On decomposing mixed-mode oscillations and their return maps

Abstract

Alternating patterns of small and large amplitude oscillations occur in a wide variety of physical, chemical, biological and engineering systems. These mixed-mode oscillations (MMOs) are often found in systems with multiple time scales. Previous differential equation modeling and analysis of MMOs has mainly focused on local mechanisms to explain the small oscillations. Numerical continuation studies reported different MMO patterns based on parameter variation. This paper aims at improving the link between local analysis and numerical simulation. Our starting point is a numerical study of a singular return map for the Koper model which is a prototypical example for MMOs that also relates to local normal form theory. We demonstrate that many MMO patterns can be understood geometrically by approximating the singular maps with affine and quadratic maps. Motivated by our numerical analysis we use abstract affine and quadratic return map models in combination with two local normal forms that generate small oscillations. Using this decomposition approach we can reproduce many classical MMO patterns and effectively decouple bifurcation parameters for local and global parts of the flow. The overall strategy we employ provides an alternative technique for understanding MMOs.

revised version, Figures 1 and 2 reduced in quality due to arXiv file size restrictions, 25 pages, 13 figures

References in corpus (1)

• Chaos at the border of criticality

Cited by in corpus (11)

• Hunting French Ducks in a Noisy Environment
• Mixed mode oscillations of the El Niño-Southern Oscillation
• Multiscale Geometry of the Olsen Model and Non-Classical Relaxation Oscillations
• Stochastic mixed-mode oscillations in a three-species predator-prey model
• Bifurcations of mixed-mode oscillations in three-timescale systems: an extended prototypical example
• Mixed mode oscillations in a conceptual climate model
• From random Poincaré maps to stochastic mixed-mode-oscillation patterns
• Entry-exit functions in fast-slow systems with intersecting eigenvalues
• Rigorous Enclosures of a Slow Manifold
• Stochasticity induced mixed-mode oscillations and distribution of recurrent outbreaks in an ecosystem
• A geometric analysis of the SIRS epidemiological model on a homogeneous network