Normal Modes, Symmetries and Stability

  • Tassos Bountis
  • Haris Skokos
Part of the Springer Series in Synergetics book series (SSSYN, volume 10)


The present Chapter studies nonlinear normal modes (NNMs) of coupled oscillators from an altogether different perspective. Focusing entirely on periodic boundary conditions and using the Fermi Pasta Ulam β (FPU − β) and FPU − α models as examples, we demonstrate the importance of discrete symmetries in locating and analyzing exactly a class of NNMs called one-dimensional “bushes”, depending on a single periodic function \(\hat{q}(t)\). Using group theoretical arguments one can similarly identify n-dimensional bushes described by \(\hat{{q}}_{1}(t),\ldots,\hat{{q}}_{n}(t)\), which represent quasiperiodic orbits characterized by n incommensurate frequencies. Expressing these solutions as linear combinations of single bushes, it is possible to simplify the linearized equations about them and study their stability analytically. We emphasize that these results are not limited to monoatomic particle chains, but can apply to more complicated molecular structures in two and three spatial dimensions, of interest to solid state physics.


Symmetry Group Normal Mode Vibrational State Discrete Symmetry Modal Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tassos Bountis
    • 1
  • Haris Skokos
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of PatrasPatrasGreece
  2. 2.MPI for the Physics of Complex SystemsDresdenGermany
  3. 3.Physics DepartmentAristotle University of ThessalonikiThessalonikiGreece

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