23 January 2025 Thu 4 pm

Studying the existence and stability of periodic traveling waves in networks of ordinary differential equations (such as discrete rings and lattices) is a challenging problem to tackle when dealing with a large number of nodes.

In this talk, we will present a theoretical and numerical framework for effectively determining the spectrum and stability of such waves in networks with $Z_n$-equivariance. In this framework, a delay-advance differential equation (master equation) is derived and used to set up a suitable two-point boundary problem (2PBVP) that captures the stability of the wave in the network. Since the defined 2PBVP is independent of the network size, it is suitable for numerical continuation. In this way, we compute a master stability curve containing the spectrum of the traveling wave, which also captures its spectrum when embedded in even larger networks with higher wavenumber.

Thanks to this stability curve, instability and multistability in networks can be understood by varying the wave number as a continuous parameter and observing crossings of the master stability curve with the imaginary axis. To showcase our framework, we consider a dissipatively coupled ring of Fitzhugh-Nagumo oscillators and study the coexistence of different attracting traveling waves at different network sizes.

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