In this paper, we study generalized adaptive synchronization of Lorenz chaotic system with parabolic partial differential equation. Systems with three uncertain parameters and the non-linear adaptive feedback control technique are considered. Moreover, a systematic design process of parameters identification and Lag synchronization of chaotic system is considered. Finally, a sufficient condition is given for Lyapunov stability.
| Published in |
American Journal of Theoretical and Applied Statistics (Volume 6, Issue 5-1)
This article belongs to the Special Issue Statistical Distributions and Modeling in Applied Mathematics |
| DOI | 10.11648/j.ajtas.s.2017060501.12 |
| Page(s) | 8-12 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2017. Published by Science Publishing Group |
Lag Synchronization, Parabolic Partial Differential Equation, Uncertain Parameters, Adaptive Technique, Lorenz Chaotic System
| [1] | Chen, Y., Chen, X., & Chen, S. (2007). Lag Synchronization of strusturally nonequivalent chaotic systems with time delays. Nonlinear Analysis, 66, 1929_1937. |
| [2] | EL-Borai, M. M. (2004). The fundamental solutions for fractionl evolution equations of parabolic type. J. of Appl. Math. Stochastic Analysis(JAMSA), 199_211. |
| [3] | El-Borai, M. M., & F., A. (1997). On the asymptotic behavior some non-linear of parabolic systems. Memories of the faculty of science Kochi University Series A Math., 9_14. |
| [4] | EL-Borai, M. M. (1973). On stochastic differential equations in a Hilbert Space. Egyptian Statistical Journal (2). |
| [5] | EL-Borai, M. M. (1973). Partial diffferetial equations in an unbounded medium. Academia Nazionalle Dei Lincei Series VIII, 657_662. |
| [6] | El-Borai, M. M., & Assad, F. (1997). On the asymptotic behavior of some nonlinear parabolic systems. Memories of the faculty of science Kochi University Series A Math., 9_14. |
| [7] | EL-Borai, M. M., & EL-Nadi, K. (2015). Aparabolic transform and some stochastic ill-posed problems. British Journal of Mathematics and Computer Science, 9(5), 418_426. |
| [8] | EL-Borai, M. M., & EL-Nadi, K. E.-S. (2015). Fractional integrated semi groups and nonlocal fractional differential equations. Journal of Progressive Research in Mathematics(JPRM), 3(22), 553_561. |
| [9] | EL-Borai, M. M., EL-Akabawy, I. G., & EL-Nadi, K. E.-S. (2007). Approximate solutions for non-linear fractional heat equations. Int. J. Contemp. Math. Sciences, 2(27), 1307_1316. |
| [10] | EL-Borai, M. M., EL-Nadi, K. E.-S., & EL-Akabawy, I. G. (2008). On the fractional optimal control. Journal of applied Mathematics and Mechanics, 4(6), 13_18. |
| [11] | EL-Borai, M. M., EL-Nadi, K. E.-S., & Foad, H. A. (2010). On some fractional stochastic delay differential equations. Computers and Mathematics with Applications, 59, 1165_1170. |
| [12] | EL-Borai, M. M., EL-Nadi, K. E.-S., Labib, O., & M., H. (2004). Volterra equations with fractional stochastic integrals. Mathematical Problems in Engineering, 5, 453_468. |
| [13] | EL-Borai, M. M., EL-Nadi, K. E.-S., Labib, O., & M., H. (2005). Numerical method for some nonlinear stochastic differential equations. Applied Math. and Comp., 168, 65_75. |
| [14] | EL-Borai, M. M., EL-Nadin, K. E.-S., & EL-Akabawy, I. G. (2008). Fractional evolution equations with non-local conditions. Journal of Applied Mathematics and Mechanics, 4(6), 1_12. |
| [15] | EL-Boraia, M. M. (2002). Some probability densities and fundamental solution of fractional evolution equations. chaoc, Solition and Fractals(14), 433_440. |
| [16] | EL-Boraia, M. M., & Assad, F. (1997). On the cauchy problem for some partial differetial equations with time delays. Memoirs of the Faculty of Science Series A. Math.(18), 49_57. |
| [17] | EL-Boraia, M. M., & Hamza, A. (1997). An abstract dirchlet problem in the hilbert space. 4. |
| [18] | Ho, M. C., Hung, Y. C., & Chou, C. H. (2002). Phase and anti-phase synchronization of two chaotic systems by using active control. Physics Letters A, 296(1), 43_48. |
| [19] | Hu, M., & Xu, Z. (2007). Nonlinear feedback Mismatch synchronization of lorenz chaotic systems. Systems Engineering and Electronics, 29(8), 1346_1348. |
| [20] | Jia, Z. (2008). Linear generalized synchronization of chaotic systems with uncertain parameters. Journal of Systems Engineering ang Electronics, 19(4), 779_784. |
| [21] | Jia, Z., Lu, J. A., & Deng, G. M. (2007). Nonlinear state feedback and adaptive synchronization of hyperchaotic Lu systems. Systems Engineering and Electronics, 29(4), 598_600. |
| [22] | Lasall, J. P. (1960). the extent of asymptoic stability. Proceedings of the National Academy of Sciences of United States of America, 46(3), 363_365. |
| [23] | Li, C., & Liao, X. (2004). Lag synchronization of Rossler system and chua circuit via a scalar signal. Physics Letters A., 329(4-5), 301_308. |
| [24] | Li, C., Liao, X., & Wong, K. (2005). Lag synchronization of hyperchaos with Application to secure communications. Chaos, Solitons & Fractals, 23(1), 183_193. |
| [25] | Li., Z., & Xu., D. (2004). A Secure Communication Scheme using Projective Chaos Synchronization. Chaos, Solitons & Fractals, 477_481. |
| [26] | Ling, C., Wu, X., & Sun, S. (1999). A General Efficient Method for Chaotic Signal Estimation. IEEE Transactions on Signal Processing, 47, 1424_1428. |
| [27] | Lorenz, E. N. (1963). Deterministic nonperiodic Flow. Journal of the Atmospheric Sciences, 20(2), 130_141. |
| [28] | Lu, J., & Cao, J. (2005). Adaptive complete synchroization of two identical or different chaotic(hyperchaotic) systems with fully unknown parameters. Chaos, 15(4). |
| [29] | Luo, R. Z. (2008). Impulsive Control and Synchronization of a New Chaotic System. Phycics Letters A., 372, 648_653. |
| [30] | Pecora, L. M., & Carrol, T. L. (1990). Synchronization in chaotic systems. Physical Review Letters A., 64(8), 821_824. |
| [31] | Qais, H. A., & Aouda, A. A. (2011). Image Encryption Based on the General Approach for Multiple Chaotic Systems. Journal of Signal and Information Processing, 2, 238_244. |
| [32] | Wang, F., & Liu, C. (2006). A New Criterion for chaos and hyperchaoc synchronization using linear feedback control. Physics Letters A., 360(2), 274_278. |
| [33] | Yassen, M. T. (2007). Controlling, Synchronization and Tracking Chaotic Liu System using Active Backstepping Design. Physics Letters A., 372, 582_587. |
| [34] | Zhang, Q. j., Lu, j. A., & Jia, Z. (2009). Global Exponential Projective Synchronization and Lag Synchronization of Hyerchaotic Lu system. Communications in Theoretical physics, 51(4), 679_683. |
| [35] | W. G. El-Sayed, Solvability of a neutral differential equation with deviated argument, J. Math. Anal. Appl. 327 (2007), 342-350. |
| [36] | W. G. El-Sayed, On the Solvability of a Functional Integral Equation, East- West J. Math. Vol. 10, No. 2(2008) pp. 145-152. |
| [37] | W. G. El-Sayed, M. A. Seddeek and F. M. El-Saedy, Projection-Iteration Method for Solving Nonlinear Integral Equation of Mixed Type, Inter. J. of Research and Reviews in Applied Sciences on "Science and Mathematics with Applications" (IJRRAS), Vol. 6(3), No. 19, Feb. 2011, 376-381. |
| [38] | W. G. El-Sayed, S. A El-Shehawy and E. M. El-Abd, On the Solvability of a Volterra Integral Equation of Quadratic Form in the Class of Continuous Functions, Journal of Natural Sciences and Mathematics (JNM), Qassim university, Vol.4, No. 2, Dec. 2010, pp 149-156. |
| [39] | W. G. El-Sayed and E. M. El-Abd, Monotonic solutions for a nonlinear functional integral equations of convolution type, Journal of Fixed Point Theory and Applications (JP) Vol.7, No. 2, 2012, (101-111). |
APA Style
Mahmoud M. El-Borai, Wagdy G. El-sayed., Aafaf E. Abduelhafid. (2017). A Generalization of Some Lag Synchronization of System with Parabolic Partial Differential Equation. American Journal of Theoretical and Applied Statistics, 6(5-1), 8-12. https://doi.org/10.11648/j.ajtas.s.2017060501.12
ACS Style
Mahmoud M. El-Borai; Wagdy G. El-sayed.; Aafaf E. Abduelhafid. A Generalization of Some Lag Synchronization of System with Parabolic Partial Differential Equation. Am. J. Theor. Appl. Stat. 2017, 6(5-1), 8-12. doi: 10.11648/j.ajtas.s.2017060501.12
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Download@article{10.11648/j.ajtas.s.2017060501.12,
author = {Mahmoud M. El-Borai and Wagdy G. El-sayed. and Aafaf E. Abduelhafid},
title = {A Generalization of Some Lag Synchronization of System with Parabolic Partial Differential Equation},
journal = {American Journal of Theoretical and Applied Statistics},
volume = {6},
number = {5-1},
pages = {8-12},
doi = {10.11648/j.ajtas.s.2017060501.12},
url = {https://doi.org/10.11648/j.ajtas.s.2017060501.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.s.2017060501.12},
abstract = {In this paper, we study generalized adaptive synchronization of Lorenz chaotic system with parabolic partial differential equation. Systems with three uncertain parameters and the non-linear adaptive feedback control technique are considered. Moreover, a systematic design process of parameters identification and Lag synchronization of chaotic system is considered. Finally, a sufficient condition is given for Lyapunov stability.},
year = {2017}
}
TY - JOUR T1 - A Generalization of Some Lag Synchronization of System with Parabolic Partial Differential Equation AU - Mahmoud M. El-Borai AU - Wagdy G. El-sayed. AU - Aafaf E. Abduelhafid Y1 - 2017/02/18 PY - 2017 N1 - https://doi.org/10.11648/j.ajtas.s.2017060501.12 DO - 10.11648/j.ajtas.s.2017060501.12 T2 - American Journal of Theoretical and Applied Statistics JF - American Journal of Theoretical and Applied Statistics JO - American Journal of Theoretical and Applied Statistics SP - 8 EP - 12 PB - Science Publishing Group SN - 2326-9006 UR - https://doi.org/10.11648/j.ajtas.s.2017060501.12 AB - In this paper, we study generalized adaptive synchronization of Lorenz chaotic system with parabolic partial differential equation. Systems with three uncertain parameters and the non-linear adaptive feedback control technique are considered. Moreover, a systematic design process of parameters identification and Lag synchronization of chaotic system is considered. Finally, a sufficient condition is given for Lyapunov stability. VL - 6 IS - 5-1 ER -
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