Spatio-temporal patterns in a reaction–diffusion system with wave instability (original) (raw)
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Standing Waves in a Two-Dimensional Reaction−Diffusion Model with the Short-Wave Instability
The Journal of Physical Chemistry A, 1999
Various patterns of standing waves are found beyond the onset of the short-wave instability in a model reactiondiffusion system. These include plain and modulated stripes, squares, and rhombi in systems with square and rectangular geometry and patterns with rotational symmetry in systems with circular geometry. We also find standing waves consisting of periodic time sequences of stripes and rhombi, stripes and squares, and 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 instance, a single oscillatory cycle can display all the basic patterns related to the aperiodic Turing instabilitysstripes, hexagons, and inverted hexagons (honeycomb)sas well as rhombi and modulated stripes.
Pattern formation arising from wave instability in a simple reaction-diffusion system
The Journal of Chemical Physics, 1995
Pattern formation is studied numerically in a three-variable reaction-diffusion model with onset of the oscillatory instability at a finite wavelength. Traveling and standing waves, asymmetric standing-traveling wave patterns, and target patterns are found. With increasing overcriticality or system length, basins of attraction of more symmetric patterns shrink, while less symmetric patterns become stable. Interaction of a defect with an impermeable boundary results in displacement of the defect. Fusion and splitting of defects are observed.
Coupled and forced patterns in reaction-diffusion systems
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008
Several reaction-diffusion systems that exhibit temporal periodicity when well mixed also display spatio-temporal pattern formation in a spatially distributed, unstirred configuration. These patterns can be travelling (e.g. spirals, concentric circles, plane waves) or stationary in space (Turing structures, standing waves). The behaviour of coupled and forced temporal oscillators has been well studied, but much less is known about the phenomenology of forced and coupled patterns. We present experimental results focusing primarily on coupled patterns in two chemical systems, the chlorine dioxide-iodine-malonic acid reaction and the Belousov-Zhabotinsky reaction. The observed behaviour can be simulated with simple chemically plausible models.
Oscillatory Turing Patterns in Reaction-Diffusion Systems with Two Coupled Layers
Physical Review Letters, 2003
A model reaction-diffusion system with two coupled layers yields oscillatory Turing patterns when oscillation occurs in one layer and the other supports stationary Turing structures. Patterns include ''twinkling eyes,'' where oscillating Turing spots are arranged as a hexagonal lattice, and localized spiral or concentric waves within spotlike or stripelike Turing structures. A new approach to generating the shortwave instability is proposed.
Spontaneous periodic travelling waves in oscillatory systems with cross-diffusion
2008
We identify a new type of pattern formation in spatially distributed active systems. We simulate one-dimensional two-component systems with predator-prey local interaction and pursuit-evasion taxis between the components. In a sufficiently large domain, spatially uniform oscillations in such systems are unstable with respect to small perturbations. This instability, through a transient regime appearing as spontanous focal sources, leads to establishment of periodic traveling waves. The traveling waves regime is established even if boundary conditions do not favor such solutions. The stable wavelength are within a range bounded both from above and from below, and this range does not coincide with instability bands of the spatially uniform oscillations.
Two- and three-dimensional standing waves in a reaction-diffusion system
Physical Review E, 2012
We observe standing waves of chemical concentration in thin layers [quasi-two-dimensional (2D)] and capillaries [three-dimensional (3D)] containing the aqueous Belousov-Zhabotinsky reaction in a reverse microemulsion stabilized by the ionic surfactant sodium bis-2-ethylhexyl sulfosuccinate (AOT) and with cyclo-octane as the continuous phase. The 3D structures are oscillatory lamellae or square-packed cylinders at high and low volume fractions, respectively, of aqueous droplets. These patterns correspond to oscillatory labyrinthine stripes and square-packed spots in the 2D configuration. Computer simulations, as well as observations in E. coli, give qualitative agreement with the observed patterns and suggest that, in contrast to Turing patterns, the structures are sensitive to the size and shape of the system.
Physical Review E, 1997
The supercritical shortwave oscillatory bifurcation is studied in finite systems using the amplitude ͑Ginzburg-Landau͒ equation. Numerical simulations show that a zero-flux boundary stabilizes sources of target patterns. As a result, stable sources attached to the boundary can exist at small overcriticality, under the condition of convective instability of the homogeneous steady state. Oscillating target patterns and alternating wave packets are formed if the coupling between left and right propagating waves is strong.
Modulated Standing Waves in a Short Reaction−Diffusion System
The Journal of Physical Chemistry, 1996
We present a study of pattern formation beyond the onset of the wave instability in a short model reactiondiffusion system whose length is between 0.5 and 1.5 times the characteristic wavelength of the wave instability. As the system length is varied, modulated standing waves, characterized by short-lived alternating nodes, are found between the domains of the half-wavelength and the one-wavelength standing waves. The spacetime two-dimensional Fourier spectra of these modulated standing waves show large side peaks. The position of these peaks differs from that of the fundamental peak by its wavenumber and by the frequency of appearance of the alternating nodes. Another region of modulated standing waves is found within the domain of standingtraveling waves.
Modulated two-dimensional patterns in reaction–diffusion systems
European Journal of Applied Mathematics, 1999
New modulation equations for hexagonal patterns in reaction–diffusion systems are derived for parameter régimes corresponding to the onset of patterns. These systems include additional nonlinearities which are not present in Rayleigh–Bénard convection or Swift–Hohenberg type models. The dynamics of hexagonal and roll patterns are studied using a combination of analytical and computational approaches which exploit the hexagonal structure of the modulation equations. The investigation demonstrates instabilities and new phenomena not found in other systems, and is applied to patterns of flame fronts in a certain model of burner stabilized flames.