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

TL;DR

This paper develops a framework for analyzing periodic solutions and bifurcations in singularly perturbed delay differential equations with state-dependent delays, introducing a new definition of singular solutions and confirming their relevance through numerical bifurcation analysis.

Contribution

It introduces a novel definition of singular solutions for state-dependent DDEs with discontinuities and demonstrates their persistence in the perturbed case through numerical bifurcation analysis.

Findings

Existence conditions for singular solutions with multiple maxima.

Identification of fold and cusp bifurcations near singular solutions.

Persistence of singular solutions and bifurcations when perturbation parameter is nonzero.

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 delay differential equation 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.