A GENERAL TRANSVERSALITY CRITERION FOR HOPF BIFURCATION IN CHARACTERISTIC EQUATIONS WITH DISTRIBUTED DELAY

Eva Kaslik^†, Loredana Flavia Gabor^‡, Maria Roxana Matei^§, Mihaela Neamțu^¶

Abstract: A general formula is given for the Hopf transversality condition associated with characteristic equations of the form R(z) = H(z,τ), where R is analytic and H(·,τ) 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 selfsimilar kernels, including the exponential and Erlang families. Several consequences for stability switching are also discussed and examples are given.

Keywords: distributed delay, Hopf bifurcation, characteristic equation, transversality condition, stability switching.

MSC: 34K18, 34K20, 34C23.

DOI  10.56082/annalsarscimath.2026.3.223

Read full article                                                           Download article

^*Accepted for publication on August 21, 2026
^†eva.kaslik@e-uvt.ro, West University of Timisoara, Romania
^‡loredana.gabor@upt.ro, Polytechnic University of Timisoara, Romania
^§maria.matei@e-uvt.ro, West University of Timisoara, Romania
^¶mihaela.neamtu@e-uvt.ro, West University of Timisoara, Romania

PUBLISHED in

Annals Academy of Romanian Scientists Series on Mathematics and Its Application,

Volume 18 no 3, 2026

ISSN ONLINE 2066 – 6594
ISSN PRINT 2066 – 5997