Multiple-timescale analysis for bifurcation from a multiple-zero eigenvalue

Multiple-zero bifurcation of a general multiparameter dynamic system is analyzed using the multiple-scale method and exploiting the close similarities with eigensolution analysis for defective systems. Because of the coalescence of the eigenvalues, the Jacobian matrix at the bifurcation is nilpotent. This entails using timescales with fractional powers of the perturbation parameter. The reconstitution method is employed to obtain an ordinary differential equation of order equal to the algebraic multiplicity of the zero eigenvalue, in the unique unknown amplitude. When the algorithm is applied to a double-zero eigenvalue, Bogdanova-Arnold's normal form for the bifurcation equation is recovered. A detailed step-by-step algorithm is described for a general system to obtain the numerical coefficients of the relevant bifurcation equation. The mechanical behavior of a nonconservative two-degree-of-freedom system is studied as an example.