An efficient center manifold technique for Hopf bifurcation of n-dimensional multi-parameter systems

Abstract

• Hopf bifurcation of a n -dimensional nonlinear multi-parametric dynamical system. • Use of the center manifold theory via a proper symbolic form. • Effective computation of the “restricted” normal form throughout the parameter space. • Construction of bifurcation portraits and evaluation of the respective limit cycles. • Application to two specific three-dimensional, three-parametric systems. The center manifold theory with respect to the simple Hopf bifurcation of a n -dimensional nonlinear multi-parametric system is treated via a proper symbolic form. Analytical expressions of the involved quantities are obtained as functions of the parameters of the system via effective algorithms based on the followed procedure and carried out using a symbolic computation software. Moreover the normal form of a codimension 1 Hopf bifurcation, as well as the corresponding Lyapunov coefficient and bifurcation portrait, can be computed for any system under consideration. Here the computational procedure is applied to two nonlinear three-dimensional, three-parametric systems and graphical results are obtained as concerns the stability regions, the bifurcation portraits, as well as emerged limit cycles with respect to both the supercritical and the subcritical case of bifurcation.