Design of Stable Signed Laplacian Matrices with Mixed Attractive–Repulsive Couplings for Complete In-Phase Synchronization

Gualberto Solis-Perales (), Aurora Espinoza-Valdez, Beatriz C. Luna-Oliveros, Jorge Rivera and Jairo Sánchez-Estrada
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Gualberto Solis-Perales: Departamento de Ciencias Computacionales, CUCEI, Universidad de Guadalajara, Av. Revolución No. 1500, Guadalajara 44430, Mexico
Aurora Espinoza-Valdez: Departamento de Ciencias Computacionales, CUCEI, Universidad de Guadalajara, Av. Revolución No. 1500, Guadalajara 44430, Mexico
Beatriz C. Luna-Oliveros: Consejo de Ciencia y Tecnología, Universidad Pedagógica Nacional, Oaxaca de Juárez 71230, Mexico
Jorge Rivera: Electrical Engineering Department, Centro de Investigación y de Estudios Avanzados, Instituto Politécnico Nacional, Zapopan 45017, Mexico
Jairo Sánchez-Estrada: Departamento de Ciencias Computacionales, CUCEI, Universidad de Guadalajara, Av. Revolución No. 1500, Guadalajara 44430, Mexico

Mathematics, 2025, vol. 13, issue 17, 1-15

Abstract: Synchronization in complex networks mainly considers positive (attractive) couplings to guarantee network stability. However, in many real-world systems or processes, negative (repulsive) interactions exist, and this poses a challenging problem. In this proposal, we present an algorithm to design stable signed Laplacian matrices with mixed attractive and repulsive couplings that ensure stability in both complete and in-phase synchronization. The main result is established through a constructive theorem that guarantees a single zero eigenvalue, while all other eigenvalues are negative, thereby preserving the diffusivity condition. The algorithm allows control over the spectral properties of the matrix by adjusting two parameters, which can be interpreted as a pole placement strategy from control theory. The approach is validated through numerical examples involving the synchronization of a network of chaotic Lorenz systems and a network of Kuramoto oscillators. In both cases, full synchronization is achieved despite the presence of negative couplings.

Keywords: negative couplings; weighted negative matrices; synchronization (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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