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

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