Computation of Hopf branches bifurcating from a class of Hopf/steady-state points

Abstract

This paper is concerned with the computation of Hopf branches bifurcating from a Hopf/steady-state point of a two parameter non-linear problem satisfying a subspace invariance condition. We will present a new approach to the theoretical and computational analysis of the bifurcating branches of Hopf points at this singular point utilizing the system used to calculate Hopf points. According to the analysis, standard continuation and branch-switching can be used to calculate these Hopf branches. In addition, a more effective method based on the extended system of the singular point is proposed for the calculation of the branch of secondary Hopf points. The use of Newton's method for solving the extended system will lead to a quadratically convergent scheme and its implementation with a suitable partitioning procedure is also discussed. Numerical examples are given.