An Explicit Theory for Pulses in Two Component, Singularly Perturbed, Reaction–Diffusion Equations

Robust Stability of Multicomponent Membranes: the Role of Glycolipids

“…This framework contains the two-component singularly perturbed systems as a subfamily that describes strongly asymmetric two-component mixtures. Previous work has exploited the asymmetry to provide explicit leading order constructions of homoclinic connections [11]. In Theorems 3.5 and 3.9 we show that the robust pearling stability condition corresponds to a natural geometric feature arising in the singular perturbation construction.…”

“…The component functions F i j obey mild regularity assumptions [11]. The resulting model can be written as a first order dynamical system in the form (1.6), which in the fast spatial variable ζ := z/δ takes the form (u 1 ) ζ = δ p 1 ,…”

“…In the singular limit, this concatenation procedure provides a homoclinic orbit precisely when the take-off curve T o intersects the slow unstable manifold W u s (0, 0); see Figure 2. When this intersection is nongenerate, transversality arguments imply that the singular orbit persists for sufficiently small 0 < δ 1; for the full analysis, see [11].…”

“…Remark 2.6. The result from [11] encompasses a larger class of systems, in particular the equilibrium a may lie on the boundary of the domain of definition of the vector field F. This necessitates additional technical assumptions on F, see [11, Assumptions (A1-4)].…”

Existence of Pulses for the System of Competition of Species

“…Miklukho-Maklaya 6, Moscow, 117198, Russia [4], [13], the Gierer-Meinhardt model [6], [14] and a three component system [2]. Singular perturbation methods to study existence and stability of pulses for a system of two equations are used in [3]. In a previous work [5] we studied the particular case where the nonlinear terms in (1) take the form: F 1 (w 1 , w 2 ) = f 1 (w 2 ) − w 1 , F 2 (w 1 , w 2 ) = f 2 (w 1 ) − w 2 .…”