Investigación

Algebraic approach for the reconstruction of Rössler system from the x₃—variable

C. Aguilar Ibáñez

Centro de Investigación en Computación del IPN, Av. Juan de Dios Bátiz s/n Esq. M. Othón de Mendizabal, Unidad Profesional Adolfo López Mateos, Col. Nueva Industrial Vallejo, México, D.F., 07700, México email: caguilar@cic.ipn.mx

Recibido el 6 de septiembre de 2005
Aceptado el 11 de noviembre de 2005

Abstract

In this paper we propose a simple method to identify the unknown parameters and to estimate the underlying variables from a given chaotic time series {x₃(t_k)}₀^k=nof the three–dimensional Rössler system (RS). The reconstruction of the RS from its x₃— variable is known to be considerably more difficult than reconstruction from its two other variables. We show that the system is observable and algebraically identifiable with respect to the auxiliary output ln(x₃), hence, a differential parameterization of the output and its time derivatives can be obtained. Based on these facts, we proceed to form an extended re–parameterized system (linear–in–the –parameters), which turns out to be invertible, allowing us to estimate the variables and missing parameters.

Keywords: Chaotic systems; inverse problem; estimation of parameters and variables.

Resumen

Este articulo se presenta un metodo sencillo para recuperar el los parámetros del modelo y para recuperar las variables no disponibles del sistema caotico de Rössler, a partir de el conocimiento de una serie de tiempo {x₃(t_k)}₀^k=n. Es muy bien sabido, que reconstruir este sistema a partir de la variable x₃ es mas difícil que tratar de reconstruirlo a partir de las otras variables. Usando el hecho que este sistema es identificable y algebraicamente observable con respecto a la transformación ln(x₃), se procede a obtener una parametrizacion diferencial de la salida. Esta parametrizacion puede ser invertible bajo ciertas condiciones. Permitiéndonos estimar parámetros y variables desconocidas del modelo.

Descriptores: Sistemas caoticos; problema inverso; estimación de parámetros y variables.

PACS: 02.60.Lj; 05.45.Gg, 05.45.Pq; 05.45+b

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Acknowledgments

This research was supported by the Coordinacion de Posgrado e Investigacion (CGPI–IPN) under research grants 20020247 and 20051306.

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