Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 104547, 15 pages
Research Article

Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum

Department of Mathematics, Prague Institute of Chemical Technology, Technicka 5, 166 28 Prague, Czech Republic

Received 2 December 2008; Revised 26 January 2009; Accepted 21 February 2009

Academic Editor: José Roberto Castilho Piqueira

Copyright © 2009 P. Pokorny. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Continuation is an efficient algorithm for finding solutions of systems of nonlinear algebraic equations where the solutions form a one-dimensional continuum. Such systems arise naturally when investigating equilibrium points and periodic solutions of ordinary differential equations with one parameter. Continuation of isolated periodic solutions of dissipative systems is a well-established technique. Less attention has been devoted to continuation of periodic solutions of conservative systems, where periodic solutions typically form a one-parameter family. To specify a single periodic solution, additional condition must be considered. However, this gives an over-determined system, which has no solution when working with approximate numerical values. We propose a simple algorithm which solves this difficulty by using singular value decomposition of the Jacobian matrix. This algorithm is applied to the conservative model of elastic pendulum. A branch of periodic solutions with constant energy is found which is born by the period doubling bifurcation of vertical oscillations.