Optimal stabilization of unstable periodic orbits embedded in chaotic systems

C.A. Cruz-Villar

Abstract


A gradient-flow-based approach is proposed in this paper for stabilizing unstable periodic orbits (UPO) embedded in chaotic systems. In order to obtain an on-line stabilizing solution, the stabilization problem is considered to be an optimal control problem, and system state sensitivities with respect to the control input are introduced. The resulting feedback controller is able to stabilize UPO embedded in both kind of systems, with or without an odd Floquet number. Moreover, the proposed approach is easily extended to identifying the period of the UPO to be stabilized when it is unknown. Simulation experiments of the proposed controller are carried out on the Rössler and the Lorenz systems.

Keywords


Optimal stabilization; sensitivity theory; gradient flow; unstable periodic orbits

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REVISTA MEXICANA DE FÍSICA, year 67, issue 2, March-April 2021. Bimonthly Journal published by Sociedad Mexicana de Física, A. C. Departamento de Física, 2º Piso, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Alcaldía Coyacán, C.P. 04510 , Ciudad de México. Apartado Postal 70-348. Tel. (+52)55-5622-4946, https://rmf.smf.mx/ojs/rmf, e-mail: rmf@ciencias.unam.mx. Chief Editor: José Alejandro Ayala Mercado. INDAUTOR Certificate of Reserve: 04-2019-080216404400-203, ISSN: 2683-2224 (on line), 0035-001X (print), both granted by Instituto Nacional del Derecho de Autor. Responsible for the last update of this issue, Technical Staff of Sociedad Mexicana de Física, A. C., Fís. Efraín Garrido Román, 2º. Piso, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, Alcaldía Coyacán, C.P. 04510 , Ciudad de México. Date of last modification, March 1st., 2021.

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