A heteroclinic surface between two saddle slow manifolds organizing sectors of rotation of mixed-mode oscillations

Abstract

We investigate the mechanism underlying the generation of mixed-mode oscillations (MMOs) in the four-dimensional Olsen model for peroxidase-oxidase reaction. MMOs are characterized by a pattern of small-amplitude oscillations (SAOs) and large-amplitude oscillations (LAOs), and they are known to occur in systems with multiple timescales. We identify many co-existing MMO periodic orbits for the Olsen model. We show that these MMOs come in pairs and exist over a sequence of staggered ranges of the timescale separation parameter. In the four-dimensional phase space of the Olsen model, we find that the SAOs come about due to the geometry of a surface of heteroclinic connections between two saddle slow manifolds. We compute the surface of heteroclinic connections through a region of phase space that exhibits a slow passage through a Hopf bifurcation, where the SAOs are generated. Moreover, we compute its intersection curves with two well-chosen hyperplanes. In this way, we show how the surface of heteroclinic connections delimits rotational sectors with given numbers of subsequent SAOs, and also determines the exit from this region of SAOs and the onset of LAOs.