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Periodic pulsating dynamics of slow–fast delayed systems with a period close to the delay

Published online by Cambridge University Press:  22 December 2017

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, USA email:,
Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX, USA email:,
Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany email: Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia


We 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.

Copyright © Cambridge University Press 2017 

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P.K. and D.R. acknowledge the support of NSF through Grant DMS-1413223.


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