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