Multiscale analysis of defective multiple-Hopf bifurcations

An adapted version of the Multiple Scale Method is formulated to analyze 1:1 resonant multiple Hopf bifurcations of discrete autonomous dynamical systems, in which, for quasi-static variations of the parameters, an arbitrary number m of critical eigenvalues simultaneously crosses the imaginary axis. The algorithm therefore requires discretizing continuous systems in advance. The method employs fractional power expansion of a perturbation parameter, both in the state variables and in time, as suggested by a formal analogy with the eigenvalue sensitivity analysis of nilpotent (defective) matrices, also illustrated in detail. The procedure leads to an order-m differential bifurcation equation in the complex amplitude of the unique critical eigenvector, which is able to capture the dynamics of the system around the bifurcation point. The procedure is then adapted to the specific case of a double Hopf bifurcation (m = 2), for which a step-by-step, computationally-oriented version of the method is furnished that is directly applicable to solve practical problems. To illustrate the algorithm, a family of mechanical systems, subjected to aerodynamic forces triggering 1:1 resonant double Hopf bifurcations is considered. By analyzing the relevant bifurcation equation, the whole scenario is described in a three-dimensional parameter space, displaying rich dynamics.