# Comb-like Turing patterns embedded in Hopf oscillations: Spatially localized states outside the 2:1 frequency locked region

## TL;DR

This paper demonstrates a new mechanism for spatially localized comb-like Turing patterns embedded in Hopf oscillations, extending frequency locking domains in oscillatory media through a local dynamics pinning mechanism.

## Contribution

It introduces a novel local dynamics pinning mechanism for Hopf fronts that enables the formation of localized Turing states outside the traditional resonance region.

## Key findings

- Localized states appear without flip-flop dynamics.
- The mechanism extends the frequency locking domain.
- Results match the core size in spiral waves.

## Abstract

A generic distinct mechanism for the emergence of spatially localized states embedded in an oscillatory background is demonstrated by using 2:1 frequency locking oscillatory system. The localization is of Turing type and appears in two space dimensions as a comb-like state in either $\pi$ phase shifted Hopf oscillations or inside a spiral core. Specifically, the localized states appear in absence of the well known flip-flop dynamics (associated with collapsed homoclinic snaking) that is known to arise in the vicinity of Hopf-Turing bifurcation in one space dimension. Derivation and analysis of three Hopf-Turing amplitude equations in two space dimensions reveals a local dynamics pinning mechanism for Hopf fronts, which in turn allows the emergence of perpendicular (to the Hopf front) Turing states. The results are shown to agree well with the comb-like core size that forms inside spiral waves. In the context of 2:1 resonance, these localized states form outside the 2:1 resonance region and thus extend the frequency locking domain for spatially extended media, such as periodically driven Belousov-Zhabotinsky chemical reactions. Implications to chlorite-iodide-malonic-acid and shaken granular media are also addressed.