Abstract
Artificial neural networks arise from the research of the configuration and function of the brain. As pointed out in [79], the brain can be regarded as a complex nonlinear parallel information processing system with a concept of neuron as a basic functional unit.
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Keywords
- Neural Network
- Periodic Solution
- Linear Matrix Inequality
- Differential Inclusion
- Cellular Neural Network
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References
D. H. Ackley, G. E. Hinton, and T. J. Sejnowski. A learning algorithm for Boltzmann machines. Cognit. Sci., 9: 147–169 1985.
I. Aleksander and H. Morton An Introduction to Neural Computing. Lodon, Chapman & Hall 1990.
J. Andres. Almost-periodic and bounded solutions of Carathèodory differential inclusions. Differential Integral Equations, 12(6): 887–912, 1999.
J. P. Aubin and A. Cellina. Differential Inclusions. Berlin, Springer-Verlag, 1984.
J. P. Aubin and H. Frankowska. Set-valued Analysis. Boston, Birhauser, 1990.
J. P. Aubin. Viability Theory. Boston, Birhauser, 1991.
A. Bacciotti, et.al. Discontinuous ordinary differential equations and stabilization. Tesi di Dottorato di Ricerca in Mathematica, Universita degli STUDI di Firenze, 2000.
A. Bacciotti. Generalized solutions of differential inclusions and stability. Ital. J. Pure Appl. Math., 17: 183–192, 2005.
J. Belair. Stability in A Model of Delayed Neural Networks. J. Dynam. Differential Equations, 5: 607–623, 1993.
J. Belair, S. A. Campbell, and P. Driessche, Van Den. Stability and delay-induced oscillations in a neural network model. SIAM J. Appl. Math., 56: 245–255, 1996.
A. Berman. Completely positive matrices. New Jersey, World Scientific Publishing, 2003.
R. Bader and W. Kryszewski. On the solution sets of differential inclusions and the periodic problem in Banach space. Nonlinear Analysis, 54: 707–754, 2003.
S. Boyd et al. Linear Matrix Inequalities in System and Control Theory. Philadelphia, SIAM, 1994.
J. Cao. On exponential stability and periodic solution of CNNs with delay. Phys. Lett. A, 267(5–6): 312–318, 2000.
J. Cao. New results concerning exponential stability and periodic solutions of delayed cellular neural networks. Phys. Lett. A, 307: 136–147, 2003.
J. Cao and J. Liang. Boundedness and stability for Cohen-Grossberg neural network with time-varying delays. J. Math. Anal. Appl., 296: 665–685, 2004.
Y. J. Cao and Q. H. Wu. A note on stability of analog neural networks with time delays. IEEE Trans. Neural Networks, 7: 1533–1535, 1996.
T. Chen. Convergence of delayed dynamical systems. Neural Proc. Lett., 10(3): 267–271, 1999.
T. Chen. Global exponential stability of delayed hopfield neural networks. Neural Networks, 14(8): 977–980, 2001.
T. Chen and S. Amari. Stability of asymmetric hopfield neural networks. IEEE T. Neural Networks, 12(1): 159–163, 2001.
T. Chen and Y. Bai. Stability of Cohen-Grossberg neural networks with nonnegative periodic solutions. Proceedings of International Joint Conference Neural Networks (IJCNN 2007), 242–247, 2007.
T. Chen and H. Chen. Universal approximation to nonlinear operators by neural networks and its applications to dynamical systems. IEEE Trans. Neural Networks, 6(4): 911–917, 1995.
T. Chen and H. Chen. Approxiamtion capability functions of several variables, nonlinear functionals and operators with radial basis function neural networks. IEEE Trans. Neural Networks, 6(4): 904–910, 1995.
A. Chen and J. Cao. Existence and attractivity of almost periodic solution for cellular neural networks with distributed delays and variable coefficients. Appl. Math. Comp., 134(1): 125–140, 2003.
P. P. Civalleri, L. M. Gilli, and L. Pabdolfi. On stability of cellular neural networks with delay. IEEE Trans.Circuits Syst., 40: 157–164, 1993.
T. Chen and W. Lu. Stability analysis of dynamical neural networks. Proceedings of International Conference Neural Networks and Signal Processing, 14–17, 2003.
T. Chen, W. Lu, and G. R. Chen. Dynamical behaviors of a large class of general delayed neural networks. Neural Comput., 17: 949–968, 2005.
T. Chen and L. Rong. Delay-independent stability analysis of Cohen-Grossberg neural networks. Phys. Lett. A, 317: 436–449, 2003.
T. Chen and L. Rong. Robust global exponential stability of Cohen-Grossberg neural networks with time delays. IEEE Trans. Neural Networks, 15(1): 203–206, 2004.
T. Chen and L. Wang. Power-rate stability of dynamical systems with unbounded time-varying delays. IEEE T. CAS-II: Express Briefs, 54(8): 705–709, (2007).
L. O. Chua, C. A. Desoer, and E. S. Kuh. Linear and Nonlinear Circuits. New York, Macgraw-Hill, 1987.
L. O. Chua and L. Yang. Cellular neural networks: Theory. IEEE Trans.Circuits Syst., 35: 1257–1272, 1988.
L. O. Chua and L. Yang. Cellular neural networks: Application. IEEE Trans.Circuits Syst., 35: 1273–1290, 1988.
F. H. Clarke. Optimization and Nonsmooth Analysis. Pliladelphia, SIAM 1983.
M. A. Cohen and S. Grossberg. Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans. on Man, Syst. Cybern., 13: 815–826, 1983.
G. Cybenko. Approximation by Superpositions of a Sigmoidal Function. Urbana, University of ILLinois, 1988.
G. Cybenko. Approximation by superpositions of a sigmoidal function. Math. Cont., Sig., Sys., 2: 303–314, 1989.
B. C. Dhage. Existence of extremal solutions for discontinuous functional integral equations. Appl. Math. Lett., 19: 881–886, 2006.
A. F. Filippov. Classical solution of differential equations with multivalued right-hand side. SIAM J. Control, 5(4): 609–621, 1967.
M. Forti and P. Nistri. Global convergence of neural networks with discontinuous neuron activations. IEEE Trans. Circuits Syst.I, 50(11): 1421–1435, 2003.
M. Forti and A. Tesi. New condition for global stability of neural networks with application to linear and quadratic programming problems. IEEE Trans. Circiuts Syst.-1, 42: 354–366, 1995.
M. Forti, P. Nistri, and D. Papini. Global exponential stability and global convergence of delayed neural networks with infinite gain. IEEE Trans. Neural Networks, 16(6): 1449–1463, 2005.
M. E. Filippakis and N. S. Papageorgiou. Periodic solutions for differential inclusions in RN. Arch. Math. (Brno), 42: 115–123, 2006.
R. E. Gaines and J. L. Mawhin. Coincidence Degree and Nonlinear Differential Equations. Lecture Notes on Mathematics, Berlin, Springer, 568: 10–35, (1977).
S. Grossberg. Biological competition: Decision rules, pattern formation, and oscillations. Proc. Natl. Acd. Sci. USA, 77(4): 2338–2342, 1980.
S. Grossberg. Nonlinear neural networks: principles, Mechanisms, and architectures. Neural Networks, 1: 17–61, 1988.
G. Haddad. Monotone viable trajectories for functional differential inclusions. J. Diff. Equ., 42: 1–24, 1981.
J. K. Hale. Theory of Functional Differential Equations. New York, Springer-Verlag 1977.
H. Harrer, J. A. Nossek, and R. Stelzl. An analog implementation of discrete-time neural networks. IEEE Trans. Neural Networks, 3: 466–476, 1992.
S. Haykin. Neural Networks: A Comprehensive Foundation. New York, Macmillan Publishing Company, 1994.
D. O. Hebb. The Orgnization of Behavior: A Neuropsuchological Theory. New York, Wiley, 1949.
R. Hecht-Nielsen. Neurocomputing, Reading. MA, Addison-Wesley, 1990.
J. J. Hopfield. Neurons with graded response have collective computational properties like thoseof of two-stage neurons. Proc. Nat. Acad. Sci -Biol., 81: 3088–3092, 1984.
X. Huang and J. Cao. Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delays. Phys. Lett. A, 314(3): 222–231, 2003.
S. Hu and N. S. Papageorgiou. On the existence of perioidic solutions for nonconvex-valued differential inclusions in R N. Proc. Amer. Math. Soc., 123(10): 3043–3050 1995.
J. J. Hopfield, and D. W. Tank. Computing neural circuits: a model. Science, 233: 625–633 1986.
A. G. Ivanov. On the equivalence of differential inclusions and controlled almost periodic systems. (Russian) Differ. Uravn. 33(7): 876–884, 1997; translation in Differ. Equ., 33(7): 879–887,1997.
M. P. Kennedy and L. O. Chua. Neural networks for nonlinear programming. IEEE Trans. Circuits Syst.-1, 35: 554–562, 1988.
W. A. Kirk and B. Sims. Handbook of Metric Fixed Point Theory. Berlin, NY, Springer-Verlag, 2001.
B. M. Levitan and V. V. Zhikov. Almost Periodic Functions and Differential Equations. New York, Cambriadge University Press, 1982.
H. Lu. On stability of nonlinear continuous-time neural networks with delays. Neural Networks, 13(10): 1135–1144, 2000.
W. Lu and T. Chen. On periodic dynamical systems. Chin. Ann. Math., 25B(4): 455–462, 2004.
W. Lu and T. P. Chen. Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions. Neural Networks, 18: 231–242, 2005.
W. Lu and T. Chen. Global exponential stability of almost periodic trajectory of a large class of delayed dynamical systems. Sci. China A: Math., 48(8): 1015–1026, 2005.
W. Lu and T. P. Chen. Dynamical Behaviors of delayed neural network systems with discontinuous activation functions. Neural Comput., 18(3): 683–708, 2006.
W. Lu and T. Chen. R n+ global stability of Cohen-Grossberg neural network system with nonnegative equilibrium. Neural Networks, 20: 714–722, 2007.
W. Lu and T. Chen. Almost periodic dynamics of a class of delayed neural networks with discontinuous activations. Neural Comput., 20: 1065–1090, 2008.
W. Lu, L. Rong, and T. Chen. Global convergence of delayed neural networks systems. Int. J. Neural Sys., 13(3): 193–204, 2003.
D. Li and P. E. Kloeden. On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions. J. Diff. Equ., 224: 1–38, 2006.
X. Liao, C. Li, and K. Wong. Criteria for exponential stability of Cohen-Grossberg neural networks. Neural Networks, 17: 1401–1414, 2004.
G Li and X. Xue. On the existence of periodic solutions for differential inclusions. J. Math. Anal. Appl., 276: 168–183, 2002.
R. K. Miller. Asymptotic behavior of nonlinear delayed-differential equations. J. Differ. Equ., 1: 293–305, 1995.
C. M. Marcus and R. M. Westervelt. Stability of analog neural networks with delay. Phys. Rev. A., 39: 347–359, 1989.
J. M. Mendel and K. S. Fu. Adaptive, Learning, and Pattern Recognition Systems: Theory and Applications. New York, Academic Press, 1970.
N. Megiddo and M. Kojima. On the existence and uniqueness of solutions in nonlinear complementarity theory. Math. Prog., 12: 110–130, 1977.
B. E. Paden and S. S. Sastry. Calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulator. IEEE Trans. Circuits Syst., 34: 73–82, 1987.
D. Papini and V. Taddei. Global exponential stability of the periodic solution of the delayed neural networks with discontinuous activations. Phys. Lett. A., 343: 117–128, 2005.
C. S. Ramòny. Histologie du système nerveux de l’homme et des vertèbrès. Paris: Maloine; Edition Francaise Revue: Tome I (1952); Tome II (1955); Madrid: Consejo Superior de Inverstigaciones Cientificas 2001.
T. Rockafellar and R.J.-B. Wets. Variational Analysis. Berlin Heidelberg, Springer-Verlag, 1998.
G. V. Smirnov. Weak asymptotic stability at first approximation for periodic differential inclusions. NoDEA, 2: 445–461, 1995.
L. Wang and X. Zou. Exponential stability of Cohen-Grossberg neural networks. Neural Networks, 15: 415–422, 2002.
K. Yoshida. Functional Analysis. Grundlehren der Mathematicchen Wissenschaften. New York: Springer-Verlag, 1978.
T. Yosizawa. Stability Theory and The Existence of Periodic Solutions and Almost Periodic Solutions. New York, Springer-Verlag, 1975.
V. I. Utkin. Sliding Modes and Their Applications in Variable Structure Systems. Moskow, MIR Publishers, 1978.
J. Zhang and X. Jin. Global stability analysis in delayed hopfield neural models. Neural Networks, 13(7): 745–753, 2000.
Y. Zheng and T. Chen. Global exponential stability of delayed periodic dynamical systems. Phys. Lett. A, 322(5–6): 344–355, 2004.
J. Zhou, Z. Liu, and G. Chen. Dynamics of delayed periodic neural networks. Neural Networks, 17(1): 87–101, 2004.
A. V. Zuev. On periodic solutions of ordinary differential equations with discontinuous right-hand side. Math. Notes, 79(4): 518–527, 2006.
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Lu, W., Chen, T. (2009). Global Convergent Dynamics of Delayed Neural Networks. In: Atay, F. (eds) Complex Time-Delay Systems. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02329-3_7
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