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Title: Effects of Non-normality on Turing Instability

Abstract: Turing mechanism describes the emergence of spatial patterns in a reaction-diffusion system of two or more species: when certain conditions are matched, a perturbation starting from a homogeneous stable state guides the system towards a nonhomogeneous one, the celebrated Turing patterns, following a diffusion-driven instability [1]. The classical linear stability analysis that describes such phenomenon is based on the spectra of involved the operators. However, such analysis may fail when the involved linear operators are non-normal, due to a transient growth [2]. Such effect is even stronger when the system is studied on a non-normal network, i.e., a network whose adjacency matrix is non-normal [3]. In a recently published work [4] we have made use of such theoretical background to extend the original theory by obtaining non-normality patterns when Turing contitions are not satisfied.

In this seminar I will go through the main steps of this itinerary, from classical Turing Instability to patterns of non-normality. Firstly, I will present qualitatively the idea of Turing, then, after having introduced the notion of network, I will talk about processes of diffusion on discrete support, I will explain a recent extention of Turing Theory on symmetric network [5] and how an asymmetric topology can affect the mechanism of pattern formation [6]. Before moving to the last part, I will introduce the concept of non-normality and its effects on the dynamics [7]. Finally, I will present our extension of Turing Theory on non-normal networks [4] and discuss some open problems and possible future developments.

References

[1] Turing, A.M., 1952. The chemical basis of morphogenesis. Phil. Trans. R. Soc. B 237, 37–72;

[2] Trefethen, L.N., Embree, M., 2005. Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, NJ;

[3] Asllani, M., Lambiotte, R., Carletti, T., 2018. Structure and dynamics of non-normal networks, Sci. Adv. 4, 1–8. Eaau9403;

[4] Muolo, R., Asllani, M., Fanelli, D., Maini, P.K., Carletti, T., 2019. Patterns of non-normality in networked systems. Journal of Theoretical Biology 480 81–91;

[5] Nakao, H., Mikhailov, A.S., 2010. Turing patterns in network-organized activator-inhibitor systems. Nat. Phys. 6, 544;

[6] Asllani, M., Challenger, J.D., Pavone, F.S., Sacconi, L., Fanelli, D., 2014. The theory of pattern formation on directed networks. Nat. Comm. 5, 4517;

[7] Asllani, M., Carletti, T., 2018. Topological resilience in non-normal networked systems. Phys. Rev. E 97, 042302.