Localization of periodic orbits of polynomial systems by ellipsoidal estimates Konstantin E. Starkov a, * ,1 , Alexander P. Krishchenko b,2 a CITEDI-IPN, Avenue del Parque 1310, Mesa de Otay, Tijuana, BC, Mexico b Bauman Moscow State Technical University, 2nd Baumanskaya Street, 5, Moscow 105005, Russia Accepted 8 June 2004 Abstract In this paper we study the localization problem of periodic orbits of multidimensional continuous-time systems in the global setting. Our results are based on the solution of the conditional extremum problem and using sign-definite quadratic and quartic forms. As examples, the Rikitake system and the LambÕs equations for a three-mode operating cavity in a laser are considered. Ó 2004 Published by Elsevier Ltd. 1. Introduction The localization problem of periodic orbits of nonlinear multidimensional continuous-time systems has been studied by many researchers during last years, see papers with analytical methods based on second order extremum conditions [1,4,6–8], with algebraic methods based on using algebraic dependent polynomials [12], with analytical methods based on high-order extremum conditions [13,14], see also [10] and others. Now it is well known that possessing of periodic orbits is one of essential features specifying the global dynamics of chaotical systems especially in domains containing attractors. For example, the description of a chaotic attractor with help of infinitely many unstable periodic orbits embedded in it is helpful in studies concerning the Lorenz system, see [3] and references therein. The main contribution of this paper consists in obtaining ellipsoidal estimates for domains containing all periodic orbits and for domains having no common points with any of periodic orbits. Our methods are based on the solution of the conditional extremum problem introduced in [7] and using sign-definite quadratic and quartic forms. As examples, we consider the Rikitake system and the LambÕs equations for a three-mode operating cavity in a laser. This paper is the reworked and enlarged version of the short conference paper [15]. 0960-0779/$ - see front matter Ó 2004 Published by Elsevier Ltd. doi:10.1016/j.chaos.2004.06.002 * Corresponding author. Tel.: +1 52 66231344; fax: +1 52 66231388. E-mail addresses: konst@citedi.mx, konstarkov@hotmail.com (K.E. Starkov), apkri@999.ru (A.P. Krishchenko). 1 Address: CITEDI-IPN, 2498 Roll Drive #757, San Diego, 92154 CA, USA. 2 Tel.: +7 095 2636750; fax: +7 095 2679893. Chaos, Solitons and Fractals 23 (2005) 981–988 www.elsevier.com/locate/chaos