Pulse-adding of Temporal Dissipative Solitons: Resonant Homoclinic Points and the Orbit Flip of Case B with Delay

TL;DR

This paper numerically investigates the bifurcation structure of temporal dissipative solitons in delay differential equations with large delay, revealing organizing centers like homoclinic bifurcations and orbit flips that govern multi-pulse solutions.

Contribution

It extends the understanding of bifurcation structures of temporal dissipative solitons in DDEs by identifying key organizing bifurcations and generalizing a prototypical model with delay.

Findings

Two-pulse TDSs branch from one-pulse TDSs at codimension-two homoclinic bifurcation points.

Non-orientable resonant homoclinic bifurcation and orbit flip of case B organize two-pulse TDS existence.

Folds of homoclinic bifurcations bound the existence region of TDSs in large delay DDEs.

Abstract

We numerically investigate the branching of temporally localized, two-pulse periodic traveling waves from one-pulse periodic traveling waves with non-oscillating tails in delay differential equations (DDEs) with large delay. Solutions of this type are commonly referred to as temporal dissipative solitons (TDSs) in applications, and we adopt this term here. We show by means of a prototypical example that -- analogous to traveling pulses in reaction-diffusion partial differential equations (PDEs) -- the branching of two-pulse TDSs from one-pulse TDSs with non-oscillating tails is organized by codimension-two homoclinic bifurcation points of a real saddle equilibrium in a corresponding traveling wave frame. We consider a generalization of Sandstede's model (a prototypical model for studying codimension-two homoclinic bifurcation points in ODEs) with an additional time-shift parameter, and use Auto07p and DDE-BIFTOOL to compute numerically the unfolding of these bifurcation points in the resulting DDE. We then interpret this model as the traveling wave equation for TDSs in a DDE with large delay by exploiting the reappearance of periodic solutions in DDEs. In doing so, we identify both the non-orientable resonant homoclinic bifurcation and the orbit flip bifurcation of case $\mathbf{B}$ as organizing centers for the existence of two-pulse TDSs in the DDE with large delay. Additionally, we discuss how folds of homoclinic bifurcations in an auxiliary system bound the existence region of TDSs in the DDE with large delay.