Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems. (14th April 2020)

• Record Type: Journal Article
• Authors: Nyman, Karl H M; Ashwin, Peter; Ditlevsen, Peter D
• Is Part Of: Nonlinearity. Volume 33:Number 6(2020)
• Page Start: 2853
• Page End: 2904
• Publication Date: 2020-04-14
• Subjects: fast-slow dynamics -- relaxation oscillation -- bifurcation -- singularity; 37G10 -- 37G15 -- 37G35; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab7292
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the fast subsystem) and associated fast dynamics, parametrized by the slow variables. Using a distinguished parameter approach we are able to classify bifurcations for one fast and one slow variable. Some of these bifurcations are associated with the critical set losing manifold structure. We also conjecture a list of generic bifurcations of the critical set for one fast and two slow variables. We further consider how the bifurcations of the critical set can be associated with generic bifurcations of attracting relaxation oscillations under an appropriate singular notion of equivalence.