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

Source abstract

A general formula is given for the Hopf transversality condition associated with characteristic equations of the form R(z) = H(z,T), where R is analytic and H(-,T) 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.