- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Advances in Mathematical Physics
Volume 2013 (2013), Article ID 657245, 9 pages
Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay
1Shandong Key Laboratory of Robotics and Intelligent Technology, College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao 266590, China
2College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China
Received 4 August 2013; Accepted 30 September 2013
Academic Editor: Changpin Li
Copyright © 2013 Xia Huang 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.
- I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
- P. L. Butzer and U. Westphal, An Introduction to Fractional Calculus, World Scientific, Singapore, 2000.
- R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Hackensack, NJ, USA, 2001.
- M. Nakagava and K. Sorimachi, “Basic characteristic of a fractance device,” IEICE Transactions on Fundamentals of Electronics, vol. 75, no. 12, pp. 1814–1818, 1992.
- A. Kiani-B, K. Fallahi, N. Pariz, and H. Leung, “A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 863–879, 2009.
- T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua's system,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485–490, 1995.
- Y. Yu, H.-X. Li, S. Wang, and J. Yu, “Dynamic analysis of a fractional-order Lorenz chaotic system,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1181–1189, 2009.
- C. Li and G. Peng, “Chaos in Chen's system with a fractional order,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 443–450, 2004.
- C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A, vol. 341, no. 1–4, pp. 55–61, 2004.
- C. P. Li and Y. T. Ma, “Fractional dynamical system and its linearization theorem,” Nonlinear Dynamics, vol. 71, no. 4, pp. 621–633, 2013.
- C. Li, Z. Gong, D. Qian, and Y. Chen, “On the bound of the Lyapunov exponents for the fractional differential systems,” Chaos, vol. 20, no. 1, Article ID 013127, 2010.
- V. Daftardar-Gejji and S. Bhalekar, “Chaos in fractional ordered Liu system,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1117–1127, 2010.
- J. G. Lu, “Chaotic dynamics and synchronization of fractional-order Arneodo's systems,” Chaos, Solitons and Fractals, vol. 26, no. 4, pp. 1125–1133, 2005.
- W.-C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1305–1314, 2008.
- M.-F. Danca, “Chaotic behavior of a class of discontinuous dynamical systems of fractional-order,” Nonlinear Dynamics, vol. 60, no. 4, pp. 525–534, 2010.
- J. H. Park, “On global stability criterion for neural networks with discrete and distributed delays,” Chaos, Solitons and Fractals, vol. 30, no. 4, pp. 897–902, 2006.
- S. Xu, J. Lam, and D. W. C. Ho, “A new LMI condition for delay-dependent asymptotic stability of delayed Hopfield neural networks,” IEEE Transactions on Circuits and Systems II, vol. 53, no. 3, pp. 230–234, 2006.
- X. Liao, K.-W. Wong, Z. Wu, and G. Chen, “Novel robust stability criteria for interval-delayed Hopfield neural networks,” IEEE Transactions on Circuits and Systems I, vol. 48, no. 11, pp. 1355–1359, 2001.
- S. Mou, H. Gao, J. Lam, and W. Qiang, “A new criterion of delay-dependent asymptotic stability for Hopfield neural networks with time delay,” IEEE Transactions on Neural Networks, vol. 19, no. 3, pp. 532–535, 2008.
- H. Hu, H. Jiang, and Z. Teng, “The boundedness of high-order Hopfield neural networks with variable delays,” Neurocomputing, vol. 73, no. 13–15, pp. 2589–2596, 2010.
- X. X. Liao, J. Wang, and Z. Zeng, “Global asymptotic stability and global exponential stability of delayed cellular neural networks,” IEEE Transactions on Circuits and Systems II, vol. 52, no. 7, pp. 403–409, 2005.
- J. L. Qiu, “Dynamics of high-order Hopfield neural networks with time delays,” Neurocomputing, vol. 73, no. 4–6, pp. 820–826, 2010.
- H. T. Lu, “Chaotic attractors in delayed neural networks,” Physics Letters A, vol. 298, no. 2-3, pp. 109–116, 2002.
- Q. Li, X.-S. Yang, and F. Yang, “Hyperchaos in Hopfield-type neural networks,” Neurocomputing, vol. 67, no. 1–4, pp. 275–280, 2005.
- M. Gilli, “Strange attractors in delayed cellular neural networks,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 11, pp. 849–853, 1993.
- B. N. Lundstrom, M. H. Higgs, W. J. Spain, and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335–1342, 2008.
- P. Arena, R. Caponetto, L. Fortuna, and D. Porto, “Bifurcation and chaos in noninteger order cellular neural networks,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 8, no. 7, pp. 1527–1539, 1998.
- I. Petráš, “A note on the fractional-order cellular neural networks,” in Proceedings of the International Joint Conference on Neural Networks, pp. 1021–1024, Sheraton Vancouver Wall Centre Hotel, Vancouver, Canada, July 2006.
- P. Arena, L. Fortuna, and D. Porto, “Chaotic behavior in noninteger-order cellular neural networks,” Physical Review E, vol. 61, no. 1, pp. 776–781, 2000.
- X. Huang, Z. Zhao, Z. Wang, and Y. Li, “Chaos and hyperchaos in fractional-order cellular neural networks,” Neurocomputing, vol. 94, pp. 13–21, 2012.
- E. Kaslik and S. Sivasundaram, “Nonlinear dynamics and chaos in fractional-order neural networks,” Neural Networks, vol. 32, pp. 245–256, 2012.
- V. Çelik and Y. Demir, “Chaotic fractional order delayed cellular neural network,” New Trends in Nanotechnology and Fractional Calculus Applications, vol. 4, pp. 313–320, 2010.
- I. Tejado, S. H. Hosseinnia, B. M. Vinagre, X. Song, and Y. Chen, “Dealing with fractional dynamics of IP network delays,” International Journal of Bifurcation and Chaos, vol. 22, no. 4, Article ID 1250089, 2012.
- K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002.
- K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms, vol. 36, no. 1, pp. 31–52, 2004.
- S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,” Computers and Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011.
- S. Abbas, “Existence of solutions to fractional order ordinary and delay differential equations and applications,” Electronic Journal of Differential Equations, vol. 2011, no. 9, pp. 1–11, 2011.