Dynamic behaviour of an autonomous system with three pairs of pure imaginary eigenvalues

Abstract

The stability and bifurcation behaviour of a non-linear autonomous system in the vicinity of a compound critical point where the jacobian of the system exhibits three distinct pairs of pure imaginary eigenvalues is discussed. The system under consideration is assumed to have three independent parameters. The solutions for the incipient Hopf bifurcations as well as the sequences of bifurcations leading to two and high dimensional tori are explored. The analysis is carried out via a recently developed approach—the ‘unification technique’ and ‘multiple-scale intrinsic harmonic balancing’. The approach enables one to obtain systematically a sel of simplified rate equations which govern the local dynamics of the system. This more comprehensive perturbation procedure leads to explicit asymptotic results concerning the bifurcation solutions and the associated stability conditions. An example drawn from electrical networks is analysed to demonstrate the applicability of the theory and analytic results.