Computation of double Hopf points for delay differential equations

Abstract

Relating to the crucial problem of branch switching, the calculation of codimension 2 bifurcation points is one of the major issues in numerical bifurcation analysis. In this paper, we focus on the double Hopf points for delay differential equations and analyze in detail the corresponding eigenspace, which enable us to obtain the finite dimensional defining system of equations of such points, instead of an infinite dimensional one that happens naturally for delay systems. We show that the double Hopf point, together with the corresponding eigenvalues, eigenvectors and the critical values of the bifurcation parameters, is a regular solution of the finite dimensional defining system of equations, and thus can be obtained numerically through applying the classical iterative methods. We show our theoretical findings by a numerical example.

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

“…To approximate the LPC parameter values, we substitute β from ( 63) into (82). The cycle period is approximated by (64) with (83). To obtain a predictor for the periodic orbit in the phase space, we set z = e iψ into (84) using the obtained α values.…”

“…Great interest has recently been shown in the analysis of degenerate Hopf bifurcations in delay differential equations (DDEs), see e.g. [1,17,18,25,35,36,39,40,48,50,51,54,57,61,64,65,66]. In the simplest case, often encountered in applications, such DDEs have the forṁ…”