Stability and bifurcation of a two-neuron network with distributed time delays

Nonlinear Analysis: Real World Applications

Volumn 11, Issue 3, 2010, Pages 1472-1490

  Hsu, Cheng Hsiung ^a   Yang, Suh Yuh ^a   Yang, Ting Hui ^b   Yang, Tzi Sheng ^c  

^a NATIONAL CENTRAL UNIVERSITY   (Taiwan)

^b TAMKANG UNIVERSITY   (Taiwan)

^c TUNGHAI UNIVERSITY   (Taiwan)

Author keywords

Characteristic equation; Distributed time delay; Hopf bifurcation; Neural network; Normal form; Pitchfork bifurcation

Indexed keywords

CHARACTERISTIC EQUATION; DISTRIBUTED TIME DELAY; DISTRIBUTED TIME DELAYS; NORMAL FORM; PITCHFORK BIFURCATIONS;

ASYMPTOTIC STABILITY; NEURAL NETWORKS; NEURONS; NONLINEAR FEEDBACK; TIME DELAY;

HOPF BIFURCATION;

EID: 77950917737     PISSN: 14681218     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.nonrwa.2009.03.004     Document Type: Article

References (21)

1. Gopalsamy K., and Leung I. Delay induced periodicity in a neural netlet of excitation and inhibition. Physica D 89 (1996) 395-426
2. Liu Z., Chen A., Cao J., and Huang L. Existence and global exponential stability of almost periodic solutions of BAM neural networks with continuously distributed delays. Phys. Lett. A 319 (2003) 305-316
3. Hale J.K., and Verduyn Lunel S.M. Introduction to Functional Differential Equations (1993), Springer-Verlag, New York
4. Hopfield J.J. Neurons with graded response have collective computational properties like two-state neurons. Proc. Natl. Acad. Sci. USA 81 (1984) 3088-3092
5. Baldi P., and Atiya A.F. How delays affect neural dynamics and learning. IEEE Trans. Neural Netw. 5 (1994) 612-621
6. Bélair J., Campbell S.A., and Van den Driessche P. Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math. 56 (1996) 245-255
7. Cao J., and Li Q. On the exponential stability and periodic solutions of delayed cellular neural networks. J. Math. Anal. Appl. 252 (2000) 50-64
8. Chen A., and Cao J. Existence and attractivity of almost periodic solutions for cellular neural networks with distributed delays and variable coefficients. Appl. Math. Comput. 134 (2003) 125-140
9. Faria T., and Magalhaes L.T. Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Differential Equations 122 (1995) 181-200
10. Faria T., and Magalhaes L.T. Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. J. Differential Equations 122 (1995) 201-224
11. Gopalsamy K., and He X.-Z. Stability in asymmetric Hopfield nets with transmission delays. Physica D 76 (1994) 344-358
12. Hsu C.-H., Yang S.-Y., Yang T.-H., and Yang T.-S. On periodic solutions of a two-neuron network system with sigmoidal activation functions. Internat. J. Bifur. Chaos 16 5 (2006) 1405-1417
13. Marcus C.M., and Westervelt R.M. Stability of analog neural networks with delay. Phys. Rev. A 39 (1989) 347-359
14. Marcus C.M., Waugh F.R., and Westervelt R.M. Nonlinear dynamics and stability of analog neural networks. Physica D 51 (1991) 234-247
15. Mastsuoka K. Stability conditions for nonlinear continuous neural networks with asymmetric connection weight. Neural Netw. 5 (1992) 495-500
16. Olien L., and Bélair J. Bifurcations, stability and monotonicity properties of a delayed neural network model. Physica D 102 (1997) 349-363
17. Shayer L.P., and Campbell S.A. Stability. bifurcation, and multistability in a system of two coupled neurons with multiple time delays. SIAM J. Appl. Math. 61 (2000) 673-700
18. Tank D.W., and Hopfield J.J. Neural computation by concentrating in information in time. Proc. Natl. Acad. Sci. USA 84 (1987) 1896-1991
19. Ye H., Michel A.N., and Wang K. Global stability and local stability of Hopfield neural networks with delays. Phys. Rev. E 50 (1994) 4206-4213
20. Ahlfors L.V. Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. 3rd ed. (1979), McGraw-Hill, New York
21. Kim A., Kwon W.H., and Volkanin L. Time-Delay System Toolbox, User's Guide (2001), Russian Academy of Sciences (Ural Branch), Russia, and Seoul National University, Korea