Stability analysis of periodic orbits in the framework of Galerkin approximations

Denis Laxalde (1)

1 Structural Dynamics and Vibration Laboratory [Montréal]

In dynamical systems, when periodic orbits are derived using approximation methods and more specifically Galerkin weighted residuals, their stability analysis usually requires a subsequent and sometimes uncorrelated treatment. The trouble is that this additional analysis may involve another level of approximation which, if not consistent with that used in the solution procedure, may lead to incorrect results. This paper suggests a general framework for stability analysis of periodic solutions derived using Galerkin approximation methods that systematically recasts the problem of finding periodic solutions into a fixed point problem in the spirit of the averaging method. It follows that approximations are consistent between the solution derivation and the stability analysis and that virtually any kind of projection basis could be used in a similar fashion. In this initial version of the paper, the approach is illustrated on a Duffing oscillator with strongly nonlinear dynamics.

Mots clés: dynamical systems; stability; periodic solutions; Galerkin methods; harmonic balance

Domaines: Vibrations [physics.class-ph]; Vibrations [physics.class-ph]; Systèmes dynamiques [math.DS]

Denis Laxalde. Stability analysis of periodic orbits in the framework of Galerkin approximations. 2011. ⟨hal-00655521⟩