Discrete Symmetry and Stability in Hamiltonian Dynamics

Details Discrete Symmetry and Stability in Hamiltonian Dynamics Discrete Symmetry and Stability in Hamiltonian Dynamics

2010-05-24

arxiv.org/abs/1005.4308v3

Nonlinear Sciences

Chaotic Dynamics

50 pages, 13 figures

Scientific paper

In this tutorial we address the existence and stability of periodic and quasiperiodic orbits in N degree of freedom Hamiltonian systems and their connection with discrete symmetries. Of primary importance in our study are the nonlinear normal modes (NNMs), i.e periodic solutions which represent continuations of the system's linear normal modes in the nonlinear regime. We examine the existence of such solutions and discuss different methods for constructing them and studying their stability under fixed and periodic boundary conditions. In the periodic case, we employ group theoretical concepts to identify a special type of NNMs called one-dimensional "bushes". We describe how to use linear combinations such NNMs to construct s(>1)-dimensional bushes of quasiperiodic orbits, for a wide variety of Hamiltonian systems and exploit the symmetries of the linearized equations to simplify the study of their destabilization. Applying this theory to the Fermi Pasta Ulam (FPU) chain, we review a number of interesting results, which have appeared in the recent literature. We then turn to an analytical and numerical construction of quasiperiodic orbits, which does not depend on the symmetries or boundary conditions. We demonstrate that the well-known "paradox" of FPU recurrences may be explained in terms of the exponential localization of the energies Eq of NNM's excited at the low part of the frequency spectrum, i.e. q=1,2,3,.... Thus, we show that the stability of these low-dimensional manifolds called q-tori is related to the persistence or FPU recurrences at low energies. Finally, we discuss a novel approach to the stability of orbits of conservative systems, the GALIk, k=2,...,2N, by means of which one can determine accurately and efficiently the destabilization of q-tori, leading to the breakdown of recurrences and the equipartition of energy, at high values of the total energy E.

Affiliated with

Also associated with

No associations

Rating

Discrete Symmetry and Stability in Hamiltonian Dynamics does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Discrete Symmetry and Stability in Hamiltonian Dynamics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Discrete Symmetry and Stability in Hamiltonian Dynamics will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-119062