---
abstract: 'We show how a variety of stable spatio-temporal periodic patterns can be created in 2D-lattices of coupled oscillators with non-homogeneous coupling delays. A “hybrid dispersion relation” is introduced, which allows studying the stability of time-periodic patterns analytically in the limit of large delay. The results are illustrated using the FitzHugh-Nagumo coupled neurons as well as coupled limit cycle (Stuart-Landau) oscillators.'
author:
- 'Markus Kantner[^1],'
- 'Serhiy Yanchuk[^2],'
- 'Eckehard Sch[ö]{}ll[^3]'
bibliography:
- '2dlattice.bib'
title: 'Delay-induced patterns in a two-dimensional lattice of coupled oscillators'
---

time delay, patterns , FitzHugh-Nagumo neurons, lattices, componentwise timeshift-transformation

34K13, 34K18, 34K20, 34K35, 34K60, 37L60, 37N25, 37N35, 93C23

Introduction\[sec: model system\]
=================================

Coupled dynamical systems with time delays arise in various applications including semiconductor lasers [@Franz2008; @Ludge2012; @Soriano2013; @Williams2013b], electronic circuits [@Reddy2000], optoelectronic oscillators [@Williams2013], neuronal networks [@Izhikevich2006; @Stepan2009; @Campbell2007], gene regulation networks [@Tiana2013], socio-economic systems [@Szalai2013; @Orosz2010] and many others [@Fiedler2007; @Flunkert2013; @Zakharova2013; @Kinzel2009; @Flunkert2010b; @Just2010]. Understanding the dynamics in such systems is a challenging task. Even a single oscillator with time delayed feedback exhibits phenomena, which are not expected in this class of systems, such as Eckhaus instability [@Wolfrum2006], coarsening [@Giacomelli2012], or chimera state [@Larger2013]. Some of them, like low frequency fluctuations 