Differential Equations and Nonlinear Mechanics
Volume 2008 (2008), Article ID 745242, 7 pages
doi:10.1155/2008/745242
Research Article
On Using Curvature to Demonstrate Stability
Department of Mathematics, Wilfrid Laurier University, 75 University Ave West, Waterloo, ON, N2L 3C5, Canada
Received 11 September 2007; Revised 28 February 2008; Accepted 2 April 2008
Academic Editor: Bhabani Dandapat
Copyright © 2008 C. Connell McCluskey. 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.
Abstract
A new approach for demonstrating the global stability of ordinary differential equations is given. It is shown that if the curvature of solutions is bounded on some set, then any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable. The approach can also be used to rule out the sudden appearance of large-amplitude periodic orbits.