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Abstract and Applied Analysis
Volume 2014, Article ID 316368, 11 pages
http://dx.doi.org/10.1155/2014/316368
Research Article

Hybrid Stability Checking Method for Synchronization of Chaotic Fractional-Order Systems

1Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Avenue Padre Tomás Pereira, Taipa 999078, Macau
2Institute for the Development and Quality, Avenue Padre Tomás Pereira, Taipa 999078, Macau
3Department of Mechanical Engineering, Hsiuping University of Science and Technology, 11 Gongye Road, Dali District, Taichung 412-80, Taiwan
4Department of Mechanical Engineering, Chung Hua University, Section 2, 707 WuFu Road, Hsinchu 30012, Taiwan

Received 24 October 2013; Accepted 28 February 2014; Published 3 April 2014

Academic Editor: Dumitru Baleanu

Copyright © 2014 Seng-Kin Lao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,” Progress of Theoretical Physics, vol. 69, no. 1, pp. 32–47, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. G. Butkovskii, S. S. Postnov, and E. A. Postnova, “Fractional integro-differential calculus and its control-theoretical applications. II. Fractional dynamic systems: modeling and hardware implementation,” Automation and Remote Control, vol. 74, no. 5, pp. 725–749, 2013. View at Google Scholar
  4. M. Berli, “Challenges in the application of fractional derivative models in capturing solute transport in porous media: darcy-scale fractional dispersion and the influence of medium properties,” Mathematical Problems in Engineering, vol. 2013, Article ID 878097, 10 pages, 2013. View at Publisher · View at Google Scholar
  5. X. J. Yang and D. Baleanu, “Fractional heat conduction problem solved by local fractional variation iteration method,” Thermal Science, vol. 17, no. 5, pp. 625–628, 2013. View at Publisher · View at Google Scholar
  6. R. L. Magin, “Fractional calculus models of complex dynamics in biological tissues,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1586–1593, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Charef, H. H. Sun, Y.-Y. Tsao, and B. Onaral, “Fractal system as represented by singularity function,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 37, no. 9, pp. 1465–1470, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. S. Tavazoei and M. Haeri, “Limitations of frequency domain approximation for detecting chaos in fractional order systems,” Nonlinear Analysis. Theory, Methods & Applications Series A: Theory and Methods, vol. 69, no. 4, pp. 1299–1320, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. K. Matsuda and H. Fujii, “H optimized wave-absorbing control: analytical and experimental results,” Journal of Guidance, Control, and Dynamics, vol. 16, no. 6, pp. 1146–1153, 1993. View at Google Scholar · View at Scopus
  10. I. Petráš, “A note on the fractional-order Chua's system,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 140–147, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. K. Diethelm, “Efficient solution of multi-term fractional differential equations using P(EC)mE methods,” Computing. Archives for Scientific Computing, vol. 71, no. 4, pp. 305–319, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. K. S. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, pp. 139–152, Nihon University, Koriyama, Japan, 1989. View at Zentralblatt MATH
  13. G.-C. Wu and D. Baleanu, “Discrete fractional logistic map and its chaos,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 75, no. 1-2, pp. 283–287, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. A. M. A. El-Sayed and M. E. Nasr, “Discontinuous dynamical systems and fractional-orders difference equations,” Journal of Fractional Calculus and Applications, vol. 4, no. 1, pp. 130–138, 2013. View at Google Scholar
  15. H. Zhao, H. K. Kwan, and J. Yu, “Fractional discrete-time chaotic map,” in Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS '06), pp. 21–24, May 2006. View at Scopus
  16. G. C. Wu, D. Baleanu, and S.-D. Zeng, “Discrete chaos in fractional sine and standard maps,” Physics Letters A, vol. 378, no. 5-6, pp. 484–487, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  17. F. Jarad, T. Abdeljawad, D. Baleanu, and K. Biçen, “On the stability of some discrete fractional nonautonomous systems,” Abstract and Applied Analysis, vol. 2012, Article ID 476581, 9 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. H. K. Chen and C. I. Lee, “Anti-control of chaos in rigid body motion,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 957–965, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. L. M. Tam and W. M. Si Tou, “Parametric study of the fractional-order Chen-Lee system,” Chaos, Solitons and Fractals, vol. 37, no. 3, pp. 817–826, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. L. J. Sheu, H. K. Chen, J. H. Chen et al., “Complete synchronization of two Chen-Lee systems,” Journal of Physics: Conference Series, vol. 96, no. 1, Article ID 012138, 2008. View at Publisher · View at Google Scholar · View at Scopus
  21. J. H. Chen, H. K. Chen, and Y. K. Lin, “Synchronization and anti-synchronization coexist in Chen-Lee chaotic systems,” Chaos, Solitons and Fractals, vol. 39, no. 2, pp. 707–716, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. J. H. Chen, “Controlling chaos and chaotification in the Chen-Lee system by multiple time delays,” Chaos, Solitons and Fractals, vol. 36, no. 4, pp. 843–852, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. L. J. Sheu, L. M. Tam, H. K. Chen, and S. K. Lao, “Alternative implementation of the chaotic Chen-Lee system,” Chaos, Solitons and Fractals, vol. 41, no. 4, pp. 1923–1929, 2009. View at Publisher · View at Google Scholar · View at Scopus
  24. L. M. Tam, S. K. Lao, H. K. Chen, and L. J. Sheu, “Hybrid projective synchronization for the fractional-order Chen-Lee system and its circuit realization,” Applied Mechanics and Materials, vol. 300, pp. 1573–1578, 2013. View at Google Scholar
  25. H.-K. Chen, L.-J. Sheu, L.-M. Tam, and S.-K. Lao, “A new finding of the existence of Feigenbaum's constants in the fractional-order Chen-Lee system,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 68, no. 4, pp. 589–599, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  26. C.-H. Chen, L.-J. Sheu, H.-K. Chen et al., “A new hyper-chaotic system and its synchronization,” Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, vol. 10, no. 4, pp. 2088–2096, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. C. I. Byrnes, A. Isidori, and J. C. Willems, “Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 36, no. 11, pp. 1228–1240, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. W. Yu, “Passive equivalence of chaos in Lorenz system,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 46, no. 7, pp. 876–878, 1999. View at Publisher · View at Google Scholar · View at Scopus
  29. S. Emiroçlu and Y. Uyaroǧlu, “Control of Rabinovich chaotic system based on passive control,” Scientific Research and Essays, vol. 5, no. 21, pp. 3298–3305, 2010. View at Google Scholar · View at Scopus
  30. F. Wang and C. Liu, “Synchronization of unified chaotic system based on passive control,” Physica D. Nonlinear Phenomena, vol. 225, no. 1, pp. 55–60, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. W. Xiang-Jun, L. Jing-Sen, and C. Guan-Rong, “Chaos synchronization of Rikitake chaotic attractor using the passive control technique,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 53, no. 1-2, pp. 45–53, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  32. X. Zhou, L. Xiong, W. Cai, and X. Cai, “Adaptive synchronization and antisynchronization of a hyperchaotic complex Chen system with unknown parameters based on passive control,” Journal of Applied Mathematics, vol. 2013, Article ID 845253, 8 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. C. J. Wu, Y. B. Zhang, and N. N. Yang, “The synchronization of a fractional order hyperchaotic system based on passive control,” Chinese Physics B, vol. 20, no. 6, Article ID 060505, 2011. View at Publisher · View at Google Scholar · View at Scopus
  34. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proceedings of the Computational Engineering in Systems and Application Multiconference, pp. 963–968, 1996.
  35. F. Q. Wang and C. X. Liu, “Study on the critical chaotic system with fractional order and circuit experiment,” Acta Physica Sinica, vol. 55, no. 8, pp. 3922–3927, 2006. View at Google Scholar