Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Symmetric functional differential equations
and neural networks with memory

Author: Jianhong Wu
Journal: Trans. Amer. Math. Soc. 350 (1998), 4799-4838
MSC (1991): Primary 34K15, 34K20, 34C25
MathSciNet review: 1451617
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry. We apply these general results to obtain the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix.

References [Enhancements On Off] (What's this?)

  • 1. J. C. Alexander, Bifurcation of zeros of parametrized functions, J. Funct. Anal. 29 (1978), 37-53. MR 80j:58023
  • 2. J. C. Alexander and G. Auchmuty, Global branching of waves, Manuscripta Math. 27 (1979), 159-166. MR 80g:58021
  • 3. -, Global bifurcations of phase-locked oscillators, Arch. Rational Mech. Anal. 93 (1986), 253-270. MR 87j:34090
  • 4. J. C. Alexander and P. M. Fitzpatrick, The homotopy of certain spaces of nonlinear operators, and its relation to global bifurcation of the fixed points of parametrized condensing operators, J. Funct. Anal. 34 (1979), 87-106. MR 81k:58023
  • 5. J. C. Alexander and J. Yorke, Global bifurcation of periodic orbits, Amer. J. Math. 100 (1978), 263-292. MR 57:14046
  • 6. J. Bélair, Stability in a model of a delayed neural network, J. Dynamics and Differential Equations 5 (1993), 607-623. MR 94i:34150
  • 7. R. D. Braddock and P. van den Driessche, On the stability of differential-difference equations, J. Austral. Math. Soc. 19(B) (1976), 358-370. MR 56:6058
  • 8. S. N. Chow and J. Mallet-Paret, The fuller index and global Hopf bifurcation, J. Differential Equations 29 (1978), 66-85. MR 58:11665
  • 9. S. N. Chow, J. Mallet-Paret and J. A. Yorke, Global Hopf bifurcation from a multiple eigenvalue, Nonlinear Anal. 2 (1978), 753-763. MR 80a:58015
  • 10. M. A. Cohn and S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Systems Man Cybernet. 13 (1983), 815-826. MR 85e:92002
  • 11. E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. London Math. Soc. 30 (1975), 76-94. MR 52:867
  • 12. G. Dylawerski, K. G[??]eba, J. Jodel and W. Marzantowicz, $S^1$-equivalent degree and the Fuller index, Ann. Polon. Math. 52 (1991), 243-280. MR 92g:58105
  • 13. L. H. Erge, K. Geba, W. Krawcewicz, and J. Wu, $S^1$-degree and global Hopf bifurcation theory of functional differential equations, J. Differential Equations 98 (1992), 277-298. MR 93f:58034
  • 14. C. Fenske, Analytische Theorie des Abbildungrades für Abbildungen in Banachraumes, Math. Nachr. 48 (1971), 279-290. MR 49:7857
  • 15. B. Fiedler, An index for global Hopf bifurcation in parabolic systems, J. Reine Angew. Math. 359 (1985), 1-36. MR 87c:58024
  • 16. -, Global Bifurcation of Periodic Solutions with Symmetry, Lecture Notes in Math., Vol. 1309, Springer-Verlag, New York, 1988. MR 90c:58125
  • 17. P. M. Fitzpatrick, Homotopy, linearization, and bifurcation, Nonlinear Anal. 12 (1988), 171-184. MR 89g:58039
  • 18. A. Frumkin and E. Moses, Physicality of the Little model, Physical Review (A), 34 (1986), 714-716.
  • 19. F. B. Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133-148. MR 35:497
  • 20. K. Geba, W. Krawcewicz and J. Wu, An equivariant degree with applications to symmetric bifurcation problems, I: construction of the degree, Proc. London Math. Soc. 69 (1994), 377-398. MR 95g:58025
  • 21. K. Geba an W.. Marzantowicz, Global bifurcation of periodic solutions, Topological Methods in Nonlinear Analysis 1 (1993), 67-93. MR 94e:58024
  • 22. E. Goles-Chacc, F. Fogelman-Soulie and D. Pellegrin, Decreasing energy functions as a tool for studying threshold networks, Discrete Appl. Math. 12 (1985), 261-277. MR 87d:68084
  • 23. E. Goles and G. Y. Vichniac, Lyapunov functions for parallel neural networks, in ``Neural Networks for Computing (J. S. Denber ed.)'', Amer. Inst. Phy., New York (1986), pp. 165-181.
  • 24. M. Golubitsky and I. N. Stewart, Hopf bifurcation in the presence of symmetry, Arch. Rational Mech. Anal. 87 (1985), 107-165. MR 86g:58034
  • 25. M. Golubitsky, I. Stewart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 2, Springer-Verlag, New York, 1988. MR 89m:58038
  • 26. G. Grinstein, C. Jayaprakash and P. He, Statistical mechanics of probabilistic automata, Phys. Rev. Lett. 55 (1985), 2527-2530. MR 87c:82029
  • 27. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York 1977. MR 58:22904
  • 28. -, Nonlinear Oscillations in Equations with Delays, in ``Nonlinear Oscillations in Biology'', pp. 157-185, Lectures in Applied Mathematics, Vol. 17, Amer. Math. Soc., Providence, RI, 1979. MR 81d:34062
  • 29. D. Hassard, N. D. Kazarinoff and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge Univ. Press, Cambridge, 1981. MR 82j:58089
  • 30. A. Herz, B. Salzer, R. Kühn and J. L. van Hemmen, Hebbian learning reconsidered: representation of static and dynamic objects in associative neural nets, Biol. Cybern. 60 (1989), 457-467.
  • 31. G. Hetzer and V. Stallbohm, Global behaviour of bifurcation branches and the essential spectrum, Math. Nachr. 86 (1978), 343-360. MR 80g:47076
  • 32. M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1-53. MR 89c:58108
  • 33. J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. 79 (1982), 2554-2558. MR 83g:92024
  • 34. J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci. 81 (1984), 3088-3092.
  • 35. J. Ize, Bifurcation theory for Fredholm operators, Mem. Amer. Math. Soc. 17 (1976). MR 54:13649
  • 36. -, Obstruction theory and multiparameter Hopf bifurcation, Trans. Amer. Math. Soc. 209 (1985), 757-792.
  • 37. J. Ize, I. Massabó and A. Vignoli, Degree theory for equivariant maps, (I). Trans. Amer. Math. Soc. 315 (1989), 433-510. MR 90a:58030
  • 38. -, Degree theory for equivariant maps, (II). The general $S^1$-action, Mem. Amer. Math. 100 (1992). MR 93d:58030
  • 39. W. Krawcewicz, P. Vivi and J. Wu, Computation formulae of an equivalent degree with applications to symmetric bifurcations, Nonlinear Studies 4 (1997), 89-120. CMP 97:14
  • 40. W. Krawcewicz and J. Wu, Theory and applications of Hopf bifurcations in symmetric functional differential equations, Nonlinear Analysis, in press.
  • 41. W. Krawcewicz, H. Xia and J. Wu, Global bifurcation theory for condensing fields and neutral equations with applications to lossless transmission problems, Canadian Appl. Math. Quarterly 1 (1993), 167-220. MR 94g:47077
  • 42. B. W. Levinger, A folk theorem in functional differential equations, J. Differential Equations 4 (1980), 612-619. MR 37:6566
  • 43. W. A. Little, Existence of persistent states in the brain, Math. Biosci. 19 (1974), 101-120.
  • 44. W. A. Little and G. L. Shaw, Analytic study of the memory storage capacity of a neural network, Math. Biosci. 39 (1978), 281-290.
  • 45. J. Mallet-Paret and R. D. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mate. Pura. Appl. 145 (1986), 33-128. MR 88e:34127
  • 46. J. Mallet-Paret and J. Yorke, Snakes: oriented families of periodic orbits, their sources, sinks and continuation, J. Differential Equations 43 (1992), 419-450. MR 84a:58071
  • 47. C. M. Marcus and R. M. Westervelt, Stability of analog neural networks with delay, Physical Review A 39 (1989), 347-359. MR 89m:92025
  • 48. R. D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal. 19 (1975), 319-338. MR 52:6516
  • 49. -, A Hopf global bifurcation theorem for retarded functional differential equations, Trans. Amer. Math. Soc. 238 (1978), 139-164. MR 81h:34082
  • 50. -, Circulant matrices and differential-delay equations, J. Differential Equations 60 (1985), 201-217. MR 87d:34112
  • 51. P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Anal. 7 (1971), 487-513. MR 46:745
  • 52. H. L. Smith, Monotone semiflows generated by functional differential equations, J. Differential Equations 66 (1987), 420-442. MR 88j:34155
  • 53. C. A. Stuart, Some bifurcation theory for $k$-set contractions, Proc. London Math. Soc. 27 (1973), 531-550. MR 48:12178
  • 54. A. Vanderbauwhede, Local Bifurcation and Symmetry, Res. Notes Math., 75, Pitman, Boston, 1982. MR 85f:58026
  • 55. J. Wu and W. Krawcewicz, Discrete waves and phase-locked oscillations in the growth of a single-species population over a patchy environment, Open Systems and Information Dynamics 1 (1992), 127-147.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 34K15, 34K20, 34C25

Retrieve articles in all journals with MSC (1991): 34K15, 34K20, 34C25

Additional Information

Jianhong Wu
Affiliation: Department of Mathematics and Statistics, York University, North York, Ontario, Canada M3J 1P3

Keywords: Periodic solution, delay differential equation, wave, symmetry, neural network, equivariant degree, global bifurcation.
Received by editor(s): September 13, 1995
Additional Notes: Research partially supported by the Natural Sciences and Engineering Research Council of Canada.
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society