phase space or to construct orbits with desired properties. This paper explains and explores geometric singular perturbation theory and its use in (biological) practice. The three main theorems due to Fenichel are the fundamental tools in the analysis, so the strategy is to state these theorems and explain their significance and applications. The theory is illustrated by many examples.

Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component.To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing–Hopf bifurcation is supercritical under realistic circumstances.In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a ‘Hopf dance’ and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.

Stabilization by Slow Diffusion in a Real Ginzburg-Landau System

SummaryThe Ginzburg-Landau equation is essential for understanding the dynamics of patterns in a wide variety of physical contexts. It governs the evolution of small amplitude instabilities near criticality. It is well known that the (cubic) Ginzburg-Landau equation has various unstable solitary pulse solutions. However, such localized patterns have been observed in systems in which there are two competing instability mechanisms. In such systems, the evolution of instabilities is described by a Ginzburg-Landau equation coupled to a diffusion equation.In this article we study the influence of this additional diffusion equation on the pulse solutions of the Ginzburg-Landau equation in light of recently developed insights into the effects of slow diffusion on the stability of pulses. Therefore, we consider the limit case of slow diffusion, i.e., the situation in which the additional diffusion equation acts on a long spatial scale. We show that the solitary pulse solution of the Ginzburg-Landau equation persists under this coupling. We use the Evans function method to analyze the effect of the slow diffusion and to show that it acts as a control mechanism that influences the (in)stability of the pulse. We establish that this control mechanism can indeed stabilize a pulse when higher order nonlinearities are taken into account.

Homoclinic Saddle-Node Bifurcations in Singularly Perturbed Systems

In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called “localized structures” in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O(ε) perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, Ws(Γ) and Wu(Γ), the stable and unstable manifolds of a slow manifold Γ. Homoclinic orbits to Γ correspond to intersections Ws(Γ)∩Wu(Γ); Ws(Γ)∩Wu(Γ)=∅ for aa*. The bifurcation at a=a* is followed by a sequence of nearby, O(ε2(logε)2) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on Γ. The structure of Ws(Γ)∩Wu(Γ) becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature.

Homoclinic orbits and chaos in three– and four–dimensional flows

We review recent work in which perturbative, geometric and topological arguments are used to prove the existence of countable sets of orbits connecting equilibria in ordinary differential equations. We first consider perturbations of a three–dimensional integrable system possessing a line of degenerate saddle points connected by a two–dimensional manifold of homoclinic loops. We show that this manifold splits to create transverse homoclinic orbits, and then appeal to geometrical and symbolic dynamic arguments to show that homoclinic bifurcations occur in which ‘simple’ connecting orbits are replaced by a countable infinity of such orbits. We discover a rich variety of connections among equilibria and periodic orbits, as well as more exotic sets, including Smale horseshoes. The second problem is a four–dimensional Hamiltonian system. Using symmetries and classical estimates, we again find countable sets of connecting orbits. There is no small parameter in this case, and the methods are non–perturbative.

Fronts and pulses in a class of reaction-diffusion equations: a geometric singular perturbation approach

In this paper we prove the existence of multiple-front solutions in a class of coupled reaction-diffusion equations with a small parameter. By a travelling wave ansatz we reduce the problem to a four-dimensional system of ordinary differential equations and prove the existence of a large variety of n-jump homoclinic and heteroclinic solutions, n = 1,2,3,... using geometric singular perturbation theory and Poincare maps. Numerical simulations of the reaction-diffusion equations indicate that several of the multi-front-type waves can be stable.