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Title: 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.

Authors:
 [1];  [2];  [3];  [1];  [2]
  1. Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105 (Israel)
  2. (Israel)
  3. Center for Nonlinear Studies, Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
Publication Date:
OSTI Identifier:
22402567
Resource Type:
Journal Article
Resource Relation:
Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 25; Journal Issue: 6; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; AMPLITUDES; BIFURCATION; CHLORINE; COMPUTERIZED SIMULATION; DIFFUSION; ILLUMINANCE; IODINE; MALONIC ACID; MATHEMATICAL SOLUTIONS; PERIODICITY; PHASE SHIFT; RESONANCE

Citation Formats

Haim, Lev, Department of Oncology, Soroka University Medical Center, Beer-Sheva 84101, Hagberg, Aric, Meron, Ehud, and Department of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 84990. Non-monotonic resonance in a spatially forced Lengyel-Epstein model. United States: N. p., 2015. Web. doi:10.1063/1.4921768.
Haim, Lev, Department of Oncology, Soroka University Medical Center, Beer-Sheva 84101, Hagberg, Aric, Meron, Ehud, & Department of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 84990. Non-monotonic resonance in a spatially forced Lengyel-Epstein model. United States. doi:10.1063/1.4921768.
Haim, Lev, Department of Oncology, Soroka University Medical Center, Beer-Sheva 84101, Hagberg, Aric, Meron, Ehud, and Department of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 84990. Mon . "Non-monotonic resonance in a spatially forced Lengyel-Epstein model". United States. doi:10.1063/1.4921768.
@article{osti_22402567,
title = {Non-monotonic resonance in a spatially forced Lengyel-Epstein model},
author = {Haim, Lev and Department of Oncology, Soroka University Medical Center, Beer-Sheva 84101 and Hagberg, Aric and Meron, Ehud and Department of Solar Energy and Environmental Physics, BIDR, Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 84990},
abstractNote = {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.},
doi = {10.1063/1.4921768},
journal = {Chaos (Woodbury, N. Y.)},
number = 6,
volume = 25,
place = {United States},
year = {Mon Jun 15 00:00:00 EDT 2015},
month = {Mon Jun 15 00:00:00 EDT 2015}
}
  • Here, 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. Furthermore, we show that strong forcing may result in a π/2 phase shift of the resonant solutions, and argue that the nonequilibrium Ising-Bloch front bifurcationmore » can be reversed. Finally, 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.« less
  • Pattern selection, localized structure formation, and front propagation are analyzed within the framework of a model for the chlorine dioxide--iodine--malonic acid reaction that represents a key to understanding recently obtained Turing structures. This model is distinguished from previously studied, simple reaction-diffusion models by producing a strongly subcritical transition to stripes. The wave number for the modes of maximum linear gain is calculated and compared with the dominant wave number for the finally selected, stationary structures grown from the homogeneous steady state or developed behind a traveling front. The speed of propagation for a front between the homogeneous steady state andmore » a one-dimensional (1D) Turing structure is obtained. This velocity shows a characteristic change in behavior at the crossover between the subcritical and supercritical regimes for the Turing bifurcation. In the subcritical regime there is an interval where the front velocity vanishes as a result of a pinning of the front to the underlying structure. In 2D, two different nucleation mechanisms for hexagonal structures are illustrated on the Lengyel-Epstein and the Brusselator model. Finally, the observation of 1D and 2D spirals with Turing-induced cores is reported.« less
  • The formation of Turing patterns was investigated in thin cylindrical layers using the Lengyel-Epstein model of the chlorine dioxide-iodine-malonic acid reaction. The influence of the width of the layer W and the diameter D of the inner cylinder on the pattern with intrinsic wavelength l were determined in simulations with initial random noise perturbations to the uniform state for W < l/2 and D ∼ l or lower. We show that the geometric constraints of the reaction domain may result in the formation of helical Turing patterns with parameters that give stripes (b = 0.2) or spots (b = 0.37) in two dimensions. For b = 0.2, the helices weremore » composed of lamellae and defects were likely as the diameter of the cylinder increased. With b = 0.37, the helices consisted of semi-cylinders and the orientation of stripes on the outer surface (and hence winding number) increased with increasing diameter until a new stripe appeared.« less
  • Multifrequency forcing of systems undergoing a Hopf bifurcation to spatially homogeneous oscillations is investigated. For weak forcing composed of frequencies near the 1 ratio 1, 1 ratio 2, and 1 ratio 3 resonances, such systems can be described systematically by a suitably extended complex Ginzburg-Landau equation. Weakly nonlinear analysis shows that, generically, the forcing function can be tuned such that resonant triad interactions with weakly damped modes stabilize subharmonic 4- and 5-mode quasipatterns. In simulations starting from random initial conditions, domains of these quasipatterns compete and yield complex, slowly ordering patterns.