Greater accuracy and broadened applicability of phase reduction using isostable coordinates

• Published in: Journal of mathematical biology 2018-01, Vol.76 (1-2), p.37-66
• Main Authors: Wilson, Dan, Ermentrout, Bard
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Dynamics; Mathematical and Computational Biology; Mathematical models; Mathematics; Mathematics and Statistics; Optimal control; Orbital mechanics; Orbits; Reduction; Strategy
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-017-1141-6

The applicability of phase models is generally limited by the constraint that the dynamics of a perturbed oscillator must stay near its underlying periodic orbit. Consequently, external perturbations must be sufficiently weak so that these assumptions remain valid. Using the notion of isostables of periodic orbits to provide a simplified coordinate system from which to understand the dynamics transverse to a periodic orbit, we devise a strategy to correct for changing phase dynamics for locations away from the limit cycle. Consequently, these corrected phase dynamics allow for perturbations of larger magnitude without invalidating the underlying assumptions of the reduction. The proposed reduction strategy yields a closed set of equations and can be applied to periodic orbits embedded in arbitrarily high dimensional spaces. We illustrate the utility of this strategy in two models with biological relevance. In the first application, we find that an optimal control strategy for modifying the period of oscillation can be improved with the corrected phase reduction. In the second, the corrected phase reduced dynamics are used to understand adaptation and memory effects resulting from past perturbations.