2018/10/31 by David Lipshutz, Lipshutz, David, Robert J. Lipshutz +1
Computer Science · Engineering · Mathematics · #34D06 #34K13 #34K20 #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #Dynamical Systems (math.DS) #FOS: Mathematics #Primary: 34K11 #Secondary: 92B25 #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1810.13398
openalex publication_date 2018/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study stability of so-called synchronous slowly oscillating periodic\nsolutions (SOPSs) for a system of identical delay differential equations (DDEs)\nwith linear decay and nonlinear delayed negative feedback that are coupled\nthrough their nonlinear term. Under a row sum condition on the coupling matrix,\nexistence of a unique SOPS for the corresponding scalar DDE implies existence\nof a unique synchronous SOPS for the coupled DDEs. However, stability of the\nSOPS for the scalar DDE does not generally imply stability of the synchronous\nSOPS for the coupled DDEs. We obtain an explicit formula, depending only on the\nspectrum of the coupling matrix, the strength of the linear decay and the\nvalues of the nonlinear negative feedback function near plus/minus infinity,\nthat determines the stability of the synchronous SOPS in the asymptotic regime\nwhere the nonlinear term is heavily weighted. We also treat the special cases\nof so-called weakly coupled systems, near uniformly coupled systems, and doubly\nnonnegative coupled systems, in the aforementioned asymptotic regime. Our\napproach is to estimate the characteristic (Floquet) multipliers for the\nsynchronous SOPS. We first reduce the analysis of the multidimensional\nvariational equation to the analysis of a family of scalar variational-type\nequations, and then establish limits for an associated family of monodromy-type\noperators. We illustrate our results with examples of systems of DDEs with\nmean-field coupling and systems of DDEs arranged in a ring.\n