Period-Doubling Cascades and Mode Interactions in Coupled Systems

The study of the asymptotic behavior of nonautonomous linear systems arising from linearization around chaotic orbits offers good prospects for understanding complex nonlinear dynamics. Knowledge of the Lyapunov exponents (and other stability spectra) plays a central role in this context and several computational techniques have been established to address the problem in finite dimension, i.e. for Ordinary Differential Equations (ODEs), basically originating from the successive re-orthonormalization of an initial small sphere of realizations in the phase space. In this talk we briefly recall the ideas behind QR-based methods for approximating the Lyapunov spectrum of ODEs and then present how they can be used in the infinitedimensional case represented by Delay Differential Equations (DDEs). The aim of the work is to develop a first systematic study (i.e. analyzing theoretical foundations, implementation and convergence) of a numerical scheme for DDEs not being a mere adaptation of ODEs methods. This is a joint work in progress with Luca Dieci from Georgia Institute of Technology (Atlanta, GA USA) and Erik Van Vleck from Department of Mathematics, Kansas University (Lawrence, KS USA).