POD Analysis for Aeroelastic Systems in a Neighborhood of a Hopf Bifurcation

The Proper Orthogonal Decomposition (POD) generally provides a direct-energy based decomposition on the phase state-space of dynamical systems. The basic idea of this work is to clarify the geometrical relation between POD objects (i.e., proper orthogonal modes, POMs, and proper orthogonal values, POVs) and geometrical characterizations of the dynamical systems in terms of invariant subspaces and invariant manifolds. The objective is to use the POD tools for analyzing the intrinsic properties of linear and nonlinear system. Specifically, the free response of the nonlinear aeroelastic system will be analyzed. Indeed, a typical aeroelastic systems is studied: a panel vibrating in a supersonic flow which exhibits a complex behavior including simply-harmonic limit cycles, not purely harmonic limit cycles and chaotic oscillations. It has been observed that once the nonlinearities contribution increases, the POD basis and the critical-mode (\ie associated to the center manifold), that in a neighborhood of a Hopf bifurcation coincide, do not coincide anymore but they differ for a rigid rotation. The first part of the analysis is conducted through a modal approximation of the system. The effect of this assumptions on the energy distribution on the state-space is analyzed performing the POD on the system response directly calculated through a time/space integration of the corresponding PDE equations and than comparing the two energy distribution. Moreover, the energy distribution along POD basis has been also related to the modal participation.