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Revista mexicana de física
versão impressa ISSN 0035-001X
Rev. mex. fis. vol.52 no.1 México Fev. 2006
Investigación
Algebraic approach for the reconstruction of Rössler system from the x3variable
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(tk)}0k=nof the threedimensional 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 reparameterized system (linearinthe 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(tk)}0k=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 (CGPIIPN) under research grants 20020247 and 20051306.
References
1. F. Takens, Dynamical Systems and Turbulence, eds. Rand, D.A. & Young, L.S. (SpringerVerlag, Berlin, 1981) p. 366. [ Links ]
2. N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, Phys. Rev. Lett. 45 (1980) 712. [ Links ]
3. T. Sauer, J. Yorke, and M. Casdagli, J. Stat. Phys. 65 (1991) 579. [ Links ]
4. G.L. Baker, J.P. Gollub, and J.A. Blackburn, American Institute of Physics Chaos 6 (1996) 528. [ Links ]
5. C. Lainscsek, C. Letellier, and I. Gorodnitsky, Physics Letters A 314 (2003) 409. [ Links ]
6. A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations (Wiley, Chichester, 1979). [ Links ]
7. J.P. Crutchfield and B.S. McNamara, Complex Systems 1 (1987) 417. [ Links ]
8. G.P King and I. Stewart, Physica D 58 (1992) 216. [ Links ]
9. H.D.I. Abarbanel, Analysis of observed chaotic data (SpringerVerlag, 1996). [ Links ]
10. U. Parlitz, R. Zöller, J. Holzfuss, and W. Lauterborn, Int. Journal of Bifurc. and Chaos 4 (1994) 1715. [ Links ]
11. K.T. Alligood, T.D. Sauer, and J.A. Yorke, Chaos: An Introduction to Dynamical Systems (SpringerVerlag, New York, 1997). [ Links ]
12. U. Feldmann, M. Hasler, and W. Schwarz, Int. J. Circuit Theory Appl. 24 (1996) 551. [ Links ]
13. P. Soderstrom and P. Stoica, System Identification (Prentice Hall 1989). [ Links ]
14. I.D. Landau, System Identification and Control Design (Prentice Hall 2000). [ Links ]
15. A.M. Dabroom and H.K. Khalil, Int. J. Control 72 (17) (1999) 1523. [ Links ]
16. L.A. Aguirre, U.S. Freitas, C. Letellier, and J. Maquet, Physica D 158 (2001) 1. [ Links ]
17. L.A. Aguirre, G.G. Rodriguez, and E.M. Mendes, Int. J. Bifurc. Chaos 7 (1997) 1411. [ Links ]
18. M. Fliess and H. SiraRamirez, ESAIM 9 (2003) 151. [ Links ]
19. M.S. SuárezCastañon, C. AguilarIbáñez, and R. BarrónFernandez, Physics Letters A 308 (2003) 47. [ Links ]
20. C. Letellier, J. Maquet, L. Le Sceller, G. Gouesbet, and L. Aguirre, J. Phys. A 31 (1998) 7913. [ Links ]
21. S. Diop, J.W. Grizzle, P.E. Moral, and A. Stefanopoulou, Interpolation and numeric differentiation for observer design, ACC (1994) 1329; (1987) 7. [ Links ]
22. O.E. Rossler, Physics Letter 57 (1976) 397. [ Links ]
23. S.H. Strogatz, Nonlinear Dynamics and Chaos (Perseus Publishing, 1994). [ Links ]
24. E. Mosekilde, Topics in Nonlinear Dynamics: Applications to Physics, Biology and Economic Systems (World Scientific, Singapore, 2003). [ Links ]
25. H. Nijmeijer and I.Mareels, An Observer look to synchronization, IEEE Transactions on Circuits and System 36 (1997) 882. [ Links ]