Multiscale analysis of defective multiple-Hopf bifurcations

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