Multiple Timescale Analysis for Bifurcation from a Multiple-Zero Eigenvalue

Angelo Luongo (1) , Angelo Di Egidio (1) , Achille Paolone (2)

1 DICEAA - Dipartimento di Ingegneria Civile, Edile-Architettura, Ambientale
2 DISG - Dipartimento di Ingegneria Strutturale e Geotecnica

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 perturbationparameter. The reconstitution method isemployed 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 coefŽ cients of the relevant bifurcation equation. The mechanical behavior of a nonconservative two-degree-of-freedom system is studied as an example.

Domaines: Autre

Angelo Luongo, Angelo Di Egidio, Achille Paolone. Multiple Timescale Analysis for Bifurcation from a Multiple-Zero Eigenvalue. AIAA Journal, 2003, 41 (6), pp.1143-1150. ⟨hal-00795124⟩