Stability and bifurcations in scalar differential equations with a general distributed delay

Eva Kaslik and Emanuel-Attila Kokovics

Applied Mathematics and Computation, 2023, vol. 454, issue C

Abstract: For a differential equation involving a general distributed time delay, a local stability and bifurcation analysis is performed, relying on fundamental properties of the characteristic function of the random variable whose probability density function is the delay distribution. Based on the root locus method, the bifurcation curves are determined in the considered parameter plane, also providing the number of unstable roots of the analyzed characteristic equation in each of the open connected regions delimited by these curves. This leads to a characterisation of the stability region of the considered equilibrium in the corresponding parameter plane. A Hopf bifurcation analysis is also completed in the general setting, and the criticality is analyzed by employing the method of multiple times scales. In contrast with some previously reported results from the literature, our analysis is accomplished in a general context and only then exemplified for particular types of delay distributions (e.g. Dirac, Gamma, uniform and triangular). The theoretical results are showcased in the framework of a simple neural model.

Keywords: Distributed delays; Stability; Instability; Stability region; Bifurcations; Multiple time scales; Delay differential equation; Hopf bifurcation (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:454:y:2023:i:c:s0096300323002692

DOI: 10.1016/j.amc.2023.128100

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