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Homoclinic bifurcations at the onset of pulse self-replication
Authors:Arjen Doelman  Tasso J Kaper
Institution:a C.W.I. Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands
b Korteweg-de Vries Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
c Department of Mathematics & Center for BioDynamics, Boston University, 111 Cummington Street, Boston, MA 02215, USA
d Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
Abstract:We establish a series of properties of symmetric, N-pulse, homoclinic solutions of the reduced Gray-Scott system: u=uv2, v=vuv2, which play a pivotal role in questions concerning the existence and self-replication of pulse solutions of the full Gray-Scott model. Specifically, we establish the existence, and study properties, of solution branches in the (α,β)-plane that represent multi-pulse homoclinic orbits, where α and β are the central values of u(x) and v(x), respectively. We prove bounds for these solution branches, study their behavior as α→∞, and establish a series of geometric properties of these branches which are valid throughout the (α,β)-plane. We also establish qualitative properties of multi-pulse solutions and study how they bifurcate, i.e., how they change along the solution branches.
Keywords:35K45  35K57  35B25  35B32  35B35  35B40  34C30  34C37  92C15  92E20
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