Low-dimensional -tori in FPU lattices: Dynamics and localization properties

Recent studies on the Fermi–Pasta–Ulam (FPU) paradox, like the theory of q -breathers and the metastability scenario, dealing mostly with the energy localization properties in the FPU space of normal modes ( q -space), motivated our first work on q -tori in the FPU problem (Christodoulidi et al., 2010)  [19]. The q -tori are low-dimensional invariant tori hosting trajectories that present features relevant to the interpretation of FPU recurrences as well as the energy localization in q -space. The present paper is a continuation of our work in Christodoulidi et al. (2010)  [19]. Our new results are: we extend a method of analytical computation of q -tori, using Poincaré–Lindstedt series, from the β to the α -FPU and we reach significantly higher expansion orders using an improved computer-algebraic program. We probe numerically the convergence properties as well as the level of precision of our computed series. We develop an additional algorithm in order to systematically locate values of the incommensurable frequencies used as an input in the PL series construction of q -tori corresponding to progressively higher values of the energy. We generalize a proposition proved in Christodoulidi et al. (2010)  [19] regarding the so-called ‘sequence of propagation’ of an initial excitation in the PL series. We show by concrete examples how the latter interprets the localization patterns found in numerical simulations. We focus, in particular, on various types of extensive initial excitations that lead to q -tori solutions with exponentially localized profiles. Finally, we discuss the relation between q -tori, q -breathers (viewed as one-dimensional q -tori), and the so-called ‘FPU-trajectories’ invoked in the original study of the FPU problem.