Segmented Waves from a Spatiotemporal Transverse Wave Instability

Lingfa Yang, Igal Berenstein, and Irving R. Epstein

Phys. Rev. Lett. 95, 038303 – Published 14 July 2005

Abstract

We observe traveling waves emitted from Turing spots in the chlorine dioxide-iodine-malonic acid reaction. The newborn waves are continuous, but they break into segments as they propagate, and the propagation of these segments ultimately gives rise to spatiotemporal chaos. We model the wave-breaking process and the motion of the chaotic segments. We find stable segmented spirals as well. We attribute the segmentation to an interaction between front rippling via a transverse instability and front symmetry breaking by a fast-diffusing inhibitor far from the codimension-2 Hopf-Turing bifurcation, and the chaos to a secondary instability of the periodic segmentation.

Received 14 March 2005

DOI:https://doi.org/10.1103/PhysRevLett.95.038303

Authors & Affiliations

Lingfa Yang, Igal Berenstein, and Irving R. Epstein^*

Department of Chemistry and Volen Center for Complex Systems, MS 015, Brandeis University, Waltham, Massachusetts 02454-9110, USA

^*Electronic address: epstein@brandeis.edu

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Issue

Vol. 95, Iss. 3 — 15 July 2005

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Images

Figure 1
Segmented waves in the CDIMA reaction (a),(b),(c). Frame (b) taken 50 s after frame (a), and frame (c) taken 115 s after (b). The spot pointed to by the arrow in (a) disappears in (b), while the arrow in (c) signals the segmentation of a wave. Frame size $15 \times 15 {mm}^{2}$ . $[I_{2}]_{0} = 0.36 mM$ , $[MA]_{0} = 1.85 mM$ , $[{ClO}_{2}]_{0} = 0.14 mM$ , $[poly (vinyl-alcohol)] = 0.85 g / L$ . Temperature inside the reactor is 4 °C; residence time is 220 s. Segmented waves represent a transient from traveling waves to Turing patterns. (d) Snapshot from a 2D simulation, size $128 \times 128 s . u .$ , showing segmented waves generated from Turing spots (see 24 for a movie).Reuse & Permissions
Figure 2
Three stable solutions from 1D simulations under periodic boundary conditions, size 128 s.u. (a) Bulk oscillation, period $T = 28.8 t . u .$ ; (b) stationary Turing pattern, wavelength $λ_{T} = 12.8 s . u .$ ; (c) wave train, propagation velocity $v = 8.9 s . u . / t . u .$ (d) propagation velocity vs interwave interval in wave trains. Velocity at interval 53 is marked for the archimedean spiral in Fig. 4c. (e) Amplitude of bulk oscillations increases with $b$ .Reuse & Permissions
Figure 3
Formation of a segmented wave and its motion in 2D simulations, size $128 \times 43 s . u .$ , periodic boundary conditions. (a) Discrete snapshots show a smooth continuous traveling wave breaking up into wave segments. (b) Space-time plot of a continuous run from (a) in the time window $t = 3370 - 3430 t . u .$ showing spatiotemporal chaos in the comoving frame at velocity $v = 9.15 t . u . / s . u$ . Marked areas show splitting (triangle), vanishing (circle), and breathing (rectangle). (c) Secondary instability destabilizes periodic segmented waves into chaos. Spatiotemporal plot in the comoving frame, $t = 20 - 80 t . u .$ (see 24 for a movie).Reuse & Permissions
Figure 4
Segmented spiral wave develops from a continuous traveling wave with a free end subject to spatial random noise ( $< 1.4 %$ ). Arrows indicate propagation directions. System size $256 \times 256 s . u .$ (a) Initial wave ( $t = 0$ ); (b) spiral begins to develop ( $t = 10$ ). (c) Developed segmented spiral ( $t = 30$ ). (d) Twenty-six superimposed snapshots in one period $T = 5 t . u$ . The spiral tip is located at the white spot. (e) Segmented target pattern arising from a “dust” ( $u = 10$ , $v = 8.7$ ) (see 24 for a movie).Reuse & Permissions
Figure 5
Dispersion relation of traveling wave transverse instability for Fig. 3 (solid line). Decreasing $b$ lowers the growth rate and shifts to the left the most positive wave number (dashed line), until the instability vanishes (dotted line).Reuse & Permissions

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