Nonlinear normal modes in a network with cubic couplings

Jean-Guy Caputo; Imene Khames; Arnaud Knippel; Jean-Guy Caputo; Imene Khames; Arnaud Knippel

doi:10.3934/math.20221127

AIMS Mathematics

2022, Volume 7, Issue 12: 20565-20578. doi: 10.3934/math.20221127

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Nonlinear normal modes in a network with cubic couplings

1.
Laboratoire de Mathématiques, INSA Rouen Normandie, 76801 Saint-Etienne du Rouvray, France
2.
Laboratoire de Mathématiques Appliquées du Havre, Université Le Havre Normandie, 76600 Le Havre, France

Received: 27 July 2022 Revised: 09 September 2022 Accepted: 15 September 2022 Published: 20 September 2022
MSC : 94C15, 70K75, 34A05

We consider a network with cubic couplings. This is related to the well known Fermi-Pasta-Ulam-Tsingou model. We show that nonlinear periodic orbits extend from particular eigenvectors of the graph Laplacian, these are termed nonlinear normal modes. We present large classes of graphs where this occurs. These are the graphs whose Laplacian eigenvectors have components in $ \{1, -1\} $ (bivalent), and $ \{1, -1, 0\} $ with a condition (soft-regular trivalent), the bipartite complete graphs and their extensions obtained by adding an edge between vertices having the same component. Finally, we study the stability of these solutions for chains and cycles.
- networks,
- graph Laplacian,
- nonlinear normal mode (NNM),
- Fermi-Pasta-Ulam-Tsingou (FPUT) model,
- graph spectra and applications
Citation: Jean-Guy Caputo, Imene Khames, Arnaud Knippel. Nonlinear normal modes in a network with cubic couplings[J]. AIMS Mathematics, 2022, 7(12): 20565-20578. doi: 10.3934/math.20221127

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Abstract

We consider a network with cubic couplings. This is related to the well known Fermi-Pasta-Ulam-Tsingou model. We show that nonlinear periodic orbits extend from particular eigenvectors of the graph Laplacian, these are termed nonlinear normal modes. We present large classes of graphs where this occurs. These are the graphs whose Laplacian eigenvectors have components in $ \{1, -1\} $ (bivalent), and $ \{1, -1, 0\} $ with a condition (soft-regular trivalent), the bipartite complete graphs and their extensions obtained by adding an edge between vertices having the same component. Finally, we study the stability of these solutions for chains and cycles.

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