2006/03/01 by Georg A. Gottwald, Lorenz Kramer · 1 citation
Computer Science · Physics and Astronomy · #Nonlinear Dynamics and Pattern Formation #Quantum optics and atomic interactions #nlin.PS #stochastic dynamics and bifurcation
paper · pdf · doi:10.1063/1.2168393
22 pages, accepted for publication in Chaos
openalex publication_date 2006/03/01 · arxiv created 2006/03/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present a normal form for traveling waves in one-dimensional excitable media in the form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behavior of single pulses in a periodic domain and also the richer behavior of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.