Abstract
This article is devoted to the analysis of the dynamics of a complex network of unstable reaction–diffusion systems. We demonstrate the existence of a non-empty parameter regime for which synchronization occurs in non-trivial attractors. We establish a lower bound of the dimension of the global attractor in an innovative manner, by proving a novel theorem of continuity of the unstable manifold, for which we invoke a principle of spectrum perturbation of non-bounded operators. Finally, we exhibit a co-dimension 2 bifurcation of the unstable manifold which shows that synchronization is compatible with instabilities.
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Notes
Consider \(X = {\mathbb {R}}\), \(A = [0,\,1]\) and \(B_m = \lbrace \tfrac{k}{m}~;~0 \le k \le m \rbrace \) for each integer \(m>0\). Then, it holds that \(\dim _H A = 1\), \(\dim _H B_m = 0\) for all \(m>0\), whereas \(\mathrm {dist}_H(A,\,B_m) \rightarrow 0\) as \(m \rightarrow \infty \).
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The authors wish to express their sincere gratitude to the anonymous reviewers for their valuable comments which greatly improved the presentation of the paper.
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Communicated by Sue Ann Campbell.
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Miranville, A., Cantin, G. & Aziz-Alaoui, M.A. Bifurcations and Synchronization in Networks of Unstable Reaction–Diffusion Systems. J Nonlinear Sci 31, 44 (2021). https://doi.org/10.1007/s00332-021-09701-9
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DOI: https://doi.org/10.1007/s00332-021-09701-9