Defects, parcellation, and renormalized negative diffusivities in non-homogeneous oscillatory media

TL;DR

This paper explores how spatial non-homogeneities influence synchronization and defect formation in oscillatory media, revealing nonlinear effects on diffusivity and phase-locked states through a detailed phase diagram analysis.

Contribution

It provides a comprehensive phase diagram of a Ginzburg-Landau model with frequency gradients, highlighting defect formation and nonlinear diffusivity renormalization in non-homogeneous oscillatory systems.

Findings

Defects originate from nonlinear renormalization of diffusivity.

Plateaus form with specific scaling of defect number and length.

Phase diagram delineates stable, defect, and phase-locked regimes.

Abstract

Spatial non-homogeneities can synchronize clusters of spatially-extended oscillators in different frequency plateaus. Motivated by physiological rhythms, we fully characterize the phase diagram of a Ginzburg-Landau (GL) model with a gradient of frequencies. For large gradients and diffusion, the rest state is stable, and the linear spectrum around it maps onto the non-Hermitian Bloch-Torrey equation. When complex pairs of eigenvalues turn unstable, precursors of plateaus grow, separated by defects where the GL amplitude vanishes. Nonlinear effects either saturate the amplitude of plateaus or lead to a phase-locked state, with saddle-node bifurcations separating the two regimes. In the region of plateaus, we trace the formation of defects to a non-linear renormalization of the diffusivity, and determine the scaling of their number and length vs dynamical parameters.