New explicit critical criterion of Hopf–Hopf bifurcation in a general discrete time system

Hopf–Hopf bifurcation is one of typical codimension-two bifurcations, which requires some rigid bifurcation conditions and occurs only in high-dimension systems. In this paper, a new critical criterion of this bifurcation is presented for a general discrete time system. Unlike the corresponding classical critical criterion (or the bifurcation definition), the new criterion is composed of a series of algebraic conditions explicitly expressed by the coefficients of the characteristic polynomial, which does not depend on eigenvalue computations of Jacobian matrix. This characteristic gives the advantage of the proposed criterion which is more convenient and efficient for detecting the existence of this type of codimension-two bifurcation or exploring the parameter mechanism of the bifurcation than the corresponding classical criterion. The equivalence between the proposed criterion and the corresponding classical criterion is rigorously proved. The bifurcation design problem of a three-degree-of-freedom vibro-impact system is used as example to show the effectiveness of the proposed criterion.