Frequency behavior of saturated nonlinear function series based on opamps
E. Ortega-Torresa, C. Sánchez-Lópeza, J. Mendoza-Lópezb
a Universidad Autónoma de Tlaxcala (UAT), Clzda Apizaquito s/n, km. 1.5, Apizaco, Tlaxcala, 70300, México.
b Microelectronics Institute of Sevilla, (IMSE-CSIC) and University of Seville, Seville 41092, Spain.
]]> Received 3 December 2012
Abstract
In multiscroll chaotic circuit design based on active devices, piece-wise linear (PWL) approaches are often used to model the behavior of nonlinear functions, thereby that the behavior of a chaotic system can be forecasted through numerical simulations. However, although PWL models are relatively easy to build, they do not include any information related on the performance parameters of the active devices to be used. This a serious shortcoming, since PWL-models introduces a level of inaccuracy into a numerical analysis which is more evident when numerical simulations and experimental results are compared. These differences are more pronounced when the chaotic waveforms to be generated are pushed to operate at high-frequency. This paper introduces experimental results on the frequency behavior of a nonlinear function called saturated nonlinear function series based on operational amplifiers. These new results are key not only on the automatic synthesis of chaotic attractors and on the synchronization schemes used in secure communication systems based on chaos, but also on the metrics used to evaluate the complexity of a chaotic system. A mathematical model to characterize the behavior of the nonlinear function is also derived, showing a better accuracy compared with the PWL approach. The theoretical derivations and related results are experimentally validated through implementations from commercially available devices.
Keywords: Chaotic systems; computer aided analysis; saturated function series; modeling; chaos.
Resumen
En el diseño de circuitos caóticos con múltiples enrollamientos basado en dispositivos activos, el comportamiento de las funciones no lineales es representado por aproximaciones lineales a trozos (PWL). Sin embargo, aunque modelos PWL son fáciles de construir, estos no incluyen ninguna información relacionada con los parámetros de desempeño de los dispositivos activos a ser usados. Este es un serio inconveniente, ya que modelos PWL introducen un nivel de inexactitud en un análisis numérico el cual llega a ser más evidente cuando las simulaciones numéricas son comparadas con resultados experimentales. Estas diferencias son más pronunciadas cuando las formas de onda caóticas a ser generadas son empujadas a operar en alta frecuencia. Este artículo introduce resultados experimentales sobre el comportamiento en frecuencia de la función no lineal llamada serie de funciones saturadas las cuales son diseñadas con amplificadores operacionales. Los nuevos resultados son clave no solamente en la síntesis automática de atractores caóticos y en los esquemas de sincronización usados en los sistemas de comunicación basados en caos, pero también en las métricas usadas para evaluar la complejidad de un sistema caótico. Un modelo analítico para caracterizar el comportamiento de la función no lineal es derivado y este es más exacto que el modelo PWL. Los resultados teóricos son validados con resultados experimentales a través del uso de dispositivos comerciales.
Descriptores: Sistemas caóticos; análisis asistido por computadora; serie de funciones saturadas; modelado; caos.
]]>PACS: 05.45.Pq; 05.45.Pq; 84.30.Ng; 07.50.Ek; 84.30.-r; 01.50.Pa
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Acknowledgments
This work has been supported in part by the projects: UAT-121AD-R and CACyPI-UATx-2013 both funded by Autonomous University of Tlaxcala, Mexico. Author J.M.L. thanks the support of the JAE-Doc program of CSIC, co-funded by the E.S.F.
References
1. H.D.I. Abarbanel, R. Brown, and M.B. Kennel, J. Nonlinear Science 1 (1991) 25. [ Links ]
2. L.O. Chua and P.M. Lin, Computer-Aided Analysis of Electronic Circuits: Algorithms and Computational Techniques. (N.J. Prentice Hall, USA 1975). [ Links ]
3. G. Chen, Controlling Chaos and Bifurcations in Engineering Systems (Boca Raton, Florida, CRC Press LLC, USA 1999). [ Links ]
4. G. Chen and T. Ueta, Chaos in Circuits and Systems (World Scientific Publishing Co. Pte. Ltd., Singapore 2002). [ Links ]
5. D. Chen Z. Sun, X. Ma, and L. Chen, Int. J. Circuit Theory Appl., Doi: 10.1002/cta.1860 (2012). [ Links ]
6. A.S. Elwakil and M.P. Kennedy, IEEE Trans. on Circuits Syst. I, Fundam. Theory Appl. 47 (2000) 4. [ Links ]
7. A.S. Elwakil, K.N. Salama, and M.P. Kennedy, Int. J. Bifurc. Chaos 12 (2002) 11. [ Links ]
8. F.I. Fazanaro, D.C. Soriano, R. Suyama, R. Attux, M.K. Madrid, and J.R. Oliveira, Chaos 23 (2013) 10. [ Links ]
9. P. Grassberger and I. Procaccia, Physical Review A 28 (1983) 3. [ Links ]
10. L. Gámez-Guzmán, C. Cruz-Hernández, R. López-Gutierrez, and E. García-Guerrero, Application to communication. Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 11. [ Links ]
11. T. Gotthans and Z. Hrubos, J. Elect. Eng. 64 (2013) 5. [ Links ]
12. J.A. Hernández, R.M. Benito, and J.C. Losada, Int. J. Bifurc. Chaos 21 (2011) 11. [ Links ]
13. A. Khan and P. Singh, Int. J. Bifurc. Chaos 18 (2008) 7. [ Links ]
14. J. Lü, G. Chen, X. Yu, and H. Leung, IEEE Trans. CAS-I 51 (2004) 15. [ Links ]
15. J. Lü and G. Chen, Int. J. Bifurc. Chaos 16 (2006) 84. [ Links ]
16. J. Lü, S. Yu, H. Leung and G. Chen, IEEE Trans. CAS-I 53 (2006) 17. [ Links ]
17. X. Liu, X. Shen, and H. Zhang, Int. J. Bifurc. Chaos 22 (2012) 15. [ Links ]
18. Y. Liu, S. Pang, and D. Chen, Math. Comp. Mod. 57 (2013) 21. [ Links ]
19. S. Ozoguz etal., Electron. Lett. 38 (2002) 2. [ Links ]
20. S. Rugonyi and K.J. Bathe, Int. J. Numerical Methods in Engineering 56 (2003) 19. [ Links ]
21. A.G. Radwan, A.M. Soliman, and A. El-Sedeek, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 50 (2003) 4. [ Links ]
22. J.A.K. Suykens and J. Vandewalle, IEEE Trans. CAS-I 40 (1993) 8. [ Links ]
23. H. Sira-Ramírez and C. Cruz-Hernández, Int. J. Bifurc. Chaos 11 (2001) 15. [ Links ]
24. C. Sánchez-López, A. Castro-Hernández, and A. Perez-Trejo, IEICE Electron. Express 5 (2008) 5. [ Links ]
25. C. Sánchez-López, R. Trejo-Guerra, J.M. Murioz-Pacheco, and E. Tlelo-Cuautle, Nonlinear Dynamics 61 (2010) 11. [ Links ]
26. C. Sánchez-López, Applied Mathematics and Computation 217 (2011) 9. [ Links ]
27. C. Sánchez-López, J.M. Muñoz-Pacheco, E. Tlelo-Cuautle, V.H. Carbajal-Gómez, and R. Trejo-Guerra, In: Proc. IEEE Int. Symp. Circuits Syst. (Rio do Janeiro, Brazil, 2011) p. 2950. [ Links ]
28. C. Sánchez-López, Rev. Mex. Fis. 58 (2012) 8. [ Links ]
29. C. Sánchez-López et al., In Proc. IEEE Int. Latin American Symp. (Circuits Syst. Cusco, Peru, 2013). p. 1-4. [ Links ]
30. H. Shao-Bo, S. Ke-Hui, and Z. Cong-Xu, Chin. Phys. B 22 (2013) 6. [ Links ]
31. R. Trejo-Guerra, E. Tlelo-Cuautle, C. Cruz-Hernández, and C. Sánchez-López, Int. J. Bifurc. Chaos 19 (2009) 10. [ Links ]
32. R. Trejo-Guerra, E. Tlelo-Cuautle, J.M. Muñoz-Pacheco, C. Sánchez-López, and C. Cruz-Hernández, Int. J. Nonlinear Sciences & Numerical Simulation 11 (2010) 8. [ Links ]
33. F.Q Wang and C.X Liu, Int. J. Modern Physics 22 (2008) 7. [ Links ]
34. C.H. Wang, H. Xu and F. Yu, Int. J. Bifurc. Chaos 23 (2013) 10. [ Links ]
35. M.E. Yalcin, J.A.K. Suykens, and J. Vandewalle, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 47 (2000) 5. [ Links ]
36. M.E. Yalcin, S. Özoguz, J.A.K. Suykens, and J. Vandewalle, Electron. Lett. 37 (2001) 2. [ Links ]
37. M.E. Yalcin, J.A.K. Suykens, J. Vandewalle, and S. (Özoguz, Int. J. Bifurc. Chaos 12 (2002) 19. [ Links ]
38. S.M. Yu, J. Lu, H. Leung, and G. Chen, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 52 (2005) 19. [ Links ]
39. S. Yu, W.K.S. Tang, and G. Chen, Int. J. Bifurc. Chaos 17 (2007) 14. [ Links ]
40. G. Zhong, K.F. Man, and G. Chen, Int. J. Bifurc. Chaos 12 (2002) 9. [ Links ]
41. H. Zhang, X. Liu, X. Shen, and J. Liu, Int. J. Bifurc. Chaos 23 (2013) 17. [ Links ]
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