| Abstract: |
| Pattern formation is a key feature of many natural and engineered systems, ranging from ecosystems to neural dynamics. Turing instability provides one of the most famous theories for pattern formation in a continuous domain, which was later extended to networked systems, where the dynamical variables interact in the nodes and flow among nodes by using network links. However, in a number of real systems, including the brain and the climate, dynamical variables are not only defined on nodes but also on links, triangles and higher-dimensional simplexes, leading to topological signals. The discrete topological Dirac operator is emerging as the key operator that allows cross-talk between signals defined on simplexes of different dimensions, for instance among nodes and links signals of a network.
Here, we propose a mathematical framework able to generate stationary and dynamical Turing patterns of topological signals defined on nodes and links of networks. This framework accounts for a rich dynamical behavior even without the (Hodge-Laplacian) diffusion term, i.e., occurring solely due to the Dirac operator. This work opens a new framework displaying a rich interplay between topology and dynamics with possible applications to brain and climate networks. |
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