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Design and control of patterns in reaction-diffusion systems
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View: Figures


Image of FIG. 1.
FIG. 1.

Trigger wave pattern observed at in a layer of aqueous solution with initial composition ; ; ; , and starch as an indicator (Ref. 21 ).

Image of FIG. 2.
FIG. 2.

Design of a CFUR. Reaction takes place in the thin layer of gel, which exchanges chemicals through the membrane with the continuous flow stirred tank reactor (CSTR) below it. Patterns are monitored with the CCD camera (Ref. 22 ).

Image of FIG. 3.
FIG. 3.

Stationary Turing patterns in the CDIMA reaction developed after illumination through a mask. The ratio between the characteristic wavelengths of the mask and the native Turing patterns is equal to 2, 3, and 4, respectively, for rows (a), (b), and (c). The numbers above the columns show the time (in min) after cessation of illumination. Frame (Ref. 44 ).

Image of FIG. 4.
FIG. 4.

Pattern memory utilizing localized spots in the BZ–AOT system. (Left column) illumination of the BZ–AOT reaction through a “face-mask” for at intensity sufficient to suppress all patterns. Right part of the reactor (narrow stripe with Turing stationary spots) remains in darkness. . (Right column) subsequent illumination without any mask for at intensity . This intensity of light prevents the emergence of new spots, but cannot suppress already existing patterns. Size of snapshots is . , , , ; volume fraction of aqueous droplets , (Ref. 46 ).

Image of FIG. 5.
FIG. 5.

Localized waves in the BZ–AOT system (a) mask. Time interval between snapshots (b) and (c) is and between snapshots (c) and (d) is . Numbers 1 and 2 mark two approaching waves. The width of the waves is . The period is . , , , , , . Vertical (Ref. 111 ).

Image of FIG. 6.
FIG. 6.

(a), (b) Standing oscillatory clusters that evolve from an initial spiral wave. Time between snapshots is . , , , , . (c)–(f) Irregular oscillatory clusters. , , , ; . catalyzed BZ reaction in a silica-gel, CSTR (Ref. 50 ).

Image of FIG. 7.
FIG. 7.

Oscillatory standing clusters in the BZ reaction with negative global feedback (Ref. 51 ).

Image of FIG. 8.
FIG. 8.

Trajectories of a spiral tip under periodic sinusoidal illumination of the BZ reaction for different ratios , where is the modulation period and is the spiral wave period. (a) Experiment, . (b) Simulations of Eqs. (1) and (2) at , , , and , , . corresponds to the background constant illumination in experiments with intensity of . The ratio gives the number of lobes per periods of modulation (Ref. 74 ).

Image of FIG. 9.
FIG. 9.

Simulations of Eqs. (1) and (2) . Tip trajectories in response to sinusoidal illumination with amplitude and period . Dashed lines indicate boundaries of entrainment bands for ratios , the number of lobes per periods of modulation. , , and . (marked by the arrow) is the intrinsic period of an unperturbed meandering spiral (Ref. 75 ).

Image of FIG. 10.
FIG. 10.

Frequency-locked regimes for the periodically illuminated BZ reaction; is the frequency of the light; is the natural frequency of bulk oscillation. Patterns in upper and bottom rows are separated in time by except for the 1:1 resonance where . The BZ system was illuminated for with constant light intensity for all experiments; at the period of forcing is , and at the period of forcing is (Ref. 23 ).

Image of FIG. 11.
FIG. 11.

(a) Shape of external light signal. During period , the intensity of illumination ; during , . (b) Dependence of system behavior on and ; symbols: (×) wave patterns, (−) small amplitude bulk oscillations, (+) high amplitude bulk oscillations, (●) two-phase standing clusters, (△) three-phase clusters, (◻) irregular clusters, (◇) localized clusters, (∗) mixed behavior—clusters and waves in different parts of the gel. Along the dotted line, . Initial reagent concentrations, of immobilized in gel; , , , in feeding chamber (Ref. 76 ).

Image of FIG. 12.
FIG. 12.

(Top row) patterns in the BZ reaction (image , false color) at increasing light intensity and slightly decreasing frequency . Values of and 0, 0; (b) 0.1, 119; (c) 0.0625, 214; (d) 0.0556, 248; (e) 0.0417, 358; (f) 0.0455, 386; (g) 0.0385, 412. (Bottom row) the complex Fourier amplitude A for each pattern: the abscissa is Re(A) and the ordinate is Im(A). Each point in the complex plane corresponds to the temporal Fourier amplitude A of a pixel in the image after frequency demodulation at (Ref. 77 ).

Image of FIG. 13.
FIG. 13.

In-phase oscillatory Turing patterns in the CDIMA reaction. (a) Series of snapshots of a region with oscillating spots. The dashed sinusoidal line is drawn to help the eye identify the oscillations. (b) Space-time plot along a line in the system. (c) Local averaged concentration for an area involving about seven spots (Ref. 88 ).

Image of FIG. 14.
FIG. 14.

Pattern formation in the -catalyzed BZ–AOT system at different temperatures. , , , , , . (a) , Turing patterns, (b) , oscillatory Turing patterns; length of black diagonal line is . (c) Space-time plot corresponding to line in (b); size is , (d) , conventional waves and spiral waves. Frame size (Ref. 112 ).

Image of FIG. 15.
FIG. 15.

(a), (b) Jumping waves in the bathoferroin-catalyzed BZ–AOT system at . , , , , , . Length of the white line in (a) is . (b) Space-time plot along the line in (a) . , wavelength of a jump; , wavelength of the wave; , period for generating a new wave from a pacemaker; , period of jumps of a single wave; , and , where is the velocity of jumping waves; is the period of bulk oscillations. (c), (d) Bubble waves in the bathoferroin-catalyzed BZ–AOT reaction , , , , , M, . . (d) (Ref. 39 ).


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Design and control of patterns in reaction-diffusion systems