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Periodic Solutions of a Singularly Perturbed Delay Differential Equation with Two State-Dependent Delays

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Abstract

Periodic orbits and associated bifurcations of singularly perturbed state-dependent delay differential equations (DDEs) are studied when the profiles of the periodic orbits contain jump discontinuities in the singular limit. A definition of singular solution is introduced which is based on a continuous parametrisation of the possibly discontinuous limiting solution. This reduces the construction of the limiting profiles to an algebraic problem. A model two state-dependent DDE is studied in detail and periodic singular solutions are constructed with one and two local maxima per period. A complete characterisation of the conditions on the parameters for these singular solutions to exist facilitates an investigation of bifurcation structures in the singular case revealing folds and possible cusp bifurcations. Sophisticated boundary value techniques are used to numerically compute the bifurcation diagram of the state-dependent DDE when the perturbation parameter is close to zero. This confirms that the solutions and bifurcations constructed in the singular case persist when the perturbation parameter is nonzero, and hence demonstrates that the solutions constructed using our singular solution definition are useful and relevant to the singularly perturbed problem. Fold and cusp bifurcations are found very close to the parameter values predicted by the singular solution theory, and we also find period-doubling bifurcations as well as periodic orbits with more than two local maxima per period, and explain the alignment between the folds on different bifurcation branches.

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Acknowledgments

Tony Humphries thanks John Mallet-Paret and Roger Nussbaum for introducing him to this problem and patiently explaining their results in the one delay case. He is also grateful to NSERC (Canada) for funding through the Discovery Grants program. Renato Calleja thanks the Department of Mathematics and Statistics at McGill for their hospitality during his time as a postdoctoral fellow and on several return visits to Montreal. He is also grateful to NSERC and the Centre de recherches mathématiques, Montréal for funding and to FQRNT, Québec for a PBEEE award. Daniel Bernucci and Michael Snarski are grateful to NSERC for Undergraduate Student Research Awards. Namdar Homayounfar thanks the Institut des Sciences Mathématiques, Montréal for an Undergraduate Summer Scholarship.

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Author notes

  1. D. A. Bernucci

    Present address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

  2. N. Homayounfar

    Present address: Department of Statistics, University of Toronto, Toronto, ON, M5S 3G3, Canada

  3. M. Snarski

    Present address: Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montreal, QC, H3A 0B9, Canada

    A. R. Humphries, D. A. Bernucci, N. Homayounfar & M. Snarski

  2. Depto. Matemáticas y Mecánica, IIMAS, Universidad Nacional Autónoma de México, 01000, Mexico, Mexico

    R. C. Calleja

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  1. A. R. Humphries
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Correspondence to A. R. Humphries.

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Humphries, A.R., Bernucci, D.A., Calleja, R.C. et al. Periodic Solutions of a Singularly Perturbed Delay Differential Equation with Two State-Dependent Delays. J Dyn Diff Equat 28, 1215–1263 (2016). https://doi.org/10.1007/s10884-015-9484-4

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