Journal of Dynamics and Differential Equations

, Volume 28, Issue 3–4, pp 1215–1263 | Cite as

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

  • A. R. HumphriesEmail author
  • D. A. Bernucci
  • R. C. Calleja
  • N. Homayounfar
  • M. Snarski


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.


State-dependent delay differential equations Bifurcation theory Periodic solutions Singularly perturbed solutions Numerical approximation 

Mathematics Subject Classification

34K18 34K13 34K26 34K28 



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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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • A. R. Humphries
    • 1
    Email author
  • D. A. Bernucci
    • 1
    • 2
  • R. C. Calleja
    • 3
  • N. Homayounfar
    • 1
    • 4
  • M. Snarski
    • 1
    • 5
  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Depto. Matemáticas y Mecánica, IIMASUniversidad Nacional Autónoma de MéxicoMexicoMexico
  4. 4.Department of StatisticsUniversity of TorontoTorontoCanada
  5. 5.Division of Applied MathematicsBrown UniversityProvidenceUSA

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