Elliptic tori in FPU non-linear chains with a small number of nodes
Autor: | Ugo Locatelli, Chiara Caracciolo |
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Rok vydání: | 2021 |
Předmět: |
Primary: 37J40
Secondary: 37N05 65P40 70H08 70H15 Fermi–Pasta–Ulam problem FOS: Physical sciences KAM theory 01 natural sciences 010305 fluids & plasmas Hamiltonian system 0103 physical sciences Applied mathematics 010306 general physics Mathematical Physics Mathematics Numerical Analysis Sequence Series (mathematics) Kolmogorov–Arnold–Moser theorem Applied Mathematics Numerical analysis Normal form methods Perturbation theory for Hamiltonian systems Mathematical Physics (math-ph) Symbolic computation Settore MAT/07 Modeling and Simulation Lower dimensional invariant tori FPU problem Hamiltonian (control theory) |
DOI: | 10.48550/arxiv.2102.05908 |
Popis: | We revisit an algorithm constructing elliptic tori, that was originally designed for applications to planetary hamiltonian systems. The scheme is adapted to properly work with models of chains of N + 1 particles interacting via anharmonic potentials, thus covering also the case of FPU chains. After having preliminarily settled the Hamiltonian in a suitable way, we perform a sequence of canonical transformations removing the undesired perturbative terms by an iterative procedure. This is done by using the Lie series approach, that is explicitly implemented in a programming code with the help of a software package, which is especially designed for computer algebra manipulations. In the cases of FPU chains with N = 4 , 8 , we successfully apply our new algorithm to the construction of elliptic tori for wide sets of the parameter ruling the size of the perturbation, i.e., the total energy of the system. Moreover, we explore the stability regions surrounding 1D elliptic tori. We compare our semi-analytical results with those provided by numerical explorations of the FPU-model dynamics, where the latter ones are obtained by using techniques based on the so called frequency analysis. We find that our procedure works up to values of the total energy that are of the same order of magnitude with respect to the maximal ones, for which elliptic tori are detected by numerical methods. |
Databáze: | OpenAIRE |
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