Non-monotonic resonance in a spatially forced Lengyel-Epstein model

Abstract

We study resonant spatially periodic solutions of the Lengyel-Epstein model modified to describe the chlorine dioxide-iodine-malonic acid reaction under spatially periodic illumination. Using multiple-scale analysis and numerical simulations, we obtain the stability ranges of 2:1 resonant solutions, i.e., solutions with wavenumbers that are exactly half of the forcing wavenumber. We show that the width of resonant wavenumber response is a non-monotonic function of the forcing strength, and diminishes to zero at sufficiently strong forcing. We further show that strong forcing may result in a π/2 phase shift of the resonant solutions, and argue that the nonequilibrium Ising-Bloch front bifurcation can be reversed. We attribute these behaviors to an inherent property of forcing by periodic illumination, namely, the increase of the mean spatial illumination as the forcing amplitude is increased.

Turing Patterning in Stratified Domains

“…In such cases, approximate solutions for the system's eigenfunctions need to be derived that are orthogonal in the patterning layer. Examples that deviate even further from the classical case are growing domains (Crampin et al 1999;Plaza et al 2004;Krause et al 2019;Sánchez-Garduño et al 2019) and spatially heterogeneous reaction-diffusion processes (Benson et al 1998;Page et al 2003Page et al , 2005Haim et al 2015;Kolokolnikov and Wei 2018), for which the canonical approach does not work. In such cases, novel approaches to pattern-forming instabilities have recently been developed for growth (Madzvamuse et al 2010;) and heterogeneity (Krause et al 2020) under certain simplifications, but such analyses are quite different to the classical case.…”

“… 2003 , 2005 ; Haim et al. 2015 ; Kolokolnikov and Wei 2018 ), for which the canonical approach does not work. In such cases, novel approaches to pattern-forming instabilities have recently been developed for growth (Madzvamuse et al.…”

Modern perspectives on near-equilibrium analysis of Turing systems

“…Numerical and asymptotic studies have shown that heterogeneity can change local instability conditions for pattern formation [103,104], modulate size and wavelength of patterns [105], and localize (or pin) spike patterns in space [106][107][108]. There is also a large literature on reaction-diffusion systems with strongly localized heterogeneities [109,110], and numerous studies exploring spatially heterogeneous reaction-diffusion systems in chemical settings [61,[111][112][113][114][115][116][117]. See [118] in this theme issue for a modern review of chemical approaches to studying Turing systems.…”

Introduction to Focus Issue: Oscillations and Dynamic Instabilities in Chemical Systems: Dedicated to Irving R. Epstein on occasion of his 70th birthday

“…18 (Two contributions on Turing patterns use this model in the issue. 19,20 ) Reversible complexation of an activator species to form an unreactive, immobile complex was shown to facilitate the formation of Turing patterns. 7 This idea is further elaborated in the work of T oth and Horv ath in this issue by immobilizing autocatalysts in ionic systems though selective binding, even for ions with equal mobilities.…”

“…29 In this issue, simulations by the group of Ehud Meron show a non-monotonic resonance in the Lengyel-Epstein model due to spatially periodic illumination. 20 Although not directly related to light modulation, but rather to the general periodic perturbation of the simplified model of the BZ reaction (the Oregonator), the Rotstein group reports complex oscillatory patterns in which middle-amplitude oscillations play important role in addition to the common small and large amplitude oscillations observed in the BZ system. 30 turn can give scroll waves in 3D.…”