Computer program uses an iterative method to construct precisely periodic orbits which dynamically approximate solutions that converge to precise dynamical solutions in the limit of the sequence. The method used is a modification of the generalized Newton-Raphson algorithm used in analyzing two point boundary problems

Bennett, A. G.

Hanafy, L. M.

Palmore, J. I.

English

NASA Technical Reports Server

June 1968	 Brief 68-10217 
NASA TECH BRIEF 
NASA Tech Briefs are issued to summarize specifiê innovations derived from the U.S. space program, 
to encourage their commercial application. Copies are available to the public at 15 cents each 
from the Clearinghouse for Federal Scientific and Technical Information, Springfield, Virginia 22151. 
Computer Program Offers New Method for Constructing

Periodic Orbits in Nonlinear Dynamical Systems 
The problem: 
To develop a method for constructing precisely 
periodic orbits which, at the start, dynamically ap-
proximates solutions that converge to precise dy-
namical solutions in the limit of the sequence. Previ-
ous methods isolated orbits with a sequence of precise 
dynamical solutions which at the start of the sequence 
has only approximate periodicity but which converges 
to a periodic solution in the limit of the sequence. 
The solution: 
A computer program that utilizes an iterative 
method for constructing periodic orbits in nonlinear 
dynamical systems. 
How it's done: 
• The method used in this program is a modification 
of the generalized Newton-Raphson algorithm used 
in analyzing two point boundary problems. This 
technique is distinct in that it constructs a sequence 
of precisely periodic but, at the start, dynamically 
approximate solutions that converge to a precise dy-
namical solution in the limit of the sequence. The 
iterates are solutions of a linear system derived from 
the dynamical equations of motion. For conservative 
Hamiltonian systems, stability of the periodic orbit is 
determined as a byproduct of the method of iterating 
an orbit. Along the periodic orbit the trace of the 
resolvent (fundamental matrix) of the equations of
variation determines uniquely the characteristic expo-
nents of the orbit. An approximation to the resolvent 
along each iterate is determined in the solution of the 
linear system. 
Notes: 
I. This program is written in Fortran H for use on the 
IBM 360, Model 65 computer. 
2. This program can be used in nonaerospace as well 
as aerospace calculations of periodic orbits in 
either circular or elliptic restricted three-body 
problems to develop qualitative information for 
space flight missions as well as detailed trajectories 
for motions of deep space stations in earth-moon 
systems. 
3. Inquiries concerning this program may be made to: 
COSMIC 
Computer Center 
University of Georgia 
Athens, Georgia 30601 
Reference: B68-10217 
Patent status: 
No patent action is contemplated by NASA. 
Source: Arthur G. Bennett, Julian I. Palmore, 
and Laurence M. Hanafy 
of TheBoeing Company 
under contract to 
Marshall Space Flight Center 
(M FS- 14654) 
Category 06 
This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States
Government assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately owned rights.
https://ntrs.nasa.gov/search.jsp?R=19680000217 2020-03-12T05:34:27+00:00Z


Computer program offers new method for constructing periodic orbits in nonlinear dynamical systems

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