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Abstract
In this article, we study canard solutions of the forced van der Pol equation in the relaxation limit for low, intermediate, and highfrequency periodic forcing. A central numerical observation made herein is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. In the lowfrequency forcing regime, we demonstrate the existence of primary maximal canards induced by folded saddle nodes of type I and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. Then, we turn to the intermediate and highfrequency forcing regimes and show that the forced van der Pol possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms the central numerical observation that the folds of the maximal canards created in the lowfrequency regime continue directly into the folds of the maximal torus canards that exist in the intermediate and highfrequency regimes. In addition, we study the secondary canards induced by the folded singularities in the lowfrequency regime and find that the fold curves of the secondary canards turn around in the intermediatefrequency regime, instead of continuing into the highfrequency regime. Also, we identify the mechanism responsible for this turning. Finally, we show that the forced van der Pol equation is a normal formtype equation for a class of singlefrequency periodically driven slow/fast systems with two fast variables and one slow variable which possess a nondegenerate fold of limit cycles. The analytic techniques used herein rely on geometric desingularisation, invariant manifold theory, Melnikov theory, and normal form methods. The numerical methods used herein were developed in Desroches et al. (SIAM J Appl Dyn Syst 7:1131–1162, 2008, Nonlinearity 23:739–765 2010).
Original language  English 

Journal  Journal of Nonlinear Science 
Volume  26 
Issue number  2 
Pages (fromto)  405451 
Number of pages  47 
ISSN  09388974 
DOIs  
Publication status  Published  2016 
Keywords
 Folded singularities
 Canards
 Torus canards
 Torus bifurcation
 Mixedmode oscillations
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 1 Finished

COFUNDPostdocDTU: COFUNDPostdocDTU
Præstrud, M. R. & Brodersen, S. W.
01/01/2014 → 31/12/2019
Project: Research