The Parameterization Method in KAM Theory

This chapter of the book is devoted to the parameterization method in KAM theory, also referred to as KAM theory without action-angle coordinates. The chapter states and proves a KAM theorem in a posteriori format, with explicit bounds suitable to be applied in an effective and quantitative way. We have included full descriptions of the derived algorithms, and applications to the standard and the Froeschle map.
Notice that during the implementation of these algorithms, we have paid attention to the clarity of the code. Some subroutines can be (and have been) improved subtantially for demanding research purposes. Nevertheless, the provided code is fast enough and has de advantage that it is easier to read. It uses a similar notation that the one used in the book and we encode objects in C++ classes. Thus, the overloading of operations will allow the reader to compare with algorithm and arguments exposed in the chapter.

 

Codes and documentation

  • README file with instructions
  • complex.h libray of complex numbers
  • grid.h library to manipulate multi-angle functions
  • matrix.h library to manipulate matrices of multi-angle functions
  • param.cc main code for computing invariant tori
  • curve_start Initial data for the standard map example
  • torus_start Initial data for the Froeschle map example
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