| Abstract: |
| We study the Turing pattern on curved surfaces. Since the seminal work by A. Turing many researchers have investigated the pattern formation on curved surfaces such as spheres and tori, in which it has been presumed that the Turing pattern is static on curved surfaces, as it is on a flat plane. We show that the Turing pattern on curved surfaces actually moves on curved surfaces. We mainly study reaction-diffusion systems on an axisymmetric surface with periodic boundary conditions, with parameters showing Turing pattern on a flat plane. Our numerical and theoretical analyses reveal that there exist propagation solutions along the azimuth direction, and the symmetries of the surface as well as pattern are involved for the initiation of the pattern propagation. By applying weakly non-linear analysis, we derive the amplitude equations and show that the intricate interactions between modes rise on curved surface and results in the initiation of pattern propagation and even more complex behaviors such as oscillation and chaos. This study provides a novel and generic mechanism of pattern propagation that is caused by surface curvature (which is not possible in 1D systems), as well as new insights into the potential role of surface geometry in pattern dynamics. |
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