From excitability to waves and back: Patterns related to subcritical finite wavenumber Hopf bifurcations

Dissipative nonlinear waves arise in many systems that comprise non-equilibrium properties, such as action potentials, calcium waves, Belousov-Zhabotinsky chemical reaction, and electrochemical oscillations. Two variable reaction-diffusion models (e.g., FitzHugh-Nagumo), are frequently employed to scrutinize the generic features and also to distinguish between three universality classes: Oscillations that arise through a linear Hopf instability, excitability which gives rise upon nonlinear localized perturbations, to pulses via homoclinic (often Shil'nikov) connections to a rest state, and fronts that bi-asymptote to distinct uniform states. However, an extension to models with a larger number of variables shows richer qualitative and counterintuitive dynamic behaviors, for example, due to a finite wavenumber Hopf bifurcation that cannot arise otherwise. To show the intriguing properties of such PDEs, I will focus on two cases: (i) Extension of Shil'nikov to multi-pulse generation by a single localized perturbation and homoclinic snaking, and (ii) jumping oscillons (JOs) which are created via a modulational instability of excitable traveling pulses.