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Symmetric functional-differential equations and neural networks with memory. (English) Zbl 0905.34034
Summary: The author establishes an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe spatial-temporal pattern, asymptotic form and global continuation of bifurcations of periodic wave solutions to functional-differential equations with symmetry. The author applies 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.

MSC:
34C23 Bifurcation theory for ordinary differential equations
34K13 Periodic solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34C25 Periodic solutions to ordinary differential equations
34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
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[1] J. C. Alexander, Bifurcation of zeroes of parametrized functions, J. Funct. Anal. 29 (1978), no. 1, 37 – 53. · Zbl 0385.47038 · doi:10.1016/0022-1236(78)90045-9 · doi.org
[2] J. C. Alexander and J. F. G. Auchmuty, Global bifurcation of waves, Manuscripta Math. 27 (1979), no. 2, 159 – 166. · Zbl 0398.35054 · doi:10.1007/BF01299293 · doi.org
[3] J. C. Alexander and Giles Auchmuty, Global bifurcations of phase-locked oscillators, Arch. Rational Mech. Anal. 93 (1986), no. 3, 253 – 270. · Zbl 0596.92010 · doi:10.1007/BF00281500 · doi.org
[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), no. 1, 87 – 106. · Zbl 0438.47061 · doi:10.1016/0022-1236(79)90027-2 · doi.org
[5] J. C. Alexander and James A. Yorke, Global bifurcations of periodic orbits, Amer. J. Math. 100 (1978), no. 2, 263 – 292. · Zbl 0386.34040 · doi:10.2307/2373851 · doi.org
[6] Jacques Bélair, Stability in a model of a delayed neural network, J. Dynam. Differential Equations 5 (1993), no. 4, 607 – 623. · Zbl 0796.34063 · doi:10.1007/BF01049141 · doi.org
[7] R. D. Braddock and P. van den Driessche, On the stability of differential-difference equations, J. Austral. Math. Soc. Ser. B 19 (1975/76), no. 3, 358 – 370. · Zbl 0356.34090 · doi:10.1017/S0334270000001211 · doi.org
[8] Shui Nee Chow and John Mallet-Paret, The Fuller index and global Hopf bifurcation, J. Differential Equations 29 (1978), no. 1, 66 – 85. · Zbl 0369.34020 · doi:10.1016/0022-0396(78)90041-4 · doi.org
[9] Shui Nee Chow, John Mallet-Paret, and James A. Yorke, Global Hopf bifurcation from a multiple eigenvalue, Nonlinear Anal. 2 (1978), no. 6, 753 – 763. · Zbl 0407.47039 · doi:10.1016/0362-546X(78)90017-2 · doi.org
[10] Michael A. Cohen and Stephen Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Systems Man Cybernet. 13 (1983), no. 5, 815 – 826. · Zbl 0553.92009 · doi:10.1016/S0166-4115(08)60913-9 · doi.org
[11] E. N. Dancer, Boundary-value problems for ordinary differential equations on infinite intervals, Proc. London Math. Soc. (3) 30 (1975), 76 – 94. · Zbl 0299.34027 · doi:10.1112/plms/s3-30.1.76 · doi.org
[12] Grzegorz Dylawerski, Kazimierz Gȩba, Jerzy Jodel, and Wacław Marzantowicz, An \?\textonesuperior -equivariant degree and the Fuller index, Ann. Polon. Math. 52 (1991), no. 3, 243 – 280. · Zbl 0723.58044
[13] L. H. Erbe, W. Krawcewicz, K. Gȩba, and J. Wu, \?\textonesuperior -degree and global Hopf bifurcation theory of functional-differential equations, J. Differential Equations 98 (1992), no. 2, 277 – 298. · Zbl 0765.34023 · doi:10.1016/0022-0396(92)90094-4 · doi.org
[14] Christian Fenske, Analytische Theorie des Abbildungsgrades für Abbildungen in Banachräumen, Math. Nachr. 48 (1971), 279 – 290 (German). · Zbl 0192.49002 · doi:10.1002/mana.19710480121 · doi.org
[15] Bernold Fiedler, An index for global Hopf bifurcation in parabolic systems, J. Reine Angew. Math. 359 (1985), 1 – 36. · Zbl 0554.35010 · doi:10.1515/crll.1985.359.1 · doi.org
[16] Bernold Fiedler, Global bifurcation of periodic solutions with symmetry, Lecture Notes in Mathematics, vol. 1309, Springer-Verlag, Berlin, 1988. · Zbl 0644.34038
[17] P. M. Fitzpatrick, Homotopy, linearization, and bifurcation, Nonlinear Anal. 12 (1988), no. 2, 171 – 184. · Zbl 0653.58028 · doi:10.1016/0362-546X(88)90033-8 · doi.org
[18] A. Frumkin and E. Moses, Physicality of the Little model, Physical Review (A), 34 (1986), 714-716.
[19] F. Brock Fuller, An index of fixed point type for periodic orbits, Amer. J. Math. 89 (1967), 133 – 148. · Zbl 0152.40204 · doi:10.2307/2373103 · doi.org
[20] Kazimierz Gȩba, Wiesław Krawcewicz, and Jian Hong Wu, An equivariant degree with applications to symmetric bifurcation problems. I. Construction of the degree, Proc. London Math. Soc. (3) 69 (1994), no. 2, 377 – 398. · Zbl 0823.47060 · doi:10.1112/plms/s3-69.2.377 · doi.org
[21] Kazimierz Gȩba and Wacław Marzantowicz, Global bifurcation of periodic solutions, Topol. Methods Nonlinear Anal. 1 (1993), no. 1, 67 – 93. · Zbl 0788.34031
[22] Eric Goles Chacc, Françoise Fogelman-Soulié, and Didier Pellegrin, Decreasing energy functions as a tool for studying threshold networks, Discrete Appl. Math. 12 (1985), no. 3, 261 – 277. · Zbl 0585.94018 · doi:10.1016/0166-218X(85)90029-0 · doi.org
[23] E. Goles and G. Y. Vichniac, Lyapunov functions for parallel neural networks, in “Neural Networks for Computing ”, Amer. Inst. Phy., New York (1986), pp. 165-181.
[24] Martin Golubitsky and Ian Stewart, Hopf bifurcation in the presence of symmetry, Arch. Rational Mech. Anal. 87 (1985), no. 2, 107 – 165. · Zbl 0588.34030 · doi:10.1007/BF00280698 · doi.org
[25] Martin Golubitsky, Ian Stewart, and David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. II, Applied Mathematical Sciences, vol. 69, Springer-Verlag, New York, 1988. · Zbl 0691.58003
[26] G. Grinstein, C. Jayaprakash, and Yu He, Statistical mechanics of probabilistic cellular automata, Phys. Rev. Lett. 55 (1985), no. 23, 2527 – 2530. · doi:10.1103/PhysRevLett.55.2527 · doi.org
[27] Jack Hale, Theory of functional differential equations, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 3. · Zbl 0352.34001
[28] Jack K. Hale, Nonlinear oscillations in equations with delays, Nonlinear oscillations in biology (Proc. Tenth Summer Sem. Appl. Math., Univ. Utah, Salt Lake City, Utah, 1978) Lectures in Appl. Math., vol. 17, Amer. Math. Soc., Providence, R.I., 1979, pp. 157 – 185.
[29] Brian D. Hassard, Nicholas D. Kazarinoff, and Yieh Hei Wan, Theory and applications of Hopf bifurcation, London Mathematical Society Lecture Note Series, vol. 41, Cambridge University Press, Cambridge-New York, 1981.
[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. · Zbl 0661.92007
[31] Georg Hetzer and Volker Stallbohm, Global behavior of bifurcation branches and the essential spectrum, Math. Nachr. 86 (1978), 347 – 360. · Zbl 0408.47046 · doi:10.1002/mana.19780860125 · doi.org
[32] Morris W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1 – 53. · Zbl 0624.58017 · doi:10.1515/crll.1988.383.1 · doi.org
[33] J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. U.S.A. 79 (1982), no. 8, 2554 – 2558. · Zbl 1369.92007
[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. · Zbl 1371.92015
[35] Jorge Ize, Bifurcation theory for Fredholm operators, Mem. Amer. Math. Soc. 7 (1976), no. 174, viii+128. · Zbl 0338.47032 · doi:10.1090/memo/0174 · doi.org
[36] -, Obstruction theory and multiparameter Hopf bifurcation, Trans. Amer. Math. Soc. 209 (1985), 757-792. · Zbl 0575.58025
[37] J. Ize, I. Massabò, and A. Vignoli, Degree theory for equivariant maps. I, Trans. Amer. Math. Soc. 315 (1989), no. 2, 433 – 510. · Zbl 0695.58006
[38] Jorge Ize, Ivar Massabò, and Alfonso Vignoli, Degree theory for equivariant maps, the general \?\textonesuperior -action, Mem. Amer. Math. Soc. 100 (1992), no. 481, x+179. · Zbl 0785.58008 · doi:10.1090/memo/0481 · doi.org
[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 · Zbl 0915.58022
[40] W. Krawcewicz and J. Wu, Theory and applications of Hopf bifurcations in symmetric functional differential equations, Nonlinear Analysis, in press. · Zbl 0917.58027
[41] W. Krawcewicz, J. Wu, and H. Xia, Global Hopf bifurcation theory for condensing fields and neutral equations with applications to lossless transmission problems, Canad. Appl. Math. Quart. 1 (1993), no. 2, 167 – 220. · Zbl 0801.34069
[42] B. W. Levinger, A folk theorem in functional differential equations, J. Differential Equations 4 (1968), 612 – 619. · Zbl 0174.13902 · doi:10.1016/0022-0396(68)90011-9 · doi.org
[43] W. A. Little, Existence of persistent states in the brain, Math. Biosci. 19 (1974), 101-120. · Zbl 0272.92011
[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. · Zbl 0395.92005
[45] John Mallet-Paret and Roger D. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl. (4) 145 (1986), 33 – 128. · Zbl 0617.34071 · doi:10.1007/BF01790539 · doi.org
[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. · Zbl 0487.34038
[47] C. M. Marcus and R. M. Westervelt, Stability of analog neural networks with delay, Phys. Rev. A (3) 39 (1989), no. 1, 347 – 359. · doi:10.1103/PhysRevA.39.347 · doi.org
[48] Roger D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Functional Analysis 19 (1975), no. 4, 319 – 338. · Zbl 0314.47041
[49] Roger D. Nussbaum, A Hopf global bifurcation theorem for retarded functional differential equations, Trans. Amer. Math. Soc. 238 (1978), 139 – 164. · Zbl 0389.34050
[50] Roger D. Nussbaum, Circulant matrices and differential-delay equations, J. Differential Equations 60 (1985), no. 2, 201 – 217. · Zbl 0622.34076 · doi:10.1016/0022-0396(85)90113-5 · doi.org
[51] Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487 – 513. · Zbl 0212.16504
[52] Hal Smith, Monotone semiflows generated by functional-differential equations, J. Differential Equations 66 (1987), no. 3, 420 – 442. · Zbl 0612.34067 · doi:10.1016/0022-0396(87)90027-1 · doi.org
[53] C. A. Stuart, Some bifurcation theory for \?-set contractions, Proc. London Math. Soc. (3) 27 (1973), 531 – 550. · Zbl 0268.47064 · doi:10.1112/plms/s3-27.3.531 · doi.org
[54] A. Vanderbauwhede, Local bifurcation and symmetry, Research Notes in Mathematics, vol. 75, Pitman (Advanced Publishing Program), Boston, MA, 1982. · Zbl 0539.58022
[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. · Zbl 0898.34064
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