On the Analysis of Internal Non-resonant Double Hopf Bifurcation with Symbolic Software

A systematic perturbation procedure is employed to derive ordered periodic and quasi-periodic solutions and their stability conditions associated with a general nonlinear autonomous system in the vicinity of a compound critical point. The critical point is characterized by two distinct pairs of pure imaginary eigenvalues of the Jacobian. The method, which is based on a Unified Intrinsic Harmonic Balancing (UIHB) approach, generates an ordered form of approximations for a solution, and makes extensive use of a symbolic manipulation computer language, namely MAPLE. The MAPLE programs developed, also incorporate a verification scheme which enables one to verify the validity of each approximation obtained in the analysis. Moreover, this UIHB method systematically leads to a set of rate equations for stability analysis of dynamic motions. The rate equations are indeed identical to the so called normal forms which are usually obtained by employing the central manifold theory [1] followed by the normal form theory [2].