Stern–Brocot trees in cascades of mixed-mode oscillations and canards in the extended Bonhoeffer–van der Pol and the FitzHugh–Nagumo models of excitable systems

Abstract

We report a systematic two-parameter study of the organization of mixed-mode oscillations and period-adding sequences observed in an extended Bonhoeffer–van der Pol and in a FitzHugh–Nagumo oscillator. For both systems, we construct isospike diagrams and show that the number of spikes of their periodic oscillations are organized in a remarkable hierarchical way, forming a Stern–Brocot tree. The Stern–Brocot tree is more general than the Farey tree. We conjecture the Stern–Brocot tree to also underlie the hierarchical structure of periodic oscillations of other systems supporting mixed-mode oscillations.