Normal form of double Hopf bifurcation in forced oscillators

Abstract

A general four-dimensional normal form of a double Hopf bifurcation is considered. As a particular case, the normal form of a forced (non-autonomous) non-linear oscillator having two frequencies, namely the linear natural frequency and the excitation frequency, is studied in detail. When these two frequencies form two purely imaginary sub-blocks of order two in the real Jordan block, the system constitutes a double Hopf bifurcation. In this paper, the normal form of the double Hopf bifurcation is reduced when the two frequencies are not in resonance. In order to use the method of normal form, the non-autonomous problem is transformed into an autonomous one by a generalized co-ordinate transformation. The method of undetermined coefficients is used to find the double Hopf bifurcation normal form. The coefficients of similar monomials rather than similar powers of ε are compared to get the normal form to various orders. The steady state periodic solutions and the bifurcation equations of the forced non-linear vibration system in the case of non-resonant are studied. A Mathematica program is designed to find the normal form. Three examples are given to use the Mathematica program and to compare them with the existing results.