Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

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

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

The postcritical behavior of a generaln-dimensional system around a resonant double Hopf bifurcation isanalyzed. Both cases in which the critical eigenvalues are in ratios of 1:2 and 1:3 are investigated. The Multiple Scale Method is employed to derive the bifurcation equations systematically in terms of thederivatives of the original vector field evaluated at the critical state. Expansions of the n-dimensional vector of state variables andof a three-dimensional vector of control parameters are performed interms of a unique perturbation parameter ε, of the order ofthe amplitude of motion. However, while resonant terms only appear at the ε3-order in the 1:3 case, they already arise at the ε2-order in the 1:2 case. Thus, by truncating theanalysis at the ε3-order in both cases, first orsecond-order bifurcation equations are respectively drawn, the latterrequiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations isstudied. The complete postcritical scenario is analyzed in terms of the three control parameters and the asymptotic results are compared with exact numerical integrations for both resonances. Branches of periodicas well as periodically modulated solutions are found and their stability analyzed.

Domaines: Génie mécanique [physics.class-ph]; Génie mécanique [physics.class-ph]

Angelo Luongo, Achille Paolone, Angelo Di Egidio. Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations. Nonlinear Dynamics, 2003, 34 (3-4), pp.269-291. ⟨hal-00797874⟩