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Travelling pulses in the Barkley model: A geometric singular perturbation approach

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  • Grifò, Gabriele
  • Iuorio, Annalisa

Abstract

In this work, we investigate travelling pulse solutions to the Barkley model, a prototypical example of excitable system with activator-inhibitor dynamics. Such patterns are numerically observed for a wide range of parameter values and show how coherent structures can be induced by mechanisms different from diffusion-driven instability. The intrinsic multiscale nature of this system allows us to apply Geometric Singular Perturbation Theory (GSPT) to constructively establish the existence of travelling pulses as homoclinic orbits in the corresponding three-dimensional phase-space. The analytical findings are corroborated by a thorough numerical investigation via direct simulation as well as continuation based on the software AUTO.

Suggested Citation

  • Grifò, Gabriele & Iuorio, Annalisa, 2025. "Travelling pulses in the Barkley model: A geometric singular perturbation approach," Chaos, Solitons & Fractals, Elsevier, vol. 201(P1).
    • Handle: RePEc:eee:chsofr:v:201:y:2025:i:p1:s0960077925013207
    DOI: 10.1016/j.chaos.2025.117307
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    References listed on IDEAS

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    1. Consolo, Giancarlo & Curró, Carmela & Grifó, Gabriele & Valenti, Giovanna, 2024. "Stationary and Oscillatory patterned solutions in three-compartment reaction–diffusion systems: Theory and application to dryland ecology," Chaos, Solitons & Fractals, Elsevier, vol. 186(C).
    2. López Garza, Gabriel & Nicolás Mata, Aurelio & Román Alonso, Graciela & Godínez Fernández, José Rafael & Castro García, Miguel Alfonso, 2022. "Cell-to-cell mathematical modeling of arrhythmia phenomena in the heart," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 153-172.
    3. Currò, C. & Grifò, G. & Valenti, G., 2023. "Turing patterns in hyperbolic reaction-transport vegetation models with cross-diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    4. Smidtaite, Rasa & Ragulskis, Minvydas, 2022. "Spiral waves of divergence in the Barkley model of nilpotent matrices," Chaos, Solitons & Fractals, Elsevier, vol. 159(C).
    5. Grifó, Gabriele & Curró, Carmela & Valenti, Giovanna, 2025. "Heteroclinic connections in a hyperbolic reaction-transport model for excitable media," Chaos, Solitons & Fractals, Elsevier, vol. 200(P3).
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