Nonlinear Oscillations of Strings and Beams

Abstract Wave and beam equations are fundamental partial differential equations used to model a wide range of mechanical systems. A central challenge in the study of such systems is to characterize their nonlinear free oscillations. Previous studies, dating back to Poincaré and the development of the celebrated KAM theory, have focused primarily on the existence of disconnected families of small-amplitude solutions bifurcating from the trivial state. Here we study the global structure of time-periodic solutions to 1D cubic wave and beam equations on an interval and uncover their intricate, fractal-like structure. In particular, we identify a new class of large-energy solutions with complex mode compositions and propose a systematic framework for their analysis. A Floquet stability study reveals that this class contains solutions that are linearly stable, indicating their importance for the nonlinear dynamics of these systems. The uncovered phenomena are ubiquitous as similar results can be observed in a broad range of systems with spatial confinement.