Spike-adding in a canonical three time scale model: superslow explosion & folded-saddle canards

Abstract : We examine the origin of complex bursting oscillations in a phenomenological ordinary differential equation model with three time scales. We show that bursting solutions in this model arise from a Hopf bifurcation followed by a sequence of spike-adding transitions, in a manner reminiscent of spike- adding transitions previously observed in systems with two time scales. However, the details of the process can be much more complex in this three-time-scale context than in two-time-scale systems. In particular, we find that spike-adding can involve canard explosions occurring on two different time scales and is associated with passage near a folded-saddle singularity. We show that the character of the bursting and the form of spike-adding transitions that occur depend on the geometry of certain singular limit systems, specifically the relative positions of the critical and superslow manifolds. We also show that, unlike the case of spike-adding in two-time-scale systems, the onset of a new spike in our model is not typically associated with a local maximum in the period of the bursting oscillation.
Document type :
Journal articles
Complete list of metadatas

Cited literature [31 references]  Display  Hide  Download

https://hal.inria.fr/hal-01652020
Contributor : Mathieu Desroches <>
Submitted on : Friday, January 12, 2018 - 3:40:26 PM
Last modification on : Friday, April 19, 2019 - 1:26:04 PM
Long-term archiving on : Sunday, May 6, 2018 - 10:16:45 AM

File

dk_tts_submitted.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Mathieu Desroches, Vivien Kirk. Spike-adding in a canonical three time scale model: superslow explosion & folded-saddle canards. SIAM Journal on Applied Dynamical Systems, Society for Industrial and Applied Mathematics, 2018, 17 (3), pp.1989-2017. ⟨10.1137/17M1143411⟩. ⟨hal-01652020⟩

Share

Metrics

Record views

353

Files downloads

338