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Localized patterns in reaction-diffusion systems
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View: Figures


Image of FIG. 1.
FIG. 1.

(Color online) Localized spots in the BZ-AOT system (Ref. 6 ). (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 the snapshots is . , , , ; volume fraction of aqueous droplets , .

Image of FIG. 2.
FIG. 2.

Oscillon in the BZ-AOT reaction (Ref. 9 ). The ring diameter is about . Time between frames, , is half the period of oscillation. , , , , , . Frame .

Image of FIG. 3.
FIG. 3.

Breathing spot in the ferrocyanide-iodate-sulfite (FIS) reaction (Ref. 46 ). (a) Minimal size, (b) maximal size, (c) time evolution of spot diameter. Feeding chamber contains , , , . Flow .

Image of FIG. 4.
FIG. 4.

(Color online) Circular interface (black) subjected to a spatial perturbation (blue) of the form .

Image of FIG. 5.
FIG. 5.

Breathing filament in a gas discharge system (Ref. 48 ). . Diameter of a spot is about .

Image of FIG. 6.
FIG. 6.

Splitting of a localized spot in a gas-discharge system (Ref. 51 ).

Image of FIG. 7.
FIG. 7.

Localized clusters in the photosensitive BZ reaction in a CFUR under global negative feedback (Ref. 52 ). Concentrations: of immobilized in a silica gel with thickness ; , , , in the feeding chamber. . Time between snapshots is (one half-period of oscillation).

Image of FIG. 8.
FIG. 8.

Superimposed snapshots of a single stabilized wave segment in the photosensitive BZ reaction under homogeneous global negative feedback of intensity (Ref. 55 ). The interval between snapshots is . Frame size is . Left and right panels correspond to lower and higher excitabilities, resulting from higher and lower light intensity, respectively. , , , , silica gel ( thickness), . , , is a threshold (slightly above the average gray level), is the Heaviside function, is the amplitude of the image at each pixel .

Image of FIG. 9.
FIG. 9.

Two hypotrochoid trajectories, seen as a superposition of snapshots taken every , of a single wave segment in the BZ reaction controlled by a global negative feedback combined with an additional imposed inhomogeneous illumination whose form confines the wave segment to the chosen trajectory (Ref. 56 ).

Image of FIG. 10.
FIG. 10.

Two photoemission electron microscopy images taken apart of a Pt(110) surface during steady-state catalytic CO oxidation. The dark objects (wave segments) are regions with enhanced coverage of adsorbed oxygen atoms. These features move along the indicated directions with constant velocity (Ref. 57 ). . The width of the pulses is about ; the length is about . Pulses of length smaller than are not stable.

Image of FIG. 11.
FIG. 11.

Moving spots and the birth of a new spot (marked by an arrow) in a gas discharge system (Ref. 58 ). The time (in s) is indicated at the bottom of each snapshot. .

Image of FIG. 12.
FIG. 12.

Localized waves in the BZ-AOT system. (a) Mask. (b)–(d) Snapshots. Time interval between snapshots (b) and (c) is and between snapshots (c) and (d). Numbers 1 and 2 mark two approaching waves. Width of the waves is . Period is . , , , , , . Vertical [A. Kaminaga, V. K. Vanag, and I. R. Epstein (unpublished)].

Image of FIG. 13.
FIG. 13.

Localized solution in model (3) and (4) in 1D with , , , , and . The width is . Solid line is and dashed line is (Ref. 65 ).

Image of FIG. 14.
FIG. 14.

(Color) Localized spots in model (5) and (6) at , , , , maps. (a) Outer radius , initial circular perturbation with expands and quickly reaches a critical radius at which it becomes metastable, but eventually loses circular stability and deforms into a labyrinthine pattern. (b) Total . Initial circular perturbation with radius shrinks and then slowly transforms to a localized labyrinthine pattern. White and red correspond to , black and purple to .

Image of FIG. 15.
FIG. 15.

Instabilities of localized spot. (a) Stationary spot—circular wave transition, (b) splitting, (c) Turing-like, (d) space-time plot for the two interfaces (left and right) of a breather in model (21), (22), and (23b) with , , , , , (Ref. 101 ). Snapshots (a)–(c) are from Ref. 6 .

Image of FIG. 16.
FIG. 16.

(Color) (a) Fast inhibitor-map snapshot of a moving spot in model (24)–(26) . Parameters: , , , , , , , , , vertical . (b) Moving spot in model (27)–(29) . Parameters: , , , , , , , . Vertical . Red (blue-purple) color corresponds to the smallest (largest) concentration.

Image of FIG. 17.
FIG. 17.

Dispersion curves [(a), (b)] for model (24)–(26) and [(c), (d)] for model (27)–(29) . [(a), (b)] Parameters as in Fig. 16 , 1, (b) 1.35. Dispersion curves for (c) SS1 at , curve 3, and , curve 3a, and (d) SS2 at . All other parameters as in Fig. 16 . In (c), all three eigenvalues (curves 1, 2, and 3) are real and negative. In (a), (b), and (d), curve 1 is , curve 2 is the corresponding , and curve 3 is the real negative eigenvalue .


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Localized patterns in reaction-diffusion systems