"Rigorous KAM results around arbitrary periodic orbits for Hamiltonian Systems."

T.Kapela, C.Simo

An instruction how to compile and run programs that perform computations for computer assisted proofs of KAM stability


Programs require CAPD-DynSys 4 library with multiprecision support (MPFR library). Source code or binary packages along with installation instructions can be found on

CAPD web page


Compiler: The program must be compiled with gcc-4.8 or newer.


  1. Unpack archive

    tar -xf KAMstability.tar.gz

  2. (optional) If CAPD is installed to the non-standard localization (/usr/local/) open file KAMstability/Makefile and edit line with


    to point to the directory with the CAPD scripts e.g. mpcapd-config. Path has to end with / e.g.

    CAPDBINDIR = /home/user/capd/bin/

  3. Change directory to KAMstability and call make

    cd KAMstability

  4. After succesfull compilation executable files H4Proof and EightProof, which perform proofs, are placed in the current directory


Main programs, which perform proofs, are

They perform the following steps:

  1. refining of approximated initial conditions of a periodic orbit
  2. proof of the periodic orbit existence and computation of rigorous enclosure of its initial conditions
  3. computations of the Taylor expansion of Poincare map at periodic orbit i.c.
  4. computations of a normal form and verification of non-degeneracy conditions

Command line arguments:

  1. input file : containing approximated initial conditions and parameters of the computations
  2. output file (optional)

Input files for proofs described in article are stored in data directory:

To perform e.g. proof of KAM stability of period 3 orbit for quartic potential run command

./H4Proof data/quartic_3.txt result.txt

The output will be written to the screen and file result.txt will contain computed normal form, rigorous estimates of initial conditions and parameters used in computations.

In the current directory file output.tex will be created with the results in the LaTeX form, similar to the one given in article.


The directory result contains example output files from proofs performed on authors computers. The actual result can slightly differ depending on operating system, processor and/or compiler used.


Time of computations for figure eight solution is ca 2 hours.

Under Linux depending on your system settings to run programs you may have to set LD_LIBRARY_PATH to point to directory with CAPD libraries e.g. by calling

export LD_LIBRARY_PATH=/usr/local/lib

or run


Programs were tested on