Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 260150, 8 pages
http://dx.doi.org/10.1155/2014/260150
Research Article

Anticontrol of Chaos for a Class of Delay Difference Equations Based on Heteroclinic Cycles Connecting Repellers

School of Science, Shandong Jianzhu University, Jinan, Shandong 250101, China

Received 1 January 2014; Revised 5 March 2014; Accepted 6 March 2014; Published 13 April 2014

Academic Editor: Ryan Loxton

Copyright © 2014 Zongcheng Li. 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. W. J. Freeman, “Chaos in the brain: possible roles in biological intelligence,” International Journal of Intelligent Systems, vol. 10, pp. 71–88, 1995. View at Google Scholar
  2. S. J. Schiff, K. Jerger, D. H. Duong, T. Chang, M. L. Spano, and W. L. Ditto, “Controlling chaos in the brain,” Nature, vol. 370, pp. 615–620, 1994. View at Google Scholar
  3. M. E. Brandt and G. R. Chen, “Bifurcation control of two nonlinear of models of cardiac activity,” IEEE Transactions on Circuits and Systems I, vol. 44, pp. 1031–1034, 1997. View at Google Scholar
  4. W. L. Ditto, M. L. Spano, J. Nelf et al., “Control of human atrial fibrillation,” International Journal of Bifurcation and Chaos, vol. 10, pp. 593–602, 2000. View at Google Scholar
  5. G. Jakimoski and L. G. Kocarev, “Chaos and cryptography: block encryption ciphers based on chaotic maps,” IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications, vol. 48, no. 2, pp. 163–169, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. L. G. Kocarev, M. Maggio, M. Ogorzalek, L. Pecora, and K. Yao, “Special issue on applications of chaos in modern communication systems,” IEEE Transactions on Circuits and Systems I, vol. 48, pp. 1385–1527, 2001. View at Google Scholar
  7. G. R. Chen and D. J. Lai, “Feedback control of Lyapunov exponents for discrete-time dynamical systems,” International Journal of Intelligent Systems, vol. 6, pp. 1341–1349, 1996. View at Google Scholar
  8. G. R. Chen and D. J. Lai, “Anticontrol of chaos via feedback,” in Proceedings of the IEEE Conference on Decision and Control, pp. 367–372, San Diego, Calif, USA, 1997.
  9. G. R. Chen and D. J. Lai, “Feedback anticontrol of discrete chaos,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 8, no. 7, pp. 1585–1590, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. G. R. Chen and Y. M. Shi, “Introduction to anti-control of discrete chaos: theory and applications,” Philosophical Transactions of the Royal Society of London A. Mathematical, Physical and Engineering Sciences, vol. 364, no. 1846, pp. 2433–2447, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. X. F. Wang and G. R. Chen, “Chaotification via arbitrarily small feedback controls: theory, method, and applications,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 10, no. 3, pp. 549–570, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Q. Chai, R. Loxton, K. L. Teo, and C. Yang, “A class of optimal state-delay control problems,” Nonlinear Analysis. Real World Applications, vol. 14, no. 3, pp. 1536–1550, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. D. Zhu and Y. P. Tian, “Necessary and sufficient conditions for stabilizability of discretetime systems via delayed feedback control,” Physics Letters A, vol. 343, pp. 95–107, 2005. View at Google Scholar
  14. X. F. Wang and G. R. Chen, “Chaotifying a stable map via smooth-amplitude high-frequency feedback control,” International Journal of Circuit Theory and Applications, vol. 28, pp. 305–312, 2000. View at Google Scholar
  15. Y. M. Shi, P. Yu, and G. R. Chen, “Chaotification of discrete dynamical systems in Banach spaces,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 9, pp. 2615–2636, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Z. C. Li, “Chaotification for linear delay difference equations,” Advances in Difference Equations, vol. 2013, article 59, 2013. View at Google Scholar
  17. Z. C. Li, Y. M. Shi, and W. Liang, “Discrete chaos induced by heteroclinic cycles connecting repellers in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 2, pp. 757–770, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. F. Giannakopoulos and A. Zapp, “Local and global Hopf bifurcation in a scalar delay differential equation,” Journal of Mathematical Analysis and Applications, vol. 237, no. 2, pp. 425–450, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. H. Jiang, Y. Shen, J. G. Jian, and X. X. Liao, “Stability, bifurcation and a new chaos in the logistic differential equation with delay,” Physics Letters A, vol. 350, no. 3-4, pp. 221–227, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Z. M. He, X. Lai, and A. Y. Hou, “Stability and Neimark-Sacker bifurcation of numerical discretization of delay differential equations,” Chaos, Solitons and Fractals, vol. 41, no. 4, pp. 2010–2017, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Y. Huang and X. Zou, “Co-existence of chaos and stable periodic orbits in a simple discrete neural network,” Journal of Nonlinear Science, vol. 15, no. 5, pp. 291–303, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. Peng, J. C. Yu, and X. J. Wang, “Complex dynamics in simple delayed two-parameterized models,” Nonlinear Analysis. Real World Applications, vol. 13, no. 6, pp. 2530–2539, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. T. Y. Li and J. A. Yorke, “Period three implies chaos,” The American Mathematical Monthly, vol. 82, no. 10, pp. 985–992, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, New York, NY, USA, 1987. View at MathSciNet
  25. M. Martelli, M. Dang, and T. Seph, “Defining chaos,” Mathematics Magazine, vol. 71, no. 2, pp. 112–122, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, CRC Press, Boca Raton, Fla, USA, 1995. View at MathSciNet
  27. S. Wiggins, Global Bifurcations and Chaos, Springer, New York, NY, USA, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  28. J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, “On Devaney's definition of chaos,” The American Mathematical Monthly, vol. 99, no. 4, pp. 332–334, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. W. Huang and X. D. Ye, “Devaney's chaos or 2-scattering implies Li-Yorke's chaos,” Topology and its Applications, vol. 117, no. 3, pp. 259–272, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. Y. M. Shi and G. R. Chen, “Chaos of discrete dynamical systems in complete metric spaces,” Chaos, Solitons and Fractals, vol. 22, no. 3, pp. 555–571, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. Y. M. Shi and G. R. Chen, “Discrete chaos in Banach spaces,” Science in China A. Mathematics, vol. 48, no. 2, pp. 222–238, 2005, Chinese version: vol. 34, pp. 595–609, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet