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May  2004, 4(2): 419-434. doi: 10.3934/dcdsb.2004.4.419

Excitability in a model with a saddle-node homoclinic bifurcation

Rui Dilão 1, and  András Volford 2,

1. 

Nonlinear Dynamics Group, Instituto Superior Técnico, Department of Physics, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
2. 

Center for Complex and Nonlinear Systems, Technical University of Budapest, H-1521 Budapest, Hungary

Received  December 2002 Revised  July 2003 Published  February 2004

In order to describe excitable reaction-diffusion systems, we derive a two-dimensional model with a Hopf and a semilocal saddle-node homoclinic bifurcation. This model gives the theoretical framework for the analysis of the saddle-node homoclinic bifurcation as observed in chemical experiments, and for the concepts of excitability and excitability threshold. We show that if diffusion drives an extended system across the excitability threshold then, depending on the initial conditions, wave trains, propagating solitary pulses and propagating pulse packets can exist in the same extended system. The extended model shows chemical turbulence for equal diffusion coefficients and presents all the known types of topologically distinct activity waves observed in chemical experiments. In particular, the approach presented here enables to design experiments in order to decide between excitable systems with sharp and finite width thresholds.
Keywords: chemical turbulence, Excitability, thresholds., reaction-diffusion equations.
Mathematics Subject Classification: 35K57, 35K60, 92C45, 92C1.
Citation: Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
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