Travelling pulses in time-delay systems

Stefan Ruschel, Northumbria University. Part of the Leeds Applied Nonlinear Dynamics seminar series.

Speaker

Stefan Ruschel, University of Newcastle

Title: 

Travelling pulses in time-delay systems

Abstract

I will review recent results on the existence, stability and bifurcations of temporally localized pulses arising in delay-coupled excitable systems that arise in optics, neuroscience and epidemiology, among others. I will motivate a convenient coordinate transformation that allows to identify such pulses with homoclinic orbits, analogous to the travelling wave ansatz for pulsed periodic travelling waves in one-dimensional reaction-diffusion partial differential equations. Using this transformation, I will discuss by means of a prototypical example how the branching of two-pulse solution from one-pulse solutions with non-oscillating tails is organized by codimension-two homoclinic bifurcation points of a real saddle equilibrium, analogous to spatially extended systems.

References:

• [1] Yanchuk, S., Ruschel, S., Sieber, J., & Wolfrum, M. (2019). Temporal dissipative solitons in time-delay feedback systems. Physical review letters, 123(5), 053901.
• [2] Ruschel, S., Krauskopf, B., & Broderick, N. G. (2020). The limits of sustained self-excitation and stable periodic pulse trains in the Yamada model with delayed optical feedback. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(9), 093101.
• [3] Giraldo, A., & Ruschel, S. (2022). Pulse-adding of Temporal Dissipative Solitons: Resonant Homoclinic Points and the Orbit Flip of Case B with Delay. arXiv preprint arXiv:2207.13547.