Topologically protected edge oscillations in nonlinear dynamical units

TL;DR

This paper demonstrates how topological protection can induce robust edge oscillations in nonlinear classical oscillator networks, inspired by condensed matter physics, with potential applications in biochemical systems.

Contribution

It introduces a novel classical oscillator model with topologically protected edge states, extending topological concepts to nonlinear and non-Hermitian dynamical systems.

Findings

Edge oscillations are robust to noise and defects.

Topological invariants explain the localization of oscillations.

System tuning controls oscillatory and dead regions.

Abstract

Many examples from quantum and classical physics are known where topological protection is responsible for the robustness of the dynamics. Less explored is the role of topological protection in the context of classical oscillatory systems. As on-site dynamics, we consider prototypical oscillator models with possible applications in biochemical systems. However, our choice of coupling geometry is inspired by models from condensed matter physics which -- in isolation -- guarantee non-trivial topology in momentum space. We choose directed couplings between units on a two-dimensional grid, alternating between weak and strong values, such that oscillations become localized at the edges of the grid while bulk units transition to oscillation-death states, resulting in a frequency chimera-like state. These patterns are resilient to parameter mismatches, additive noise, and structural defects. To explain the robustness of these edge oscillations we use topological characteristics and calculate Zak phases. As it turns out, the edge-localized oscillations result from a bulk-boundary correspondence that applies to our system even though our derived effective Hamiltonian is non-Hermitian. By appropriately tuning the system parameters, it is possible to control which regions of the two-dimensional grid are in an oscillatory state and which settle to oscillation-death states.