Mathematical Problems in Engineering
Volume 2004 (2004), Issue 3, Pages 277-290

Localization of periodic orbits of autonomous systems based on high-order extremum conditions

Konstantin E. Starkov1,2

1CITEDI-IPN, avenue del Parque 1310, Mesa de Otay, Tijuana, Baja California 22510, Mexico
2CITEDI-IPN, 2498 Roll Drive 757, San Diego 92154, CA, USA

Received 12 November 2003; Revised 16 March 2004

Copyright © 2004 Konstantin E. Starkov. 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.


This paper gives localization and nonexistence conditions of periodic orbits in some subsets of the state space. Mainly, our approach is based on high-order extremum conditions, on high-order tangency conditions of a nonsingular solution of a polynomial system with an algebraic surface, and on some ideas related to algebraically-dependent polynomials. Examples of the localization analysis of periodic orbits are presented including the Blasius equations, the generalized mass action (GMA) system, and the mathematical model of the chemical reaction with autocatalytic step.