A General Transversality Criterion for Hopf Bifurcation in Characteristic Equations with Distributed Delay

E. Kaslik, L. F. Gabor, M. R. Matei, M. Neamțu

Annals Academy of Romanian Scientists

Hopf bifurcations Delay Differential Equations

Abstract

A general formula is given for the Hopf transversality condition associated with characteristic equations of the form $R(z)=H(z,\tau)$, where $R$ is analytic and $H(\cdot,\tau)$ is the Laplace transform of a distributed-delay kernel. Our main result expresses the crossing direction of a simple purely imaginary characteristic root and provides a geometric interpretation of the transversality condition. We obtain as special cases the classical discrete-delay formula and explicit criteria for self-similar kernels, including the exponential and Erlang families. Several consequences for stability switching are also discussed, and examples are given.

BibTeX

@article{KaslikGaborMateiNeamtu_AARC_2026,
  title         = {
    A General Transversality Criterion for {Hopf} Bifurcation in Characteristic
    Equations with Distributed Delay
  },
  author        = {E. Kaslik and L. F. Gabor and M. R. Matei and M. Neamțu},
  year          = 2026,
  journal       = {Annals Academy of Romanian Scientists},
  series        = {Series on Science of Mathematics},
  volume        = 18,
  number        = {3/2026},
  pages         = {213–230},
  issn          = {2066-6594},
  abstract      = {
    A general formula is given for the Hopf transversality condition associated
    with characteristic equations of the form $R(z)=H(z,\tau)$, where $R$ is
    analytic and $H(\cdot,\tau)$ is the Laplace transform of a
    distributed-delay kernel. Our main result expresses the crossing direction
    of a simple purely imaginary characteristic root and provides a geometric
    interpretation of the transversality condition. We obtain as special cases
    the classical discrete-delay formula and explicit criteria for self-similar
    kernels, including the exponential and Erlang families. Several
    consequences for stability switching are also discussed, and examples are
    given.
  },
}