Period adding and broken Farey tree sequence of bifurcations for mixed-mode oscillations and chaos in the simplest three-variable nonlinear system

Abstract

A detailed study of the simplest three-variable model exhibiting mixed-mode oscillations and chaos is presented. We show that mixed-mode oscillations appear due to a sequence of bifurcations which is characterized by a combination of the Farey tree that is broken by chaotic windows and period adding. This scenario is supported by a family of one-dimensional return maps. The model also exhibits hysteresis between stable steady state and mixed modes.