A Program for Computing Periodic Orbits in Hamiltonian Systems Based on Multiple Shooting Algorithms

Stavros C. Farantos

Institute of Electronic Structure and Laser,
Foundation for Research and Technology - Hellas,


Department of Chemistry
University of Crete,
Iraklion, Crete 711 10, Greece.

Published in "Computer Physics Communications Vol. 108, p. 240, 1998."


POMULT is a FORTRAN code for locating Periodic Orbits and Equilibrium Points in Hamiltonian systems based on 2-point boundary value solvers which use multiple shooting algorithms. The code has mainly been developed for locating periodic orbits in molecular Hamiltonian systems with many degrees of freedom and it utilizes a damped Newton-Raphson method and a secant method. Graphical User Interface has also been written in tcl-tk script language for interactively manipulating the input and output data. POMULT provides routines for a general analysis of a dynamical system such as fast Fourier transform of the trajectories, Poincare surfaces of sections, maximum Lyapunov exponents and evaluation of the classical autocorrelation functions and power spectra.
Keywords: molecular dynamics and spectra, periodic orbits, multiple shooting algorithm, damped Newton-Raphson method

Title of the program: POMULT (Periodic Orbit MULTishooting)
Catalogue number: POMT
Program obtainable form: CPC Program Library, Queen's University of Belfast, N. Ireland
Licensing provisions: Numerical Recipes
Computer: Tested on workstations HP-9000/735, IBM-7030/3CT, PC-Linux
Installation: IESL-FORTH, Iraklion, Crete, Greece
Operating system: UNIX
Programming language used: FORTRAN 77 with extensions, lower case, implicit, include
Memory required to execute with typical data: 5 Mbytes
No. of bits in a word: 32
No. of bytes in distributed program, included test data etc: 2211840
Distribution format: gzip compressed tar file
Keywords: molecular dynamics and spectra, periodic orbits, multiple shooting algorithm, damped Newton-Raphson method
Nature of the physical problem:
Given a multidimensional highly coupled molecular potential energy surface we want to compute families of periodic solutions of Hamilton equations. These families of periodic orbits reveal the structure of the classical phase space by detecting the regions of phase space with regular and chaotic motions. Furthermore, periodic orbits point out possible localization of the quantum wavefunctions, and explain/predict spectroscopic features.
Method of solution:
The location of periodic orbits is based on damped Newton-Raphson methods or secant-Quasi Newton methods. Simple or Multiple shooting algorithms are employed which are robust in cases of long period or highly unstable periodic orbits.
Restrictions on the complexity of the problem:
The program has been tested with 2-, 3-, 5-, and 6-dimensional molecular potential functions. Limitations are observed in cases of high instability or in regions of phase space densely occupied by periodic orbits. The above difficulties cause also limitations in the continuation of a family of periodic orbits with a parameter.
Typical running time:
This depends on the complexity of the potential function, the period and the number of periodic orbits which are computed, and whether the equations of motion are stiff or not.
Standard numerical actions like integration of ordinary differential equations and solution of linear algebraic equations are carried out with routines from the package ``Numerical Recipes''. The program can be interfaced with ODESSA or other available programs which carry out sensitivity analysis of differential or algebraic equations. Generally, the program has been written in such a way that the user can incorporate his/her own favorable subroutines. A Makefile, a README file as well as a help file are provided for the installation of the program and the explanation of the input data. Graphical User Interface for the input data has been written in a tcl-tk script language. The user should ensure that the libraries versions tcl7.0 and tk4.0 or higher are installed in her/his system.


Tue Dec 16 20:32:24 EET 1997