2008/02/13 by Georg A. Gottwald, G. A. Gottwald
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Bifurcation #Bifurcation diagram #Bifurcation theory #Center manifold #Chaos control and synchronization #Classical mechanics #Homoclinic bifurcation #Hopf bifurcation #Instability #Mathematical analysis #Mathematics #Mechanics #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Physics #Pitchfork bifurcation #Quantum mechanics #Saddle-node bifurcation #Transcritical bifurcation #nlin.PS #q-bio.OT #stochastic dynamics and bifurcation
paper · pdf · doi:10.1063/1.2890430
to be published in Chaos
arxiv created 2008/02/13 · openalex publication_date 2008/03/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We present a bifurcation analysis of a normal form for traveling waves in one-dimensional excitable media. The normal form that has been recently proposed on phenomenological grounds is given in the form of a differential delay equation. The normal form exhibits a symmetry-preserving Hopf bifurcation that may coalesce with a saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans, which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.