Periodic Pulsating Dynamics of Slow-Fast Delayed Systems with a Period Close to the Delay

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dc.contributor.authorKravetc, Pavel
dc.contributor.authorRachinskii, Dmitry I.
dc.contributor.authorVladimirov, A.
dc.contributor.utdAuthorKravetc, Pavel
dc.contributor.utdAuthorRachinskii, Dmitry I.
dc.date.accessioned2020-10-02T15:52:18Z
dc.date.available2020-10-02T15:52:18Z
dc.date.issued2017-12-22
dc.descriptionDue to copyright restrictions and/or publisher's policy full text access from Treasures at UT Dallas is limited to current UTD affiliates (use the provided Link to Article).
dc.description.abstractWe consider slow-fast delayed systems and discuss pulsating periodic solutions, which are characterised by specific properties that (a) the period of the periodic solution is close to the delay, and (b) these solutions are formed close to a bifurcation threshold. Such solutions were previously found in models of mode-locked lasers. Through a case study of population models, this work demonstrates the existence of similar solutions for a rather wide class of delayed systems. The periodic dynamics originates from the Hopf bifurcation on the positive equilibrium. We show that the continuous transformation of the periodic orbit to the pulsating regime is simultaneous with multiple secondary almost resonant Hopf bifurcations, which the equilibrium undergoes over a short interval of parameter values. We derive asymptotic approximations for the pulsating periodic solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realisation of the bifurcation scenario is highlighted.
dc.description.departmentSchool of Natural Sciences and Mathematics
dc.description.sponsorshipNSF Grant DMS-1413223
dc.identifier.bibliographicCitationKravetc, P., D. Rachinskii, and A. Vladimirov. 2019. "Periodic pulsating dynamics of slow-fast delayed systems with a period close to the delay." European Journal of Applied Mathematics 30(1): 39-62, doi: 10.1017/S0956792517000377
dc.identifier.issn0956-7925
dc.identifier.issue1
dc.identifier.urihttps://dx.doi.org/10.1017/S0956792517000377
dc.identifier.urihttps://hdl.handle.net/10735.1/8973
dc.identifier.volume30
dc.language.isoen
dc.publisherCambridge Univ Press
dc.rights©2017 Cambridge University Press
dc.source.journalEuropean Journal of Applied Mathematics
dc.subjectBifurcation theory
dc.subjectSingular perturbations (Mathematics)
dc.subjectFunctional differential equations
dc.subjectPredation (Biology)
dc.subjectDifferential equations
dc.subjectLasers
dc.titlePeriodic Pulsating Dynamics of Slow-Fast Delayed Systems with a Period Close to the Delay
dc.type.genrearticle

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Rachinskiy, Dmitry I.