Advertisement

Lyapunov–Schmidt Reduction

  • Shangjiang Guo
  • Jianhong Wu
Chapter
  • 2.7k Downloads
Part of the Applied Mathematical Sciences book series (AMS, volume 184)

Abstract

The main objective of this chapter is to introduce the Lyapunov–Schmidt reduction method and show how this reduction can be performed in a way compatible with symmetries.

Keywords

Periodic Solution Transmission Line Hopf Bifurcation Wave Train Implicit Function Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Abolinia, V.E., Mishkis, A.D.: A mixed problem for a linear hyperbolic system on the plane. Latvijas Valsts Univ. Zinatn. Raksti 20, 87–104 (1958)MathSciNetGoogle Scholar
  2. 2.
    Abolinia, V.E., Mishkis, A.D.: Mixed problems for quasi-linear hyperbolic systems in the plane. Mat. Sb. (N.S.) 50, 423–442 (1960)Google Scholar
  3. 3.
    Adimy, M.: Integrated semigroups and delay differential equations. J. Math. Anal. Appl. 177, 125–134 (1993)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Afraimovich, V., Shil’nikov, L.: On singular trajectories of dynamical systems. Usp. Mat. Nauk 5, 189–190 (1972) (in Russian)Google Scholar
  5. 5.
    Ait Babram, M.: An algorithmic scheme for approximating center manifolds and normal forms for functional differential equations. In: Arino, O., Hbid, M.L., Ait Dads, E. (eds.) Delay Differential Equations and Applications. NATO Sci. Ser. II Math. Phys. Chem., vol. 205, pp. 193–226. Springer, Dordrecht (2006)Google Scholar
  6. 6.
    Ait Babram, M., Arino, O., Hbid, M.L.: Computational scheme of a center manifold for neutral functional differential equations. J. Math. Anal. Appl. 258(2), 396–414 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Ait Babram, M., Hbid, M.L., Arino, O.: Approximation scheme of a center manifold for functional-differential equations. J. Math. Anal. Appl. 213(2), 554–572 (1997)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Alexander, J.C.: Bifurcation of zeros of parametrized functions. J. Funct. Anal. 29, 37–53 (1978)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Alexander, J.C., Fitzpatrick, P.M.: The homotopy of a certain spaces of nonlinear equations, and its relation to global bifurcation of the fixed points of parametrized condensing operators. J. Funct. Anal. 34, 87–106 (1979)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Alexander, J.C., Yorke, J.A.: Global bifurcations of periodic orbits. Am. J. Math. 100, 263–292 (1978)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Algaba, A., Merino, M., Freire, E., Gamero, E., Rodrguez-Luis, A.J.: Some results on Chua’s equation near a triple-zero linear degeneracy. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 13(3), 583–608 (2003)zbMATHGoogle Scholar
  12. 12.
    Arino, O.: Contribution á l’étude des comportements des solutions d’équation différentielle á retard par des méthodes de monotonie et de bifurcation. Thése d’état, Université de Bordeaux 1 (1980)Google Scholar
  13. 13.
    an der Heiden, U.: Periodic solutions of a nonlinear second-order differential equations with delay. J. Math. Anal. Appl. 70, 599–609 (1979)Google Scholar
  14. 14.
    Andronov, A.A.: Application of Poincaré’s theorem on “bifurcation points” and “change in stability” to simple auto-oscillatory systems. C. R. Acad. Sci. Paris 189(15), 559–561 (1929)Google Scholar
  15. 15.
    Andronov, A.A., Leontovich, E.: Some cases of dependence of limit cycles on a parameter. J. State Univ. Gorki 6, 3–24 (1937) (in Russian)Google Scholar
  16. 16.
    Andronov, A.A., Pontryagin, L.: Systémes grossiéres. Dokl. Akad. Nauk SSSR 14, 247–251 (1937) (in Russian).Google Scholar
  17. 17.
    Arino, O., Hbid, M.L.: Existence of periodic solutions for a delay differential equation via the Poincaré procedure. Differ. Equat. Dyn. Syst. 4(2), 125–148 (1996)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Arino, O., Sánchez, E.: A variation of constants formula for an abstract functional-differential equation of retarded type. Differ. Integr. Equat. 9(6), 1305–1320 (1996)zbMATHGoogle Scholar
  19. 19.
    Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1983)zbMATHGoogle Scholar
  20. 20.
    Arnold, V.I.: Lectures on bifurcations in versal families. Russ. Math. Surv. 27, 54–123 (1972)Google Scholar
  21. 21.
    Arrowsmith, D.K., Place, C.M.: An Introduction to Dynamical Systems. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  22. 22.
    Ashkenazi, M., Chow, S.N.: Normal forms near critical points for differential equations and maps. IEEE Trans. Circuits Syst. 35, 850–862 (1988)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Aubin, J.P.: Applied Functional Analysis. Wiley, New York (1979)zbMATHGoogle Scholar
  24. 24.
    Balanov, Z., Krawcewicz, W.: Remarks on the equivariant degree theory. Topol. Methods Nonlinear Anal. 13, 91–103 (1999)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Balanov, Z., Krawcewicz, W., Steinlein, H.: Reduced SO(3) ×S 1-equivariant degree with applications to symmetric bifurcations problems. Nonlinear Anal. 47, 1617–1628 (2001)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Bélair, J.: Population models with state-dependent delays. Lect. Notes Pure Appl. Math. 131, 165–176 (1991)Google Scholar
  27. 27.
    Bélair, J., Campbell, S.A.: Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM J. Appl. Math. 54, 1402–1424 (1994)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Bélair, J., Campbell, S.A., van den Driessche, P.: Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math. 56, 245–255 (1996)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Bélair, J., Dufour, S.: Stability in a three-dimensional system of delay-differential equations. Can. Appl. Math. Q. 4(2), 135–156 (1996)zbMATHGoogle Scholar
  30. 30.
    Bellman, R., Cooke, K.L.: Differential Difference Equations. Academic Press, New York (1963)zbMATHGoogle Scholar
  31. 31.
    Bernfeld, S.R., Negrini, P., Salvadori, L.: Generalized Hopf bifurcation and h-asymptotic stability. J. Nonlinear Anal. Theor. Meth. Appl. 4, 109–1107 (1980)Google Scholar
  32. 32.
    Bernfeld, S.R., Negrini, P., Salvadori, L.: Quasi-invariant manifolds stability and generalized Hopf bifurcation. Ann. Math. Pura Appl. 4, 105–119 (1982)MathSciNetGoogle Scholar
  33. 33.
    Birkhoff, G.D.: Dynamical Systems. AMS, Providence (1927)zbMATHGoogle Scholar
  34. 34.
    Birkhoff, G.D.: Nouvelles recherches sur les systèmes dynamiques. Memoriae Pont. Acad. Sci. Novi. Lincaei Ser. 3 1, 85–216 (1935)MathSciNetGoogle Scholar
  35. 35.
    Bogdanov, R.: Versal deformations of a singular point on the plane in the case of zero eigenvalues. In: Proceedings of Petrovskii Seminar, Moscow State University, vol. 2, pp. 37–65 (1976) (in Russian) (English translation: Selecta Math. Soviet. 1(4), 389–421, 1981)Google Scholar
  36. 36.
    Braaksma, B.L.J., Broer, H.W.: Quasiperiodic flow near a codimension one singularity of a divergence free vector field in dimension four. In: Bifurcation, Ergodic Theory and Applications (Dijon, 1981). Astérisque, vol. 98–99, pp. 74–142. Soc. Math. France, Paris (1982)Google Scholar
  37. 37.
    Brayton, R.K.: Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type. Q. Appl. Math. 24, 215–224 (1966)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Brayton, R.K.: Nonlinear oscillations in a distributed network. Q. Appl. Math. 24, 289–301 (1967)zbMATHGoogle Scholar
  39. 39.
    Brayton, R.K., Miranker, W.L.: A stability theory for nonlinear mixed initial boundary value problems. Arch. Ration. Mech. Anal. 17, 358–376 (1964)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Brayton, R.K., Moser, J.K.: A theory of nonlinear networks. I. Q. Appl. Math. 22, 1–33 (1964)MathSciNetGoogle Scholar
  41. 41.
    Bredon, G.E.: Introduction to Compact Transformation Groups. Academic, New York (1972)zbMATHGoogle Scholar
  42. 42.
    Broer, H.W.: Coupled Hopf-bifurcations: persistent examples of n-quasiperiodicity determined by families of 3-jets. Geometric methods in dynamics. I. Astérisque 286, xix, 223–229 (2003)Google Scholar
  43. 43.
    Broer, H.W.: Quasiperiodicity in local bifurcation theory. In: Bruter, C.P., Aragnol, A., Lichnérowicz, A. (eds.) Bifurcation Theory, Mechanics and Physics. Mathematics and Its Applications, pp. 177–208. Reidel, Dordrecht (1983)Google Scholar
  44. 44.
    Broer, H.W., Vegter, G.: Subordinate Sil’nikov bifurcations near some singularities of vector fields having low codimension. Ergod. Theor. Dyn. Syst. 4, 509–525 (1984)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Brokate, M., Colonius, F.: Linearizing equations with state-dependent delays. Appl. Math. Optim. 21, 45–52 (1990)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Brouwder, F.E.: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. 1, 1–39 (1983)Google Scholar
  47. 47.
    Brouwer, L.E.J.: Über Abbildung der Mannigfaltigkeiten. Math. Ann. 70, 97–115 (1912)Google Scholar
  48. 48.
    Bruno, A.D.: Local Method of Nonlinear Analysis of Differential Equations (in Russian). Izdatel’stvo Nauka, Moscow (1979)Google Scholar
  49. 49.
    Buono, P.L., Bélair, J.: Restrictions and unfolding of double Hopf bifurcation in functional differential equations. J. Differ. Equat. 189, 234–266 (2003)zbMATHGoogle Scholar
  50. 50.
    Busenberg, S., Huang, W.: Stability and Hopf bifurcation for a population delay model with diffusion effects. J. Differ. Equat. 124(1), 80–107 (1996)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Busenberg, S., Travis, C.C.: On the use of reducible functional differential equations. J. Math. Anal. Appl. 89, 46–66 (1982)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Campbell, S.A.: Time delays in neural systems. In: McIntosh, R., Jirsa, V.K. (eds.) Handbook of Brain Connectivity. Springer, New York (2007)Google Scholar
  53. 53.
    Carr, J.: Applications of Centre Manifold Theory. Applied Mathematical Sciences, vol. 35. Springer, New York (1981)Google Scholar
  54. 54.
    Chafee, N.: A bifurcation problem for a functional differential equation of finitely retarded type. J. Math. Anal. Appl. 35, 312–348 (1971)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Chafee, N.: Generalized Hopf bifurcation and perturbation in a full neighborhood of a given vector field. Indiana Univ. Math. J. 27, 173–194 (1978)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Chen, G., Della Dora, J.: Rational normal form for dynamical systems via Carleman linearization. In: Proceeding of ISSAC-99, pp. 165–172. ACM Press–Addison Wesley, Vancouver (1999)Google Scholar
  57. 57.
    Chen, G., Della Dora, J.: Further reduction of normal forms for dynamical systems. J. Differ. Equat. 166, 79–106 (2000)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Chen, Y.: Existence and unstable sets of oscillating periodic orbits for delayed excitatory networks of two neurons. Differ. Equat. Dyn. Syst. 9, 169–185 (2001)zbMATHGoogle Scholar
  59. 59.
    Chen, Y., Wu, J.: Existence and attraction of a phase-locked oscillation in a delayed network of two neurons. Differ. Integr. Equat. 14, 1181–1236 (2001)zbMATHGoogle Scholar
  60. 60.
    Chen, Y., Wu, J.: Slowly oscillating periodic solutions for a delayed frustrated network of two neurons. J. Math. Anal. Appl. 259, 188–208 (2001)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Chen, Y., Wu, J., Krisztin, T.: Connecting orbits from synchronous periodic solutions to phase-locked periodic solutions in a delay differential system. J. Differ. Equat. 163, 130–173 (2000)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Chossat, P., Lauterbach, R.: Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific, Singapore (2000)zbMATHGoogle Scholar
  63. 63.
    Chow, S.N.: Existence of periodic solutions of autonomous functional differential equations. J. Differ. Equat. 15, 350–378 (1974)zbMATHGoogle Scholar
  64. 64.
    Chow, S.-N., Diekmann, O., Mallet-Paret, J.: Multiplicity of symmetric periodic solutions of a nonlinear Volterra integral equation. Jpn. J. Appl. Math. 2, 433–469 (1985)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Chow, S.N., Hale, J.: Methods of Bifurcation Theory. Springer, New York (1982)zbMATHGoogle Scholar
  66. 66.
    Chow, S.-N., Li, C., Wang, D.: Normal Forms and Bifurcations of Planar Vector Fields. Cambridge University Press, Cambridge (1994)Google Scholar
  67. 67.
    Chow, S.-N., Lin, X.-L., Mallet-Paret, J.: Transition layers for singularly perturbed delay differential equations with monotone nonlinearities. J. Dynam. Differ. Equat. 1, 3–43 (1989)MathSciNetzbMATHGoogle Scholar
  68. 68.
    Chow, S.N., Mallet-Paret, J.: Integral averaging and bifurcation. J. Differ. Equat. 26, 112–159 (1977)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Chow, S.N., Mallet-Paret, J.: The Fuller index and global Hopf bifurcation. J. Differ. Equat. 29, 66–85 (1978)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Chow, S.-N., Mallet-Paret, J.: Singularly perturbed delay differential equations. In: Chandra, J., Scott, A. (eds.) Coupled Oscillators, pp. 7–12. North-Holland, Amsterdam (1983)Google Scholar
  71. 71.
    Chow, S.N., Mallet-Paret, J., Yorke, J.A.: Global Hopf bifurcation from a multiple eigenvalue. Nonlinear Anal. 2, 753–763 (1978)MathSciNetzbMATHGoogle Scholar
  72. 72.
    Cooke, K.L., Huang, W.Z.: On the problem of linearization for state-dependent delay differential equations. Proc. Am. Math. Soc. 124, 1417–1426 (1996)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations. Health, Boston (1965)zbMATHGoogle Scholar
  74. 74.
    Crandall, M.G., Rabinowitz, P.H.: The Hopf bifurcation theorem in infinite dimension. Arch. Ration. Mech. Anal. 67, 53–72 (1977/78)Google Scholar
  75. 75.
    Cicogna, G.: Symmetry breakdown from bifurcation. Lettere al Nuovo Cimento 31, 600–602 (1981)MathSciNetGoogle Scholar
  76. 76.
    Cushing, J.M.: Integrodifferential Equations and Delay Models in Population Dynamics. Lecture Notes in Biomathematics, vol. 20. Springer, New York (1977)Google Scholar
  77. 77.
    Cushman, R., Sanders, J.A.: Nilpotent normal forms and representation theory of sl(2, R). In: Golubitsky, M., Guckenheimer, J. (eds.) Multiparameter Bifurcation Theory. Contemporary Mathematics, vol. 56, pp. 31–51. AMS, Providence (1986)Google Scholar
  78. 78.
    Cushman, R., Sanders, J.A.: Splitting algorithm for nilpotent normal forms. Dynam. Stabil. Syst. 2(3–4), 235–246 (1988)MathSciNetGoogle Scholar
  79. 79.
    Cushman, R., Sanders, J.A.: A survey of invariant theory applied to normal forms of vector fields with nilpotent linear part. In: Proceedings of Invariant Theory, pp. 82–106. Springer, New York (1990)Google Scholar
  80. 80.
    de Oliveira, J.C., Hale, J.K.: Dynamic behavior from the bifurcation function. Tôhoku Math. J. 32, 577–592 (1980)zbMATHGoogle Scholar
  81. 81.
    Diekmann, O., van Gils, S.A.: The center manifold for delay equations in the light of suns and stars. In: Roberts, M., Stewart, I.N. (eds.) Singularity Theory and Its Application, Warwick, 1989, Part II, Springer LMN 1463, pp. 122–141. Springer, New York (1991)Google Scholar
  82. 82.
    Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay Equations, Functional-, Complex-, and Nonlinear Analysis. Springer, New York (1995)zbMATHGoogle Scholar
  83. 83.
    Dormayer, P.: Smooth bifurcation of symmetric periodic solutions of functional-differential equations. J. Differ. Equat. 82, 109–155 (1989)MathSciNetzbMATHGoogle Scholar
  84. 84.
    Dumortier, F., Ibáñez, S.: Singularities of vector fields on \({\mathbb{R}}^{3}\). Nonlinearity 11, 1037–1047 (1998)MathSciNetzbMATHGoogle Scholar
  85. 85.
    Dylawerski, G., Gȩba, K., Jodel, J., Marzantowicz, W.: S 1-equivalent degree and the Fuller index. Ann. Polon. Math. 52, 243–280 (1991)MathSciNetzbMATHGoogle Scholar
  86. 86.
    Eichmann, M.: A local Hopf bifurcation theorem for differential equations with state-dependent delays. Ph.D. Dissertation, Justus-Liebig University in Giessen (2006)Google Scholar
  87. 87.
    Elphick, C., Tirapegui, E., Brachet, M.E., Coullet, P., Iooss, G.: A simple global characterization for normal forms of singular vector fields. Phys. D 29, 95–127 (1987)MathSciNetzbMATHGoogle Scholar
  88. 88.
    Erbe, L.H., Krawcewicz, W., Geba, K., Wu, J.: S 1-degree and global Hopf bifurcation theory of functional differential equations. J. Differ. Equat. 98, 227–298 (1992)MathSciNetGoogle Scholar
  89. 89.
    Erbe, L.H., Krawcewicz, W., Peschke, G.: Bifurcations of a parametrized family of boundary value problems for second order differential inclusions. Ann. Math. Pura Appl. 165, 169–195 (1993)MathSciNetzbMATHGoogle Scholar
  90. 90.
    Erbe, L.H., Krawcewicz, W., Wu, J.: Leray-Schauder degree for semilinear Fredholm maps and periodic boundary value problems of neutral equations. Nonlinear Anal. 15, 747–764 (1990)MathSciNetzbMATHGoogle Scholar
  91. 91.
    Faria, T., Magalhães, L.T.: Normal forms for retarded functional-differential equations with parameters and applications to Hopf bifurcation. J. Differ. Equat. 122(2), 181–200 (1995)zbMATHGoogle Scholar
  92. 92.
    Faria, T., Magalhães, L.T.: Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity. J. Differ. Equat. 122(2), 201–224 (1995)zbMATHGoogle Scholar
  93. 93.
    Faria, T., Magalhães, L.T.: Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities. J. Dynam. Differ. Equat. 8, 35–70 (1996)zbMATHGoogle Scholar
  94. 94.
    Feigenbaum, M.J.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19(1), 25–52 (1978)MathSciNetzbMATHGoogle Scholar
  95. 95.
    Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)MathSciNetzbMATHGoogle Scholar
  96. 96.
    Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equat. 31, 53–98 (1979)MathSciNetzbMATHGoogle Scholar
  97. 97.
    Fermi, E., Pasta, J., Ulam, S.: Los Alamos Report LA-1940 (E. Fermi, Collected Papers II (1955)), pp. 977–988. University of Chicago Press, Chicago (1965)Google Scholar
  98. 98.
    Field, M.J.: Lectures on Bifurcations, Dynamics and Symmetry. Pitman Research Notes in Mathematics, vol. 356. Longman, Harlow (1996)Google Scholar
  99. 99.
    Field, M.J., Melbourne, I., Nicol, M.: Symmetric attractors for diffeomorphisms and flows. Proc. Lond. Math. Soc. 72, 657–696 (1996)MathSciNetzbMATHGoogle Scholar
  100. 100.
    Fiedler, B.: Global Hopf bifurcation in porous catalysts. In: Knobloch, H.W., Schmidt, K. (eds.) Proceedings Equadiff 82. Lecture Notes in Mathematics 1017, pp. 177–184. Springer, New York (1983)Google Scholar
  101. 101.
    Fiedler, B.: An index for global Hopf bifurcation in parabolic systems. J. Reine Angew. Math. 359, 1–36 (1985)MathSciNetzbMATHGoogle Scholar
  102. 102.
    Fiedler, B.: Global Bifurcation of Periodic Solutions with Symmetry. Lecture Notes in Mathematics, vol. 1309. Springer, New York (1988)Google Scholar
  103. 103.
    Fiedler, M.: Additive compound matrices and inequality for eigenvalues of stochastic matrices. Czech. Math. J. 99, 392–402 (1974)MathSciNetGoogle Scholar
  104. 104.
    Filip, A.M., Venakides, S.: Existence and modulation of traveling waves in particle chains. Comm. Pure Appl. Math. 52, 693–735 (1999)MathSciNetGoogle Scholar
  105. 105.
    Freire, E., Gamero, E., Rodríguez-Luis, A.J., Algaba, A.: A note on the triplezero linear degeneracy: normal forms, dynamical and bifurcation behaviors of an unfolding. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 12, 2799–820 (2002)zbMATHGoogle Scholar
  106. 106.
    Gamero, E., Freire, E., Rodríguez-Luis, A.J.: Hopf-zero bifurcation: normal form calculation and application to an electronic oscillator. In: International Conference on Differential Equations, vol. 1, 2 (Barcelona, 1991), pp. 517–524. World Scientific, River Edge, NJ (1993)Google Scholar
  107. 107.
    Gaspard, P.: Local birth of homoclinic chaos. Phys. D 62, 94–122 (1993)MathSciNetzbMATHGoogle Scholar
  108. 108.
    Gavrilov, N.: On some bifurcations of an equilibrium with one zero and a pair of pure imaginary roots. In: Methods of Qualitative Theory of Differential Equations (in Russian). GGU, Gorkii (1978)Google Scholar
  109. 109.
    Gavrilov, N.: Bifurcations of an equilibrium with two pairs of pure imaginary roots. In: Methods of Qualitative Theory of Differential Equations (in Russian). GGU, Gorkii (1980)Google Scholar
  110. 110.
    Gavrilov, N.K., Shil’nikov, L.P.: On three-dimensional systems close to systems with a structurally unstable homoclinic curve: II. Math. USSR-Sb. 19, 139–156 (1973)Google Scholar
  111. 111.
    Geba, K., Marzantowicz, W.: Global bifurcation of periodic solutions. Topol. Methods Nonlinear Anal. 1, 67–93 (1993)MathSciNetzbMATHGoogle Scholar
  112. 112.
    Geba, K., Krawcewicz, W., Wu, J.: An equivariant degree with applications to symmetric bifurcation problems 1: construction of the degree. Bull. Lond. Math. Soc. 69, 377–398 (1994)MathSciNetzbMATHGoogle Scholar
  113. 113.
    Giannakopoulos, F., Zapp, A.: Local and global Hopf bifurcation in a scalar delay differential equation. J. Math. Anal. Appl. 237(2), 425–450 (1999)MathSciNetzbMATHGoogle Scholar
  114. 114.
    Giannakopoulos, F., Zapp, A.: Bifurcations in a planar system of differential delay equations modeling neural activity. Phys. D 159, 215–232 (2001)MathSciNetzbMATHGoogle Scholar
  115. 115.
    Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York (1973)Google Scholar
  116. 116.
    Golubitsky, M., Marsden, J., Stewart, I., Dellnitz, M.: The constrained Lyapunov-Schmidt procedure and periodic orbits. Field. Inst. Comm. 4, 81–127 (1995)MathSciNetGoogle Scholar
  117. 117.
    Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 1. Springer, New York (1985)zbMATHGoogle Scholar
  118. 118.
    Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 2. Springer, New York (1988)zbMATHGoogle Scholar
  119. 119.
    Govaerts, W., Pryce, J.: Mixed block elimination for linear systems with wider borders. IMA J. Numer. Anal. 13, 161–180 (1993)MathSciNetzbMATHGoogle Scholar
  120. 120.
    Grabosch, A., Moustakas, U.: A semigroup approach to retarded differential equations. In: Nagel, R. (ed.) One-parameter Semigroups of Positive Operators. Lecture Notes in Mathematics, vol. 1184, pp. 219–232. Springer, Berlin (1986)Google Scholar
  121. 121.
    Grafton, R.B.: A periodicity theorem for autonomous functional differential equations. J. Differ. Equat. 6, 87–109 (1969)MathSciNetzbMATHGoogle Scholar
  122. 122.
    Grimmer, R.: Existence of periodic solutions of functional differential equations. J Math. Anal. Appl. 72(2), 666–673 (1979)MathSciNetzbMATHGoogle Scholar
  123. 123.
    Grobman, D.: Homeomorphisms of systems of differential equations. Dokl. Akad. Nauk SSSR 128, 880 (1959)MathSciNetzbMATHGoogle Scholar
  124. 124.
    Guckenheimer, J.: On a codimension two bifurcation. In: Rand, D.A., Young, L.-S. (eds.) Dynamical Systems and Turbulence. Warwick 1980 (Coventry, 1979/1980), vol. 898 of Lecture Notes in Mathematics, pp. 99–142. Springer, Berlin (1981)Google Scholar
  125. 125.
    Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations: Dynamical System and Bifurcations of Vector Fields. Springer, New York (1983)Google Scholar
  126. 126.
    Guckenheimer, J.: Multiple bifurcation problems of codimension two. SIAM J. Math. Anal. 15, 1–49 (1984)MathSciNetzbMATHGoogle Scholar
  127. 127.
    Gumowski, I.: Sur le calcul des solutions périodiques de l’équation de Cherwell-Wright. C.R. Acad. Sci. Paris Ser. A-B 268, 157–159 (1969)MathSciNetzbMATHGoogle Scholar
  128. 128.
    Guo, S.: Equivariant normal forms for neutral functional differential equations. Nonlinear Dyn. 61(1), 311–329 (2010)zbMATHGoogle Scholar
  129. 129.
    Guo, S.: Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay. Nonlinearity 18, 2391–2407 (2005)MathSciNetzbMATHGoogle Scholar
  130. 130.
    Guo, S.: Zero singularities in a ring network with two delays. Z. Angew. Math. Phys. 64(2), 201–222 (2013)MathSciNetzbMATHGoogle Scholar
  131. 131.
    Guo, S., Chen, Y., Wu, J.: Equivariant normal forms for parameterized delay differential equations with applications to bifurcation theory. Acta Math. Sin. Engl. Ser. 28(4), 825–856 (2012)MathSciNetGoogle Scholar
  132. 132.
    Guo, S., Chen, Y., Wu, J.: Two-parameter bifurcations in a network of two neurons with multiple delays. J. Differ. Equat. 244, 444–486 (2008)MathSciNetzbMATHGoogle Scholar
  133. 133.
    Guo, S., Huang, L.: Hopf bifurcating periodic orbits in a ring of neurons with delays. Phys. D 183(1–2), 19–44 (2003)MathSciNetzbMATHGoogle Scholar
  134. 134.
    Guo, S., Huang, L.: Global continuation of nonlinear waves in a ring of neurons. Proc. Math. Roy. Soc. Edinb. 135A, 999–1015 (2005)MathSciNetGoogle Scholar
  135. 135.
    Guo, S., Huang, L.: Stability of nonlinear waves in a ring of neurons with delays. J. Differ. Equat. 236, 343–374 (2007)MathSciNetzbMATHGoogle Scholar
  136. 136.
    Guo, S., Lamb, J.S.W.: Equivariant Hopf bifurcation for neutral functional differential equations. Proc. Am. Math. Soc. 136, 2031–2041 (2008)MathSciNetzbMATHGoogle Scholar
  137. 137.
    Guo, S., Lamb, J.S.W., Rink, B.W.: Branching patterns of wave trains in the FPU lattice. Nonlinearity 22, 283–299 (2009)MathSciNetzbMATHGoogle Scholar
  138. 138.
    Guo, S., Man, J.: Center manifolds theorem for parameterized delay differential equations with applications to zero singularities. Nonlinear Anal. Theor. Meth. Appl. 74(13), 4418–4432 (2011)MathSciNetzbMATHGoogle Scholar
  139. 139.
    Guo, S., Man, J.: Patterns in hierarchical networks of neuronal oscillators with D3 xZ3 symmetry. J. Differ. Equat. 254, 3501–3529 (2013)MathSciNetzbMATHGoogle Scholar
  140. 140.
    Guo, S., Yuan, Y.: Pattern formation in a ring network with delay. Math. Model. Meth. Appl. Sci. 19(10), 1797–1852 (2009)MathSciNetzbMATHGoogle Scholar
  141. 141.
    Guo, S., Wu, J.: Generalized Hopf bifurcation in delay differential equations (in Chinese). Sci. Sin. Math. 42, 91–105 (2012)Google Scholar
  142. 142.
    Gurney, W.S.C., Blythe, S.P., Nisbee, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)Google Scholar
  143. 143.
    Hadeler, K.P., Tomiuk, J.: Periodic solutions of difference differential equations. Arch. Ration. Anal. 1, 87–95 (1977)MathSciNetGoogle Scholar
  144. 144.
    Hale, J.K.: Linear Functional-Differential Equations with Constant Coefficients. Contributions to Differential Equations II, pp. 291–317. Research Institute for Advanced Studies, Baltimore (1963)Google Scholar
  145. 145.
    Hale, J.K.: Critical cases for neutral functional differential equations. J. Differ. Equat. 10, 59–82 (1971)MathSciNetzbMATHGoogle Scholar
  146. 146.
    Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)zbMATHGoogle Scholar
  147. 147.
    Hale, J.K.: Flows on centre manifolds for scalar functional differential equations. Proc. Math. Roy. Soc. Edinb. 101A, 193–201 (1985)MathSciNetGoogle Scholar
  148. 148.
    Hale, J.K.: Large diffusivity and asymptotic behavior in parabolic systems. J. Differ. Equat. 118, 455–466 (1986)MathSciNetzbMATHGoogle Scholar
  149. 149.
    Hale, J.K.: Partial neutral functional-differential equations. Rev. Roum. Math. Pure. Appl. 39, 339–344 (1994)MathSciNetzbMATHGoogle Scholar
  150. 150.
    Hale, J.K.: Diffusive coupling, dissipation, and synchronization. J. Dynam. Differ. Equat. 9(1), 1–52 (1997)MathSciNetzbMATHGoogle Scholar
  151. 151.
    Hale, J.K., Huang, W.: Period doubling in singularly perturbed delay equations. J. Differ. Equat. 114, 1–23 (1994)MathSciNetzbMATHGoogle Scholar
  152. 152.
    Hale, J.K., Kocak, H.: Dynamics and Bifurcations. Springer, New York (1991)zbMATHGoogle Scholar
  153. 153.
    Hale, J.K., Tanaka, S.M.: Square and pulse waves with two delays. J. Dynam. Differ. Equat. 12, 1–30 (2000)MathSciNetzbMATHGoogle Scholar
  154. 154.
    Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)zbMATHGoogle Scholar
  155. 155.
    Hale, J.K., Weedermann, M.: On perturbations of delay differential equations with periodic orbits. J. Differ. Equat. 197, 219–246 (2004)MathSciNetzbMATHGoogle Scholar
  156. 156.
    Hartung, F.: Linearized stability in periodic functional differential equations with state-dependent delays. J. Comput. Appl. Math. 174, 201–211 (2005)MathSciNetzbMATHGoogle Scholar
  157. 157.
    Hartung, F., Turi, J.: On differentiability of solutions with respect to parameters in state-dependent delay equations. J. Differ. Equat. 135, 192–237 (1997)MathSciNetzbMATHGoogle Scholar
  158. 158.
    Hartung, F., Krisztin, T., Walther, H.-O., Wu, J.: Functional differential equations with state-dependent delays: theory and applications. In: Canada, A. (ed.) Handbook of Differential Equations: Ordinary Differential Equations, vol. 3. Elsevier, North Holland (2006)Google Scholar
  159. 159.
    Hassard, B.D., Wan, Y.H.: Bifurcation formulae derived from center manifold theory. J. Math. Appl. Math. 42, 297–260 (1978)MathSciNetGoogle Scholar
  160. 160.
    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  161. 161.
    Hartman, P.: A lemma in the theory of structural stability of differential equations. Proc. Am. Math. Soc. 11, 610–620 (1960)MathSciNetzbMATHGoogle Scholar
  162. 162.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)zbMATHGoogle Scholar
  163. 163.
    Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra. Academic, New York (1974)zbMATHGoogle Scholar
  164. 164.
    Hirsch, M.W., Push, C.C., Shub, M.: Invariant Manifolds. Springer Lecture Notes in Mathematics, vol. 583. Springer, New York (1977)Google Scholar
  165. 165.
    Hirschberg, P., Knobloch, E: Silnikov-Hopf bifurcation. Phys. D 62, 202–216 (1993)MathSciNetzbMATHGoogle Scholar
  166. 166.
    Holmes, P.J.: Unfolding a degenerate nonlinear oscillators: a codimension two bifurcation. In: Helleman, R.H.G. (ed.) Nonlinear Dynamics, pp. 473–488. New York Academy of Science, New York (1980)Google Scholar
  167. 167.
    Hopf, E.: Abzweigung einer periodischen lösung eines Differential Systems. Berichen Math. Phys. Kl. Säch. Akad. Wiss. Leipzig 94, 1–22 (1942)Google Scholar
  168. 168.
    Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci. U.S.A. 81, 3088–3092 (1984)Google Scholar
  169. 169.
    Hsu, I.D., Kazarinoff, N.D.: An applicable Hopf bifurcation formula and instability of small periodic solutions of the Field-Noyes model. J. Math. Anal. Appl. 55, 61–89 (1976)MathSciNetGoogle Scholar
  170. 170.
    Hu, Q., Wu, J.: Global Hopf bifurcation for differential equations with state-dependent delay. J. Differ. Equat. 248, 2801–2840 (2010)MathSciNetzbMATHGoogle Scholar
  171. 171.
    Hu, Q., Wu, J.: Global continua of rapidly oscillating periodic solutions of state-dependent delay differential equations. J. Dynam. Differ. Equat. 22, 253–284 (2010)MathSciNetzbMATHGoogle Scholar
  172. 172.
    Hu, Q., Wu, J., Zou, X.: Estimates of periods and global continua of periodic solutions of differential equations with state-dependent delay. SIAM J. Math. Anal. 44, 2401–2427 (2012)MathSciNetzbMATHGoogle Scholar
  173. 173.
    Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1978)zbMATHGoogle Scholar
  174. 174.
    Iooss, G.: Travelling waves in the Fermi-Pasta-Ulam lattice. Nonlinearity 13, 849–866 (2000)MathSciNetzbMATHGoogle Scholar
  175. 175.
    Iooss, G., Adelmeyer, M.: Topics in Bifurcation Theory and Applications. World Scientific, Singapore (1992)zbMATHGoogle Scholar
  176. 176.
    Iooss, G., Langford, W.F.: Conjectures on the routes to turbulence via bifurcation. In: Helleman, R.H.G. (ed.) Nonlinear Dynamics, pp. 489–505. New York Academy of Science, New York (1980)Google Scholar
  177. 177.
    Ize, J., Bifurcation Theory for Fredholm Operators, vol. 174. Memoirs of the American Mathematical Society, Providence (1976)Google Scholar
  178. 178.
    Ize, J.: Obstruction theory and multiparameter Hopf bifurcation. Trans. Am. Math. Soc. 289, 757–792 (1985)MathSciNetzbMATHGoogle Scholar
  179. 179.
    Ize, J., Massabó, I., Vignoli, V.: Degree theory for equivariant maps, I. Trans. Am. Math. Soc. 315, 433–510 (1989)zbMATHGoogle Scholar
  180. 180.
    Ize, J., Massabó, I., Vignoli, V.: Degree theory for equivariant maps, the \({\mathbb{S}}^{1}\)-action. Memoirs of the American Mathematical Society, vol. 418. American Mathematical Society, Providence (1992)Google Scholar
  181. 181.
    Ize, J., Vignoli, A.: Equivariant degree for abelian actions, Part I; equivariant homotopy groups. Topol. Methods Nonlinear Anal. 2, 367–413 (1993)MathSciNetzbMATHGoogle Scholar
  182. 182.
    Ize, J., Vignoli, A.: Equivariant degree for abelian actions, Part II; Index computations. Topol. Methods Nonlinear Anal. 7, 369–430 (1996)MathSciNetzbMATHGoogle Scholar
  183. 183.
    Jolly, M.S., Rosa, R.: Computation of non-smooth local centre manifolds. IMA J. Numer. Anal. 25(4), 698–725 (2005)MathSciNetzbMATHGoogle Scholar
  184. 184.
    Joseph, D.D., Sattinger, D.H.: Bifurcating time periodic solutions and their stability. Arch. Ration. Mech. Anal. 45, 79–109 (1972)MathSciNetzbMATHGoogle Scholar
  185. 185.
    Kaplan, L., Yorke, J.A.: Ordinary differential equations which yield periodic solutions of differential delay equations. J. Math. Anal. Appl. 48(2), 317–324 (1974)MathSciNetzbMATHGoogle Scholar
  186. 186.
    Kato, T.: Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, vol. 132. Springer, New York (1976)Google Scholar
  187. 187.
    Keener, J.: Infinite period bifurcation and global bifurcation branches. SIAM J. Appl. Math. 41, 127–144 (1981)MathSciNetzbMATHGoogle Scholar
  188. 188.
    Keller, H.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz, P. (ed.) Applications of Bifurcation Theory, pp. 359–384. Academic, New York (1977)Google Scholar
  189. 189.
    Kelley, A.: The stable, center-stable, center, center-unstable and unstable manifolds. J. Differ. Equat. 3, 546–570 (1967)MathSciNetzbMATHGoogle Scholar
  190. 190.
    Kielhöfer, H.: Hopf bifurcation at multiple eigenvalues. Arch. Ration. Mech. Anal. 69, 53–83 (1979)zbMATHGoogle Scholar
  191. 191.
    Krasnosel’skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, New York (1964)Google Scholar
  192. 192.
    Krawcewicz, W., Vivi, P.: Normal bifurcation and equivariant degree. Indian J. Math. 42, 55–68 (2000)MathSciNetzbMATHGoogle Scholar
  193. 193.
    Krawcewicz, W., Wu, J.: Theory of Degrees with Applications to Bifurcations and Differential Equations. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)zbMATHGoogle Scholar
  194. 194.
    Krawcewicz, W., Wu, J.: Theory and applications of Hopf bifurcations in symmetric functional-differential equations. Nonlinear Anal. Theor. Meth. Appl. 35(7), 845–870 (1999)MathSciNetzbMATHGoogle Scholar
  195. 195.
    Krawcewicz, W., Wu, J., Xia, H.: Global Hopf bifurcation theory for condensing fields and neutral equations with applications to lossless transmission problems. Can. Appl. Math. Q. 1, 167–220 (1993)MathSciNetzbMATHGoogle Scholar
  196. 196.
    Krisztin, T.: A local unstable manifold for differential equations with state-dependent delay. Discrete Contin. Dyn. Syst. 9, 993–1028 (2003)MathSciNetzbMATHGoogle Scholar
  197. 197.
    Krisztin, T., Walther, H.-O., Wu, J.: Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback. The Fields Institute Monograph Series. American Mathematical Society, Providence (1999)zbMATHGoogle Scholar
  198. 198.
    Kuang, K.: Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology. Jpn. J. Ind. Appl. Math. 9, 205–238 (1992)MathSciNetzbMATHGoogle Scholar
  199. 199.
    Kulenovic, M.R.S., Ladas, G.: Linearized oscillations in population dynamics. Bull. Math. Biol. 49, 615–627 (1987)MathSciNetzbMATHGoogle Scholar
  200. 200.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol. 112, 2nd edn. Springer, Berlin (1998)Google Scholar
  201. 201.
    Kuznetsov, Y.A.: Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODEs. SIAM J. Numer. Anal. 36, 1104–1124 (1999)MathSciNetzbMATHGoogle Scholar
  202. 202.
    Langford, W.F.: Periodic and steady-state mode interactions lead to tori. SIAM J. Appl. Math. 37, 649–686 (1979)MathSciNetGoogle Scholar
  203. 203.
    Langford, W.F.: Chaotic dynamics in the unfoldings of degenerate bifurcations. In: Proceedings of the International Symposium on Applied Mathematics and Information Science, Kyoto University, Japan, pp. 241–247 (1982)Google Scholar
  204. 204.
    Langford, W.F.: A review of interactions of Hopf and steady-state bifurcations. In: Barenblatt, G.I., Iooss, G., Joseph, D.D. (eds.) Nonlinear Dynamics and Turbulence, pp. 215–237. Pitman Advanced Publishing Program, Boston (1983)Google Scholar
  205. 205.
    Langford, W.F.: Hopf bifurcation at a hysteresis point. In: Szõkefalvi-Nagy, B., Hatvani, L. (eds.) Differential Equations: Qualitative Theory, Colloq. Math. Soc. János Bolyai, vol. 47, pp. 649–686. North Holland, Amsterdam (1987)Google Scholar
  206. 206.
    Lenhart, S.N., Travis, C.C.: Stability of functional partial differential equations. J. Differ. Equat. 58, 212–227 (1985)MathSciNetzbMATHGoogle Scholar
  207. 207.
    Leray, J., Schauder, J.: Topologie et équations fonctionnelles. Ann. Sci. Ecole. Norm. Sup. 51, 45–78 (1934)MathSciNetGoogle Scholar
  208. 208.
    Levinger, B.W.: A Folk theorem in functional differential equations. J. Differ. Equat. 4, 612–619 (1968)MathSciNetzbMATHGoogle Scholar
  209. 209.
    Li, S., Liao, X., Li, C., Wong, K.-W.: Hopf bifurcation of a two-neuron network with different discrete time delays. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 15, 1589–1601 (2005)MathSciNetzbMATHGoogle Scholar
  210. 210.
    Li, M.Y., Muldowney, J.S.: On Bendixson’s criterion. J. Differ. Equat. 106, 27–39 (1993)MathSciNetzbMATHGoogle Scholar
  211. 211.
    Ma, T., Wang, S.: Bifurcation theory and applications. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 53. World Scientific, Hackensack, NJ (2005)Google Scholar
  212. 212.
    Mallet-Paret, J.: Generic periodic solutions of functional differential equation. J. Differ. Equat. 25, 163–183 (1977)MathSciNetzbMATHGoogle Scholar
  213. 213.
    Mallet-Paret, J.: Morse decomposition for delay differential equations. J. Differ. Equat. 72, 270–315 (1988)MathSciNetzbMATHGoogle Scholar
  214. 214.
    Mallet-Paret, J., Nussbaum, R.: Global continuation and asymptotic behavior for periodic solutions of a delay differential equation. Ann. Math. Pura Appl. 145, 33–128 (1986)MathSciNetzbMATHGoogle Scholar
  215. 215.
    Mallet-Paret, J., Nussbaum, R.D., Paraskevopoulos, P.: Periodic solutions for functional-differential equations with multiple state-dependent time lags. Topol. Methods Nonlinear Anal. 3, 101–162 (1994)MathSciNetzbMATHGoogle Scholar
  216. 216.
    Mallet-Paret, J., Yorke, J.A.: Snakes: oriented families of periodic orbits, their sources, sinks and continuation. J. Differ. Equat. 43, 419–450 (1982)MathSciNetzbMATHGoogle Scholar
  217. 217.
    Marsden, J., McCracken, M.: The Hopf Bifurcation and Its Applications. Applied Mathematical Sciences, vol. 19. Springer, New York (1976)Google Scholar
  218. 218.
    Medvedev, V.: On a new type of bifurcations on manifolds. Mat. Sbornik 113, 487–492 (1980) (in Russian)Google Scholar
  219. 219.
    Memory, M.C.: Bifurcation and asymptotic behaviour of solutions of a delay-differential equation with diffusion. SIAM J. Math. Anal. 20, 533–546 (1989)MathSciNetzbMATHGoogle Scholar
  220. 220.
    Memory, M.C.: Stable and unstable manifolds for partial functional differential equations. Nonlinear Anal. 16, 131–142 (1991)MathSciNetzbMATHGoogle Scholar
  221. 221.
    Memory, M.C.: Invariant manifolds for partial functional differential equations. In: Arino, O., Axelrod, D.E., Kimmel, M. (eds.) Mathematical Population Dynamics, pp. 223–232. Marcel Dekker, New York (1991)Google Scholar
  222. 222.
    Metz, J.Z., Diekmann, O.: The Dynamics of Physiologically Structured Populations. Springer, New York (1986)zbMATHGoogle Scholar
  223. 223.
    Michel, L.: Points critiques des fonctions G-invariantes. Note aux Comptes-Rendus Acad. Sci. Paris sér. A-B 272, A433–A436 (1971)Google Scholar
  224. 224.
    Milton, J.: Dynamics of Small Neural Populations. American Mathematical Society, Providence, RI (1996)zbMATHGoogle Scholar
  225. 225.
    Morita, Y.: Destablization of periodic solutions arising in delay-diffusion systems in several space dimensions. Jpn. J. Appl. Math. 1, 39–65 (1984)zbMATHGoogle Scholar
  226. 226.
    Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky Mt. J. Math. 20, 857–871 (1990)MathSciNetzbMATHGoogle Scholar
  227. 227.
    Munkres, J.: Topology, 2nd edn. Prentice Hall, Englewood Cliffs (1975)zbMATHGoogle Scholar
  228. 228.
    Negrini, P., Salvadori, L.: Attractivity and Hopf bifurcation. Nonlinear Anal. 3, 87–99 (1979)MathSciNetGoogle Scholar
  229. 229.
    Neimark, J.I.: Motions close to doubly-asymptotic motion. Soviet Math. Dokl. 8, 228–231 (1967)Google Scholar
  230. 230.
    Newhouse, S., Palis, J., Takens, F.: Bifurcations and stability of families of diffeomorphisms. Publ. Math. Inst. Hautes Etud. Sci. 57, 5–71 (1983)MathSciNetzbMATHGoogle Scholar
  231. 231.
    Nicholson, A.J.: An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–65 (1954)Google Scholar
  232. 232.
    Nussbaum, R.D.: Periodic solutions of some nonlinear functional differential equations. Ann. Math. Pura Appl. 101, 263–338 (1974)MathSciNetzbMATHGoogle Scholar
  233. 233.
    Nussbaum, R.D.: A global bifurcation theory with application to functional differential equations. J. Funct. Anal. 19, 319–338 (1975)MathSciNetzbMATHGoogle Scholar
  234. 234.
    Nussbaum, R.D.: Global bifurcation of periodic solutions of some autonomous functional differential equations. J. Math. Anal. Appl. 55, 699–725 (1976)MathSciNetzbMATHGoogle Scholar
  235. 235.
    Nussbaum, R.D.: The range of periods of periodic solutions of x′(t) =  − αf(x(t − 1)). J. Math. Anal. Appl. 58, 280–292 (1977)MathSciNetzbMATHGoogle Scholar
  236. 236.
    Nussbaum, R.D.: A global Hopf bifurcation theorem of functional differential systems. Trans. Am. Math. Soc. 238, 139–164 (1978)MathSciNetzbMATHGoogle Scholar
  237. 237.
    Nussbaum, R.D.: Circulant matrices and differential-delay equations. J. Differ. Equat. 60, 201–217 (1985)MathSciNetzbMATHGoogle Scholar
  238. 238.
    Olien, L., Bélair, J.: Bifurcations, stability, and monotonicity properties of a delayed neural network model. Phys. D 102, 349–363 (1997)MathSciNetzbMATHGoogle Scholar
  239. 239.
    Oster, G., Ipaktchi, A.: Population cycles. In: Eyring, H. (ed.) Periodicities in Chemistry and Biology, pp. 111–132. Academic, New York (1978)Google Scholar
  240. 240.
    Palais, R.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)MathSciNetzbMATHGoogle Scholar
  241. 241.
    Palis, J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, New York (1982)zbMATHGoogle Scholar
  242. 242.
    Palis, J., Pugh, C.: Fifty problems in dynamical systems. In: Manning, A. (ed.) Dynamical Systems – Warwick 1974, vol. 468 of Lecture Notes in Mathematics, pp. 345–353. Springer, Berlin (1975)Google Scholar
  243. 243.
    Palis, J., Takens, F.: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations: Fractal Dimensions and Infinitely Many Attractors, vol. 35 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1993)Google Scholar
  244. 244.
    Peixoto, M.M.: Structural stability on two-dimensional manifolds. Topology 1, 101–120 (1962)MathSciNetzbMATHGoogle Scholar
  245. 245.
    Pliss, V.: Principal reduction in the theory of stability of motion. Izv. Akad. Nauk. SSSR Math. Ser. 28, 1297–1324 (1964) (in Russian)MathSciNetzbMATHGoogle Scholar
  246. 246.
    Poincaré, H.: Sur les propriétés des fonctions définies par les équations aux différences partielles. Thése. Gauthier-Villars, Paris (1879)Google Scholar
  247. 247.
    Poincaré, H.: Mémoire sur les courbes définis par une equation différentielle IV. J. Math. Pures Appl. 1, 167–244 (1885)zbMATHGoogle Scholar
  248. 248.
    Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste, vol. I. Cauthier-Villars, Paris (1892)Google Scholar
  249. 249.
    Pontryagin, L.: On the dynamical systems close to Hamiltonian systems. J. Exp. Theor. Phys. 4, 234–238 (1934) (in Russian)Google Scholar
  250. 250.
    Poore, A.B.: On the theory and application of the Hopf-Friedrichs bifurcation theory. Arch. Ration. Mech. Anal. 60, 371–393 (1976)MathSciNetzbMATHGoogle Scholar
  251. 251.
    Rabinowitz, P.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)MathSciNetzbMATHGoogle Scholar
  252. 252.
    Ruan, S., Filfil, R.F.: Dynamics of a two-neuron system with discrete and distributed delays. Phys. D 191, 323–342 (2004)MathSciNetzbMATHGoogle Scholar
  253. 253.
    Ruan, S., Wei, J.: Periodic solutions of planar systems with two delays. Proc. Math. Roy. Soc. Edinb. 129, 1017–1032 (1999)MathSciNetzbMATHGoogle Scholar
  254. 254.
    Rudin, W.: Functional Analysis. McGraw-Hill Science, New York (1991)zbMATHGoogle Scholar
  255. 255.
    Ruelle, D.: Bifurcations in the presence of a symmetry group. Arch. Ration. Mech. Anal. 51, 136–152 (1973)MathSciNetzbMATHGoogle Scholar
  256. 256.
    Ruelle, D., Takens, F.: On the nature of turbulence. Comm. Math. Phys. 20, 167–192, and 23, 343–344 (1971)Google Scholar
  257. 257.
    Rustichini, A.: Hopf bifurcation for functional differential equations of mixed type. J. Dynam. Differ. Equat. 1, 145–177 (1989)MathSciNetzbMATHGoogle Scholar
  258. 258.
    Sacker, R.: On invariant surfaces and bifurcations of periodic solutions of ordinary differential equations. Report IMM-NYU 333, New York University (1964)Google Scholar
  259. 259.
    Sanders, J.: On the computation of normal forms. Computational aspects of Lie group representations and related topics. In: Cohen, A.M. (ed.) Proceedings of the 1990 Computational Algebra Seminar, CWI Tracts 84, Amsterdam, pp. 129–142 (1991)Google Scholar
  260. 260.
    Sattinger, D.H.: Bifurcation of periodic solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 41, 66–80 (1971)MathSciNetzbMATHGoogle Scholar
  261. 261.
    Sattinger, D.H.: Bifurcation and symmetry breaking in applied mathematics. Bull. Am. Math. Soc. 3, 779–819 (1980)MathSciNetzbMATHGoogle Scholar
  262. 262.
    Shayer, L.P., Campbell, S.A.: Stability, bifurcation, and multistability in a system of two coupled neurons with multiple time delays. SIAM J. Appl. Math. 61, 673–700 (2000)MathSciNetzbMATHGoogle Scholar
  263. 263.
    Shil’nikov, L.P.: On a Poincaré-Birkhoff problem. Math. USSR-Sb. 3, 353–371 (1967)Google Scholar
  264. 264.
    Shu, Y., Wang, L., Wu, J.: Global dynamics of the Nicholson’s blowflies equation revisited: onset and termination of nonlinear oscillations (preprint)Google Scholar
  265. 265.
    Sieberg, H.W.: Some historical remarks concerning degree theory. Am. Math. Mon. 88, 125–139 (1981)Google Scholar
  266. 266.
    Sijbrand, J.: Properties of center manifolds. Trans. Am. Math. Soc. 289, 431–469 (1985)MathSciNetzbMATHGoogle Scholar
  267. 267.
    Skinner, F.K., Bazzazi, H., Campbell, S.A.: Two-cell to N-cell heterogeneous, inhibitory networks: precise linking of multistable and coherent properties. J. Comput. Neurosci. 18, 343–352 (2005)MathSciNetGoogle Scholar
  268. 268.
    Smale, S.: Diffeomorphisms with many periodic points. In: Carins, S. (ed.) Differential and Combinatorial Topology, pp. 63–80. Princeton University Press, Princeton, NJ (1963)Google Scholar
  269. 269.
    Smith, H.L.: Hopf bifurcation in a system of functional equations modelling the spread of infectious disease. SIAM J. Appl. Math. 43, 370–385 (1983)MathSciNetzbMATHGoogle Scholar
  270. 270.
    Staffans, O.J.: Hopf bifurcation for functional and functional differential equations with infinite delay. J. Differ. Equat. 70, 114–151 (1987)MathSciNetzbMATHGoogle Scholar
  271. 271.
    Stech, H.: Hopf bifurcation calculations for functional differential equations. J. Math. Anal. Appl. 1109, 472–491 (1985)MathSciNetGoogle Scholar
  272. 272.
    Takens, F.: A nonstabilizable jet of a singularity of a vector field. In: Dynamical Systems (Proceedings Symposium, University of Bahia, Salvador, 1971), pp. 583–597. Academic, New York (1973)Google Scholar
  273. 273.
    Takens, F.: Normal forms for certain singularities of vector fields. Ann. Inst. Fourier (Grenoble) 23, 163–195 (1973)Google Scholar
  274. 274.
    Takens, F.: Singularities of vector fields. Publ. Math. IHES 43, 47–100 (1974)MathSciNetGoogle Scholar
  275. 275.
    Thom, R.: Topological models in biology. Topology 8, 313–335 (1969)MathSciNetzbMATHGoogle Scholar
  276. 276.
    Thom, R.: Stabilité structurelle et morphogénése. Benjamin, New York (1972)Google Scholar
  277. 277.
    Tsiligiannis, C.A., Lyberatos, G.: Normal forms, resonance and bifurcation analysis via the Carleman linearization. J. Math. Anal. Appl. 139, 123–138 (1989)MathSciNetzbMATHGoogle Scholar
  278. 278.
    Tu, F., Liao, X., Zhang, W.: Delay-dependent asymptotic stability of a two-neuron system with different time delays. Chaos Solitons Fractals 28, 437–447 (2006)MathSciNetzbMATHGoogle Scholar
  279. 279.
    Turaev, D., Shil’nikov, L.: Blue sky catastrophes. Dokl. Math. 51, 404–407 (1995)zbMATHGoogle Scholar
  280. 280.
    Ushiki, S.: Normal forms for singularities of vector fields. Jpn. J. Appl. Math. 1, 1–34 (1984)MathSciNetzbMATHGoogle Scholar
  281. 281.
    van der Pol, B.: A theory of the amplitude of free and forced triode vibrations. Radio Rev. 1, 701–710, 754–762 (1920)Google Scholar
  282. 282.
    van Gils, S.A., Valkering, T.: Hopf bifurcation and symmetry: standing and traveling waves in a circular–chain. Jpn. J. Appl. Math. 3, 207–222 (1986)zbMATHGoogle Scholar
  283. 283.
    Vanderbauwhede, A.: Symmetry and bifurcation near families of solutions. J. Diff. Equat. 36, 173–178 (1980)MathSciNetzbMATHGoogle Scholar
  284. 284.
    Vanderbauwhede, A.: Local Bifurcation and Symmetry. Research Notes in Mathematics, vol. 75. Pitman, London (1982)Google Scholar
  285. 285.
    Vanderbauwhede, A.: Center manifolds, normal forms and elementary bifurcations. Dynamics Reported, vol. 2. Wiley, New York (1989)Google Scholar
  286. 286.
    Vanderbauwhede, A., Iooss, G.: Center manifold theory in infinite dimension. Dynam. Report. Exposition Dynam. Syst. (N.S.) 1, 125–163 (1992)Google Scholar
  287. 287.
    Vidossich, G.: On the structure of periodic solutions of differential equations. J. Differ. Equat. 21, 263–278 (1976)MathSciNetzbMATHGoogle Scholar
  288. 288.
    Walther, H.-O.: A theorem on the amplitudes of periodic solutions of differential delay equations with application to bifurcation. J. Differ. Equat. 29, 396–404 (1978)MathSciNetzbMATHGoogle Scholar
  289. 289.
    Walther, H.-O.: Bifurcation from periodic solutions in functional differential equations. Math. Z. 182, 269–290 (1983)MathSciNetzbMATHGoogle Scholar
  290. 290.
    Walther, H.-O.: The solution manifold and C 1 smoothness of solution operators for differential equations with state-dependent delay. J. Differ. Equat. 195, 46–65 (2003)MathSciNetzbMATHGoogle Scholar
  291. 291.
    Walther, H.-O.: Bifurcation of periodic solutions with large periods for a delay differential equation. Ann. Math. Pura Appl. 185(4), 577–611 (2006)MathSciNetzbMATHGoogle Scholar
  292. 292.
    Weedermann, M.: Normal forms for neutral functional differential equations. Field. Inst. Comm. 29, 361–368 (2001)MathSciNetGoogle Scholar
  293. 293.
    Weedermann, M.: Hopf bifurcation calculations for scalar delay differential equations. Nonlinearity 19, 2091–2102 (2006)MathSciNetzbMATHGoogle Scholar
  294. 294.
    Wei, J., Li, M.Y.: Hopf bifurcation analysis in a delayed Nicholson blowflies equation. Nonlinear Anal. 60, 1351–1367 (2005)MathSciNetzbMATHGoogle Scholar
  295. 295.
    Wei, J., Ruan, S.: Stability and bifurcation in a neural network model with two delays. Phys. D 130, 255–272 (1999)MathSciNetzbMATHGoogle Scholar
  296. 296.
    Wei, J.J., Velarde, M.G.: Bifurcation analysis and existence of periodic solutions in a simple neural network with delays. Chaos 14, 940–953 (2004)MathSciNetzbMATHGoogle Scholar
  297. 297.
    Wiggins, S.: Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Springer, New York (1994)zbMATHGoogle Scholar
  298. 298.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, New York (2003)zbMATHGoogle Scholar
  299. 299.
    Wittenberg, R.W., Holmes, P.: The limited effectiveness of normal forms: a critical review and extension of local bifurcation studies of the Brusselator PDE. Phys. D 100, 1–40 (1997)MathSciNetzbMATHGoogle Scholar
  300. 300.
    Wright, E.M.: A nonlinear differential difference equation. J. Reine Angew. Math. 194, 66–87 (1955)MathSciNetzbMATHGoogle Scholar
  301. 301.
    Wu, J.: Global continua of periodic solutions to some differential equations of neutral type. Tôhoku Math J. 45, 67–88 (1993)zbMATHGoogle Scholar
  302. 302.
    Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)zbMATHGoogle Scholar
  303. 303.
    Wu, J.: Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998)zbMATHGoogle Scholar
  304. 304.
    Wu, J.: Introduction to Neural Dynamics and Signal Transmission Delay. Walter de Gruyter, Berlin (2001)zbMATHGoogle Scholar
  305. 305.
    Wu, J., Xia, H.: Self-sustained oscillations in a ring array of coupled lossless transmission lines. J. Differ. Equat. 124 247–278 (1996)MathSciNetzbMATHGoogle Scholar
  306. 306.
    Wu, J., Xia, H.: Rotating waves in neutral partial functional-differential equations. J. Dynam. Differ. Equat. 11, 209–238 (1999)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Shangjiang Guo
    • 1
  • Jianhong Wu
    • 2
  1. 1.College of Mathematics and EconometricsHunan UniversityChangshaChina, People’s Republic
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations