Fronts and patterns in a spatially forced CDIMA reaction

Physical Review E, 2013

Spatial periodic forcing can entrain a pattern-forming system in the same way as temporal periodic forcing can entrain an oscillator. The forcing can lock the pattern's wave number to a fraction of the forcing wave number within tonguelike domains in the forcing parameter plane, it can increase the pattern's amplitude, and it can also create patterns below their onset. We derive these results using a multiple-scale analysis of a spatially forced Swift-Hohenberg equation in one spatial dimension. In two spatial dimensions the one-dimensional forcing can induce a symmetry-breaking instability that leads to two-dimensional (2D) patterns, rectangular or oblique. These patterns resonate with the forcing by locking their wave-vector component in the forcing direction to half the forcing wave number. The range of this type of 2:1 resonance overlaps with the 1:1 resonance tongue of stripe patterns. Using a multiple-scale analysis in the overlap region we show that the 2D patterns can destabilize the 1:1 resonant stripes even at exact resonance. This result sheds new light on the use of spatial periodic forcing for controlling patterns.

Physical Chemistry Chemical Physics, 2011

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns with spatial periodic forcing. Wavenumber-locked stripe patterns are the typical resonant structures that labyrinthine patterns exhibit in response to one-dimensional forcing by illumination when images of stripes are projected on a working medium. Our experimental results reveal that segmented oblique, hexagonal and rectangular patterns can also be obtained. However, these two-dimensional resonant structures only develop in a relatively narrow range of forcing parameters, where the unforced stripe pattern is in close proximity to the domain of hexagonal patterns. Numerical simulations based on a model that incorporates the forcing by illumination using an additive term reproduce well the experimental observations. These findings confirm that additive one-dimensional forcing can generate a two-dimensional resonant response. However, such a response is considerably less robust than the effect of multiplicative forcing.

Physical Review E, 2004

Various resonant and near-resonant patterns form in a light-sensitive Belousov-Zhabotinsky (BZ) reaction in response to a spatially homogeneous time-periodic perturbation with light. The regions (tongues) in the forcing frequency and forcing amplitude parameter plane where resonant patterns form are identified through analysis of the temporal response of the patterns. Resonant and near-resonant responses are distinguished. The unforced BZ reaction shows both spatially uniform oscillations and rotating spiral waves, while the forced system shows patterns such as standing-wave labyrinths and rotating spiral waves. The patterns depend on the amplitude and frequency of the perturbation, and also on whether the system responds to the forcing near the uniform oscillation frequency or the spiral wave frequency. Numerical simulations of a forced FitzHugh-Nagumo reaction-diffusion model show both resonant and near-resonant patterns similar to the BZ chemical system.

2015

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.

Arxiv preprint nlin/0703059, 2007

Abstract: Multi-frequency forcing of systems undergoing a Hopf bifurcation to spatially homogeneous oscillations is investigated using a complex Ginzburg-Landau equation that systematically captures weak forcing functions that simultaneously hit the 1: 1-, the 1: 2-, ...

Physica D: Nonlinear Phenomena, 2004

We use the forced complex Ginzburg-Landau (CGL) equation to study resonance in oscillatory systems periodically forced at approximately twice the natural oscillation frequency. The CGL equation has both resonant spatially uniform solutions and resonant two-phase standing-wave pattern solutions such as stripes or labyrinths. The spatially uniform solutions form a tongueshaped region in the parameter plane of the forcing amplitude and frequency. But the parameter range of resonant standing-wave patterns does not coincide with the tongue of spatially uniform oscillations. On one side of the tongue the boundary of resonant patterns is inside the tongue and is formed by the nonequilibrium Ising Bloch bifurcation and the instability to traveling waves. On the other side of the tongue the resonant patterns extend outside the tongue forming a parameter region in which standing-wave patterns are resonant but uniform oscillations are not. The standing-wave patterns in that region appear similar to those inside the tongue but the mechanism of their formation is different. The formation mechanism is studied using a weakly nonlinear analysis near a Hopf-Turing bifurcation. The analysis also gives the existence and stability regions of the standing-wave patterns outside the resonant tongue. The analysis is supported by numerical solutions of the forced complex Ginzburg-Landau equation.

Physical Chemistry Chemical Physics, 2012

We use the photosensitive chlorine dioxide-iodine-malonic acid reaction-diffusion system to study wavenumber locking of Turing patterns to two-dimensional ''square'' spatial forcing, implemented as orthogonal sets of bright bands projected onto the reaction medium. Various resonant structures emerge in a broad range of forcing wavelengths and amplitudes, including square lattices and superlattices, one-dimensional stripe patterns and oblique rectangular patterns. Numerical simulations using a model that incorporates additive two-dimensional spatially periodic forcing reproduce well the experimental observations.

Physical Review E, 2006

The effects of a spatially periodic forcing on an oscillating chemical reaction as described by the Lengyel-Epstein model are investigated. We find a surprising competition between two oscillating patterns, where one is harmonic and the other subharmonic with respect to the spatially periodic forcing. The occurrence of a subharmonic pattern is remarkable as well as its preference up to rather large values of the modulation amplitude. For small modulation amplitudes we derive from the model system a generic equation for the envelope of the oscillating reaction that includes an additional forcing contribution, compared to the amplitude equations known from previous studies in other systems. The analysis of this amplitude equation allows the derivation of analytical expressions even for the forcing corrections to the threshold and to the oscillation frequency, which are in a wide range of parameters in good agreement with the numerical analysis of the complete reaction equations. In the nonlinear regime beyond threshold, the subharmonic solutions exist in a finite range of the control parameter that has been determined by solving the reaction equations numerically for various sets of parameters.

Chemical Engineering Science, 2000

We utilize a simple three-variable reaction}di!usion model to study patterns that emerge beyond the onset of the (short-)wave instability. We have found various wave patterns including standing waves, traveling waves, asymmetric standing}traveling waves and target patterns. We employ both periodic and zero #ux boundary conditions in the simulations, and we analyze the patterns using space}time two-dimensional Fourier spectra. A fascinating pattern of waves which periodically change their direction of propagation along a ring is found for very short systems. A related pattern of modulated standing waves is found for systems with zero #ux boundary conditions. In a two-dimensional system with small overcriticality we observe a wide variety of standing wave patterns. These include plain and modulated stripes, squares and rhombi. We also "nd standing waves consisting of periodic time sequences of stripes, rhombi and hexagons. The short-wave instability can lead to a much greater variety of spatio-temporal patterns than the aperiodic Turing and the long-wave oscillatory instabilities. For example, a single oscillatory cycle may display all the basic patterns related to the aperiodic Turing instability * stripes, hexagons and inverted hexagons (honeycomb) * as well as rhombi and modulated stripes. A rich plethora of patterns is seen in a system with cylindrical geometry * examples include rotating patterns of standing waves and counter-propagating waves.

Physical Review E, 2005

We investigate the response of two-dimensional pattern forming systems with a broken up-down symmetry, such as chemical reactions, to spatially resonant forcing and propose related experiments. The nonlinear behavior immediately above threshold is analyzed in terms of amplitude equations suggested for a 1 : 2 and 1 : 1 ratio between the wavelength of the spatial periodic forcing and the wavelength of the pattern of the respective system. Both sets of coupled amplitude equations are derived by a perturbative method from the Lengyel-Epstein model describing a chemical reaction showing Turing patterns, which gives us the opportunity to relate the generic response scenarios to a specific pattern forming system. The nonlinear competition between stripe patterns and distorted hexagons is explored and their range of existence, stability and coexistence is determined. Whereas without modulations hexagonal patterns are always preferred near onset of pattern formation, single mode solutions (stripes) are favored close to threshold for modulation amplitudes beyond some critical value. Hence distorted hexagons only occur in a finite range of the control parameter and their interval of existence shrinks to zero with increasing values of the modulation amplitude. Furthermore depending on the modulation amplitude the transition between stripes and distorted hexagons is either sub-or supercritical.