Topologically protected edge oscillations in nonlinear dynamical units (2506.18699v1)

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.

• The paper demonstrates that robust edge-localized oscillations emerge in nonlinear oscillator grids through an SSH-inspired coupling that ensures topological protection.
• It employs numerical simulations and non-Hermitian Hamiltonian analysis to identify parameter regimes where edge oscillations persist despite noise and defects.
• The study highlights potential applications in designing resilient biochemical oscillators and synthetic biological networks with robust synchronization.

Topologically Protected Edge Oscillations in Nonlinear Dynamical Units

The paper "Topologically protected edge oscillations in nonlinear dynamical units" explores the emergence and dynamics of edge-localized oscillations within two-dimensional grids of nonlinear dynamical units. This paper extends the concept of topological protection, commonly observed in quantum mechanical systems, to classical nonlinear oscillatory systems, offering novel insights and potential applications, particularly in biological systems.

Background and Analytical Approach

Topological protection has been a pivotal concept in solid-state physics, where it ensures the robustness of certain system properties against perturbations. This robustness is typically associated with topological invariants that characterize complex behaviors, such as edge states in the quantum Hall effect. Extending these ideas to classical systems involves analyzing coupled networks of oscillators with specific on-site dynamics and coupling geometries that give rise to non-trivial topology.

The work begins with an examination of the Su-Schrieffer-Heeger (SSH) model, known for its alternating hopping parameters that lead to edge modes protected by topological invariants. Inspired by this, the authors consider nonlinear oscillators with on-site dynamics reminiscent of those in biochemical systems, specifically selecting Stuart-Landau oscillators, activator-repressor models, and brusselators. The coupling geometry is designed to mimic the SSH model, consisting of directed alternating weak and strong couplings, which induces localization of oscillations at the grid's edges.

Numerical and Analytical Findings

The simulations reveal that, under the appropriate parameter ratios for coupling strengths, oscillations become confined to the grid's boundaries, while the bulk oscillators transition to oscillation-death states. These edge oscillations are identified as a form of frequency chimera - synchronized in frequency but not in phase. Through numerical simulations, the research identifies the conditions necessary for this robust behavior and demonstrates its resilience to perturbations such as parameter mismatches, additive noise, and structural defects.

An effective Hamiltonian is derived to explore the topological properties of the system in momentum space. This non-Hermitian Hamiltonian unveils non-trivial topological characteristics through the computation of Zak phases for the bulk bands. The paper shows that the topological nature of these phases corresponds with the emergence of edge-localized oscillations - a clear manifestation of bulk-boundary correspondence, even in a non-Hermitian framework.

Broader Implications and Future Directions

This paper extends the scope of topological considerations into the field of nonlinear and biological systems, opening up avenues for creating robust oscillatory states in complex systems where classical modes of protection might not suffice. The results imply potential applications in designing biochemical oscillators that function reliably under biological fluctuations and noise, with implications for cellular and genetic networks that require precise timing mechanisms.

Future research could explore the complex interplay of topological protection and nonlinear dynamics in other systems and geometries. In particular, exploring the consequences of introducing additional nonlinear interactions, such as repulsive coupling, might offer richer dynamics and potential for novel states like chiral edge modes. Understanding these phenomena could foster innovation in designing synthetic biological networks and enhancing the robustness of synchronization in artificial systems.