ABSTRACT. In this paper we study the existence and stability of asymptotically large stationary multi-pulse solutions in a family of singularly perturbed reaction-diffusion equations. This family includes the generalized Gierer-Meinhardt equation. The existence of N-pulse homoclinic orbits (N ≥ 1) is established by the methods of geometric singular perturbation theory. A theory, called the NLEP (=NonLocal Eigenvalue Problem) approach, is developed, by which the stability of these patterns can be studied explicitly. This theory is based on the ideas developed in our earlier work on the Gray-Scott model. It is known that the Evans function of the linear eigenvalue problem associated to the stability of the pattern can be decomposed into the product of a slow and a fast transmission function. The NLEP approach determines explicit leading order approximations of these transmission functions. It is shown that the zero/pole cancellation in the decomposition of the Evans function, called the NLEP paradox, is a phenomenon that occurs naturally in singularly perturbed eigenvalue problems. It follows that the zeroes of the Evans function, and thus the spectrum of the stability problem, can be studied by the slow transmission function. The key ingredient of the analysis of this expression is a transformation of the associated nonlocal eigenvalue problem into an inhomogeneous hypergeometric differential equation. By this transformation it is possible to determine both the number and the position of all elements in the discrete spectrum of the linear eigenvalue problem. The method is applied to a special case that corresponds to the classical model proposed by Gierer and Meinhardt. It is shown that the one-pulse pattern can gain (or lose) stability through a Hopf bifurcation at a certain value µ Hopf of the main parameter µ. The NLEP approach not only yields a leading order approximation of µ Hopf , but it also shows that there is another bifurcation value, µ edge , at which a new (stable) eigenvalue bifurcates from the edge of the essential spectrum. Finally, it is shown that the N-pulse patterns are always unstable when N ≥ 2. 443

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In this work, we analyse a pair of one-dimensional coupled reaction-diffusion equations known as the Gray-Scott model, in which self-replicating patterns have been observed. We focus on stationary and travelling patterns, and begin by deriving the asymptotic scaling of the parameters and variables necessary for the analysis of these patterns. Single-pulse and multi-pulse stationary waves are shown to exist in the appropriately scaled equations on the infinite line. A (single) pulse is a narrow interval in which the concentration U of one chemical is small, while that of the second, V , is large, and outside of which the concentration U tends (slowly) to the homogeneous steady state U ≡ 1, while V is everywhere close to V ≡ 0. In addition, we establish the existence of a plethora of periodic steady states consisting of periodic arrays of pulses interspersed by intervals in which the concentration V is exponentially small and U varies slowly. These periodic states are spatially inhomogeneous steady patterns whose length scales are determined exclusively by the reactions of the chemicals and their diffusions, and not by other mechanisms such as boundary conditions. A complete bifurcation study of these solutions is presented. We also establish the non-existence of travelling solitary pulses in this system. This non-existence result reflects the system's degeneracy and indicates that some event, for example pulse splitting, 'must' occur when two pulses move apart from each other (as has been observed in simulations): these pulses evolve towards the non-existent travelling solitary pulses. The main mathematical techniques employed in this analysis of the stationary and travelling patterns are geometric singular perturbation theory and adiabatic Melnikov theory. Finally, the theoretical results are compared to those obtained from direct numerical simulation of the coupled partial differential equations on a 'very large' domain, using a moving grid code. It has been checked that the boundaries do not influence the dynamics. A subset of the family of stationary single pulses appears to be stable. This subset determines the boundary of a region in parameter space in which the self-replicating process takes place. In that region, we observe that the core of a time-dependent self-replicating pattern turns out to be precisely a stationary periodic pulse pattern of the type that we construct. Moreover, the simulations reveal some other essential components of the pulse-splitting process and provide an important guide to further analysis.

The long-term dynamics of many dynamical systems evolve on an attracting, invariant "slow manifold" that can be parameterized by a few observable variables. Yet a simulation using the full model of the problem requires initial values for all variables. Given a set of values for the observables parameterizing the slow manifold, one needs a procedure for finding the additional values such that the state is close to the slow manifold to some desired accuracy. We consider problems whose solution has a singular perturbation expansion, although we do not know what it is nor have any way to compute it. We show in this paper that, under some conditions, computing the values of the remaining variables so that their (m + 1)st time derivatives are zero provides an estimate of the unknown variables that is an mth-order approximation to a point on the slow manifold in sense to be defined. We then show how this criterion can be applied approximately when the system is defined by a legacy code rather than directly through closed form equations.

In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781-3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.