Skip to main content
SpringerLink
Log in

Lyapunov–Schmidt Reduction

  • Chapter
  • First Online:
Book cover Bifurcation Theory of Functional Differential Equations

Part of the book series: Applied Mathematical Sciences ((AMS,volume 184))

  • 3881 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    L is a Fredholm operator if (i) the kernel \(\mathrm{Ker}\mathcal{L}\) is finite-dimensional, (ii) the range \(\mathrm{Ran}\mathcal{L}\) is closed, and (iii) \(\mathrm{Ran}\mathcal{L}\) has finite codimension in Y. The index of a Fredholm operator \(\mathcal{L}\) is defined to be the integer \(\mathrm{Ind}\mathcal{L} =\dim \mathrm{ Ker}\mathcal{L}-\mathrm{ codim}\mathrm{Ran}\mathcal{L}\).

References

  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)

    MathSciNet  Google Scholar 

  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. Adimy, M.: Integrated semigroups and delay differential equations. J. Math. Anal. Appl. 177, 125–134 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Afraimovich, V., Shil’nikov, L.: On singular trajectories of dynamical systems. Usp. Mat. Nauk 5, 189–190 (1972) (in Russian)

    Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  8. Alexander, J.C.: Bifurcation of zeros of parametrized functions. J. Funct. Anal. 29, 37–53 (1978)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  10. Alexander, J.C., Yorke, J.A.: Global bifurcations of periodic orbits. Am. J. Math. 100, 263–292 (1978)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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. 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. 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. 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. Andronov, A.A., Pontryagin, L.: Systémes grossiéres. Dokl. Akad. Nauk SSSR 14, 247–251 (1937) (in Russian).

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  19. Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1983)

    MATH  Google Scholar 

  20. Arnold, V.I.: Lectures on bifurcations in versal families. Russ. Math. Surv. 27, 54–123 (1972)

    Google Scholar 

  21. Arrowsmith, D.K., Place, C.M.: An Introduction to Dynamical Systems. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  22. Ashkenazi, M., Chow, S.N.: Normal forms near critical points for differential equations and maps. IEEE Trans. Circuits Syst. 35, 850–862 (1988)

    MathSciNet  MATH  Google Scholar 

  23. Aubin, J.P.: Applied Functional Analysis. Wiley, New York (1979)

    MATH  Google Scholar 

  24. Balanov, Z., Krawcewicz, W.: Remarks on the equivariant degree theory. Topol. Methods Nonlinear Anal. 13, 91–103 (1999)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  26. Bélair, J.: Population models with state-dependent delays. Lect. Notes Pure Appl. Math. 131, 165–176 (1991)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  29. Bélair, J., Dufour, S.: Stability in a three-dimensional system of delay-differential equations. Can. Appl. Math. Q. 4(2), 135–156 (1996)

    MATH  Google Scholar 

  30. Bellman, R., Cooke, K.L.: Differential Difference Equations. Academic Press, New York (1963)

    MATH  Google Scholar 

  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. Bernfeld, S.R., Negrini, P., Salvadori, L.: Quasi-invariant manifolds stability and generalized Hopf bifurcation. Ann. Math. Pura Appl. 4, 105–119 (1982)

    MathSciNet  Google Scholar 

  33. Birkhoff, G.D.: Dynamical Systems. AMS, Providence (1927)

    MATH  Google Scholar 

  34. Birkhoff, G.D.: Nouvelles recherches sur les systèmes dynamiques. Memoriae Pont. Acad. Sci. Novi. Lincaei Ser. 3 1, 85–216 (1935)

    MathSciNet  Google Scholar 

  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. 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. Brayton, R.K.: Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type. Q. Appl. Math. 24, 215–224 (1966)

    MathSciNet  MATH  Google Scholar 

  38. Brayton, R.K.: Nonlinear oscillations in a distributed network. Q. Appl. Math. 24, 289–301 (1967)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  40. Brayton, R.K., Moser, J.K.: A theory of nonlinear networks. I. Q. Appl. Math. 22, 1–33 (1964)

    MathSciNet  Google Scholar 

  41. Bredon, G.E.: Introduction to Compact Transformation Groups. Academic, New York (1972)

    MATH  Google Scholar 

  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. 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. 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)

    MathSciNet  MATH  Google Scholar 

  45. Brokate, M., Colonius, F.: Linearizing equations with state-dependent delays. Appl. Math. Optim. 21, 45–52 (1990)

    MathSciNet  MATH  Google Scholar 

  46. Brouwder, F.E.: Fixed point theory and nonlinear problems. Bull. Am. Math. Soc. 1, 1–39 (1983)

    Google Scholar 

  47. Brouwer, L.E.J.: Über Abbildung der Mannigfaltigkeiten. Math. Ann. 70, 97–115 (1912)

    Google Scholar 

  48. Bruno, A.D.: Local Method of Nonlinear Analysis of Differential Equations (in Russian). Izdatel’stvo Nauka, Moscow (1979)

    Google Scholar 

  49. Buono, P.L., Bélair, J.: Restrictions and unfolding of double Hopf bifurcation in functional differential equations. J. Differ. Equat. 189, 234–266 (2003)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  51. Busenberg, S., Travis, C.C.: On the use of reducible functional differential equations. J. Math. Anal. Appl. 89, 46–66 (1982)

    MathSciNet  MATH  Google Scholar 

  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. Carr, J.: Applications of Centre Manifold Theory. Applied Mathematical Sciences, vol. 35. Springer, New York (1981)

    Google Scholar 

  54. Chafee, N.: A bifurcation problem for a functional differential equation of finitely retarded type. J. Math. Anal. Appl. 35, 312–348 (1971)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. Chen, G., Della Dora, J.: Further reduction of normal forms for dynamical systems. J. Differ. Equat. 166, 79–106 (2000)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  62. Chossat, P., Lauterbach, R.: Methods in Equivariant Bifurcations and Dynamical Systems. World Scientific, Singapore (2000)

    MATH  Google Scholar 

  63. Chow, S.N.: Existence of periodic solutions of autonomous functional differential equations. J. Differ. Equat. 15, 350–378 (1974)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  65. Chow, S.N., Hale, J.: Methods of Bifurcation Theory. Springer, New York (1982)

    MATH  Google Scholar 

  66. Chow, S.-N., Li, C., Wang, D.: Normal Forms and Bifurcations of Planar Vector Fields. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  68. Chow, S.N., Mallet-Paret, J.: Integral averaging and bifurcation. J. Differ. Equat. 26, 112–159 (1977)

    MathSciNet  MATH  Google Scholar 

  69. Chow, S.N., Mallet-Paret, J.: The Fuller index and global Hopf bifurcation. J. Differ. Equat. 29, 66–85 (1978)

    MathSciNet  MATH  Google Scholar 

  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. Chow, S.N., Mallet-Paret, J., Yorke, J.A.: Global Hopf bifurcation from a multiple eigenvalue. Nonlinear Anal. 2, 753–763 (1978)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  73. Coppel, W.A.: Stability and Asymptotic Behavior of Differential Equations. Health, Boston (1965)

    MATH  Google Scholar 

  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. Cicogna, G.: Symmetry breakdown from bifurcation. Lettere al Nuovo Cimento 31, 600–602 (1981)

    MathSciNet  Google Scholar 

  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. 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. Cushman, R., Sanders, J.A.: Splitting algorithm for nilpotent normal forms. Dynam. Stabil. Syst. 2(3–4), 235–246 (1988)

    MathSciNet  Google Scholar 

  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. de Oliveira, J.C., Hale, J.K.: Dynamic behavior from the bifurcation function. Tôhoku Math. J. 32, 577–592 (1980)

    MATH  Google Scholar 

  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. Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay Equations, Functional-, Complex-, and Nonlinear Analysis. Springer, New York (1995)

    MATH  Google Scholar 

  83. Dormayer, P.: Smooth bifurcation of symmetric periodic solutions of functional-differential equations. J. Differ. Equat. 82, 109–155 (1989)

    MathSciNet  MATH  Google Scholar 

  84. Dumortier, F., Ibáñez, S.: Singularities of vector fields on \({\mathbb{R}}^{3}\). Nonlinearity 11, 1037–1047 (1998)

    MathSciNet  MATH  Google Scholar 

  85. Dylawerski, G., Gȩba, K., Jodel, J., Marzantowicz, W.: S 1-equivalent degree and the Fuller index. Ann. Polon. Math. 52, 243–280 (1991)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  94. Feigenbaum, M.J.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19(1), 25–52 (1978)

    MathSciNet  MATH  Google Scholar 

  95. Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)

    MathSciNet  MATH  Google Scholar 

  96. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equat. 31, 53–98 (1979)

    MathSciNet  MATH  Google Scholar 

  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. Field, M.J.: Lectures on Bifurcations, Dynamics and Symmetry. Pitman Research Notes in Mathematics, vol. 356. Longman, Harlow (1996)

    Google Scholar 

  99. Field, M.J., Melbourne, I., Nicol, M.: Symmetric attractors for diffeomorphisms and flows. Proc. Lond. Math. Soc. 72, 657–696 (1996)

    MathSciNet  MATH  Google Scholar 

  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. Fiedler, B.: An index for global Hopf bifurcation in parabolic systems. J. Reine Angew. Math. 359, 1–36 (1985)

    MathSciNet  MATH  Google Scholar 

  102. Fiedler, B.: Global Bifurcation of Periodic Solutions with Symmetry. Lecture Notes in Mathematics, vol. 1309. Springer, New York (1988)

    Google Scholar 

  103. Fiedler, M.: Additive compound matrices and inequality for eigenvalues of stochastic matrices. Czech. Math. J. 99, 392–402 (1974)

    MathSciNet  Google Scholar 

  104. Filip, A.M., Venakides, S.: Existence and modulation of traveling waves in particle chains. Comm. Pure Appl. Math. 52, 693–735 (1999)

    MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  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. Gaspard, P.: Local birth of homoclinic chaos. Phys. D 62, 94–122 (1993)

    MathSciNet  MATH  Google Scholar 

  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. 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. 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. Geba, K., Marzantowicz, W.: Global bifurcation of periodic solutions. Topol. Methods Nonlinear Anal. 1, 67–93 (1993)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  114. Giannakopoulos, F., Zapp, A.: Bifurcations in a planar system of differential delay equations modeling neural activity. Phys. D 159, 215–232 (2001)

    MathSciNet  MATH  Google Scholar 

  115. Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Graduate Texts in Mathematics, vol. 14. Springer, New York (1973)

    Google Scholar 

  116. Golubitsky, M., Marsden, J., Stewart, I., Dellnitz, M.: The constrained Lyapunov-Schmidt procedure and periodic orbits. Field. Inst. Comm. 4, 81–127 (1995)

    MathSciNet  Google Scholar 

  117. Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 1. Springer, New York (1985)

    MATH  Google Scholar 

  118. Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 2. Springer, New York (1988)

    MATH  Google Scholar 

  119. Govaerts, W., Pryce, J.: Mixed block elimination for linear systems with wider borders. IMA J. Numer. Anal. 13, 161–180 (1993)

    MathSciNet  MATH  Google Scholar 

  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. Grafton, R.B.: A periodicity theorem for autonomous functional differential equations. J. Differ. Equat. 6, 87–109 (1969)

    MathSciNet  MATH  Google Scholar 

  122. Grimmer, R.: Existence of periodic solutions of functional differential equations. J Math. Anal. Appl. 72(2), 666–673 (1979)

    MathSciNet  MATH  Google Scholar 

  123. Grobman, D.: Homeomorphisms of systems of differential equations. Dokl. Akad. Nauk SSSR 128, 880 (1959)

    MathSciNet  MATH  Google Scholar 

  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. Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations: Dynamical System and Bifurcations of Vector Fields. Springer, New York (1983)

    Google Scholar 

  126. Guckenheimer, J.: Multiple bifurcation problems of codimension two. SIAM J. Math. Anal. 15, 1–49 (1984)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  128. Guo, S.: Equivariant normal forms for neutral functional differential equations. Nonlinear Dyn. 61(1), 311–329 (2010)

    MATH  Google Scholar 

  129. Guo, S.: Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay. Nonlinearity 18, 2391–2407 (2005)

    MathSciNet  MATH  Google Scholar 

  130. Guo, S.: Zero singularities in a ring network with two delays. Z. Angew. Math. Phys. 64(2), 201–222 (2013)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  133. Guo, S., Huang, L.: Hopf bifurcating periodic orbits in a ring of neurons with delays. Phys. D 183(1–2), 19–44 (2003)

    MathSciNet  MATH  Google Scholar 

  134. Guo, S., Huang, L.: Global continuation of nonlinear waves in a ring of neurons. Proc. Math. Roy. Soc. Edinb. 135A, 999–1015 (2005)

    MathSciNet  Google Scholar 

  135. Guo, S., Huang, L.: Stability of nonlinear waves in a ring of neurons with delays. J. Differ. Equat. 236, 343–374 (2007)

    MathSciNet  MATH  Google Scholar 

  136. Guo, S., Lamb, J.S.W.: Equivariant Hopf bifurcation for neutral functional differential equations. Proc. Am. Math. Soc. 136, 2031–2041 (2008)

    MathSciNet  MATH  Google Scholar 

  137. Guo, S., Lamb, J.S.W., Rink, B.W.: Branching patterns of wave trains in the FPU lattice. Nonlinearity 22, 283–299 (2009)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  139. Guo, S., Man, J.: Patterns in hierarchical networks of neuronal oscillators with D3 xZ3 symmetry. J. Differ. Equat. 254, 3501–3529 (2013)

    MathSciNet  MATH  Google Scholar 

  140. Guo, S., Yuan, Y.: Pattern formation in a ring network with delay. Math. Model. Meth. Appl. Sci. 19(10), 1797–1852 (2009)

    MathSciNet  MATH  Google Scholar 

  141. Guo, S., Wu, J.: Generalized Hopf bifurcation in delay differential equations (in Chinese). Sci. Sin. Math. 42, 91–105 (2012)

    Google Scholar 

  142. Gurney, W.S.C., Blythe, S.P., Nisbee, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)

    Google Scholar 

  143. Hadeler, K.P., Tomiuk, J.: Periodic solutions of difference differential equations. Arch. Ration. Anal. 1, 87–95 (1977)

    MathSciNet  Google Scholar 

  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. Hale, J.K.: Critical cases for neutral functional differential equations. J. Differ. Equat. 10, 59–82 (1971)

    MathSciNet  MATH  Google Scholar 

  146. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

    MATH  Google Scholar 

  147. Hale, J.K.: Flows on centre manifolds for scalar functional differential equations. Proc. Math. Roy. Soc. Edinb. 101A, 193–201 (1985)

    MathSciNet  Google Scholar 

  148. Hale, J.K.: Large diffusivity and asymptotic behavior in parabolic systems. J. Differ. Equat. 118, 455–466 (1986)

    MathSciNet  MATH  Google Scholar 

  149. Hale, J.K.: Partial neutral functional-differential equations. Rev. Roum. Math. Pure. Appl. 39, 339–344 (1994)

    MathSciNet  MATH  Google Scholar 

  150. Hale, J.K.: Diffusive coupling, dissipation, and synchronization. J. Dynam. Differ. Equat. 9(1), 1–52 (1997)

    MathSciNet  MATH  Google Scholar 

  151. Hale, J.K., Huang, W.: Period doubling in singularly perturbed delay equations. J. Differ. Equat. 114, 1–23 (1994)

    MathSciNet  MATH  Google Scholar 

  152. Hale, J.K., Kocak, H.: Dynamics and Bifurcations. Springer, New York (1991)

    MATH  Google Scholar 

  153. Hale, J.K., Tanaka, S.M.: Square and pulse waves with two delays. J. Dynam. Differ. Equat. 12, 1–30 (2000)

    MathSciNet  MATH  Google Scholar 

  154. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)

    MATH  Google Scholar 

  155. Hale, J.K., Weedermann, M.: On perturbations of delay differential equations with periodic orbits. J. Differ. Equat. 197, 219–246 (2004)

    MathSciNet  MATH  Google Scholar 

  156. Hartung, F.: Linearized stability in periodic functional differential equations with state-dependent delays. J. Comput. Appl. Math. 174, 201–211 (2005)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. Hassard, B.D., Wan, Y.H.: Bifurcation formulae derived from center manifold theory. J. Math. Appl. Math. 42, 297–260 (1978)

    MathSciNet  Google Scholar 

  160. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  161. Hartman, P.: A lemma in the theory of structural stability of differential equations. Proc. Am. Math. Soc. 11, 610–620 (1960)

    MathSciNet  MATH  Google Scholar 

  162. Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)

    MATH  Google Scholar 

  163. Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems and Linear Algebra. Academic, New York (1974)

    MATH  Google Scholar 

  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. Hirschberg, P., Knobloch, E: Silnikov-Hopf bifurcation. Phys. D 62, 202–216 (1993)

    MathSciNet  MATH  Google Scholar 

  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. Hopf, E.: Abzweigung einer periodischen lösung eines Differential Systems. Berichen Math. Phys. Kl. Säch. Akad. Wiss. Leipzig 94, 1–22 (1942)

    Google Scholar 

  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. 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)

    MathSciNet  Google Scholar 

  170. Hu, Q., Wu, J.: Global Hopf bifurcation for differential equations with state-dependent delay. J. Differ. Equat. 248, 2801–2840 (2010)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  173. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Springer, New York (1978)

    MATH  Google Scholar 

  174. Iooss, G.: Travelling waves in the Fermi-Pasta-Ulam lattice. Nonlinearity 13, 849–866 (2000)

    MathSciNet  MATH  Google Scholar 

  175. Iooss, G., Adelmeyer, M.: Topics in Bifurcation Theory and Applications. World Scientific, Singapore (1992)

    MATH  Google Scholar 

  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. Ize, J., Bifurcation Theory for Fredholm Operators, vol. 174. Memoirs of the American Mathematical Society, Providence (1976)

    Google Scholar 

  178. Ize, J.: Obstruction theory and multiparameter Hopf bifurcation. Trans. Am. Math. Soc. 289, 757–792 (1985)

    MathSciNet  MATH  Google Scholar 

  179. Ize, J., Massabó, I., Vignoli, V.: Degree theory for equivariant maps, I. Trans. Am. Math. Soc. 315, 433–510 (1989)

    MATH  Google Scholar 

  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. Ize, J., Vignoli, A.: Equivariant degree for abelian actions, Part I; equivariant homotopy groups. Topol. Methods Nonlinear Anal. 2, 367–413 (1993)

    MathSciNet  MATH  Google Scholar 

  182. Ize, J., Vignoli, A.: Equivariant degree for abelian actions, Part II; Index computations. Topol. Methods Nonlinear Anal. 7, 369–430 (1996)

    MathSciNet  MATH  Google Scholar 

  183. Jolly, M.S., Rosa, R.: Computation of non-smooth local centre manifolds. IMA J. Numer. Anal. 25(4), 698–725 (2005)

    MathSciNet  MATH  Google Scholar 

  184. Joseph, D.D., Sattinger, D.H.: Bifurcating time periodic solutions and their stability. Arch. Ration. Mech. Anal. 45, 79–109 (1972)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  186. Kato, T.: Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften, vol. 132. Springer, New York (1976)

    Google Scholar 

  187. Keener, J.: Infinite period bifurcation and global bifurcation branches. SIAM J. Appl. Math. 41, 127–144 (1981)

    MathSciNet  MATH  Google Scholar 

  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. Kelley, A.: The stable, center-stable, center, center-unstable and unstable manifolds. J. Differ. Equat. 3, 546–570 (1967)

    MathSciNet  MATH  Google Scholar 

  190. Kielhöfer, H.: Hopf bifurcation at multiple eigenvalues. Arch. Ration. Mech. Anal. 69, 53–83 (1979)

    MATH  Google Scholar 

  191. Krasnosel’skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon, New York (1964)

    Google Scholar 

  192. Krawcewicz, W., Vivi, P.: Normal bifurcation and equivariant degree. Indian J. Math. 42, 55–68 (2000)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  196. Krisztin, T.: A local unstable manifold for differential equations with state-dependent delay. Discrete Contin. Dyn. Syst. 9, 993–1028 (2003)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  199. Kulenovic, M.R.S., Ladas, G.: Linearized oscillations in population dynamics. Bull. Math. Biol. 49, 615–627 (1987)

    MathSciNet  MATH  Google Scholar 

  200. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol. 112, 2nd edn. Springer, Berlin (1998)

    Google Scholar 

  201. Kuznetsov, Y.A.: Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODEs. SIAM J. Numer. Anal. 36, 1104–1124 (1999)

    MathSciNet  MATH  Google Scholar 

  202. Langford, W.F.: Periodic and steady-state mode interactions lead to tori. SIAM J. Appl. Math. 37, 649–686 (1979)

    MathSciNet  Google Scholar 

  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. 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. 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. Lenhart, S.N., Travis, C.C.: Stability of functional partial differential equations. J. Differ. Equat. 58, 212–227 (1985)

    MathSciNet  MATH  Google Scholar 

  207. Leray, J., Schauder, J.: Topologie et équations fonctionnelles. Ann. Sci. Ecole. Norm. Sup. 51, 45–78 (1934)

    MathSciNet  Google Scholar 

  208. Levinger, B.W.: A Folk theorem in functional differential equations. J. Differ. Equat. 4, 612–619 (1968)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  210. Li, M.Y., Muldowney, J.S.: On Bendixson’s criterion. J. Differ. Equat. 106, 27–39 (1993)

    MathSciNet  MATH  Google Scholar 

  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. Mallet-Paret, J.: Generic periodic solutions of functional differential equation. J. Differ. Equat. 25, 163–183 (1977)

    MathSciNet  MATH  Google Scholar 

  213. Mallet-Paret, J.: Morse decomposition for delay differential equations. J. Differ. Equat. 72, 270–315 (1988)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  217. Marsden, J., McCracken, M.: The Hopf Bifurcation and Its Applications. Applied Mathematical Sciences, vol. 19. Springer, New York (1976)

    Google Scholar 

  218. Medvedev, V.: On a new type of bifurcations on manifolds. Mat. Sbornik 113, 487–492 (1980) (in Russian)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  220. Memory, M.C.: Stable and unstable manifolds for partial functional differential equations. Nonlinear Anal. 16, 131–142 (1991)

    MathSciNet  MATH  Google Scholar 

  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. Metz, J.Z., Diekmann, O.: The Dynamics of Physiologically Structured Populations. Springer, New York (1986)

    MATH  Google Scholar 

  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. Milton, J.: Dynamics of Small Neural Populations. American Mathematical Society, Providence, RI (1996)

    MATH  Google Scholar 

  225. Morita, Y.: Destablization of periodic solutions arising in delay-diffusion systems in several space dimensions. Jpn. J. Appl. Math. 1, 39–65 (1984)

    MATH  Google Scholar 

  226. Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky Mt. J. Math. 20, 857–871 (1990)

    MathSciNet  MATH  Google Scholar 

  227. Munkres, J.: Topology, 2nd edn. Prentice Hall, Englewood Cliffs (1975)

    MATH  Google Scholar 

  228. Negrini, P., Salvadori, L.: Attractivity and Hopf bifurcation. Nonlinear Anal. 3, 87–99 (1979)

    MathSciNet  Google Scholar 

  229. Neimark, J.I.: Motions close to doubly-asymptotic motion. Soviet Math. Dokl. 8, 228–231 (1967)

    Google Scholar 

  230. Newhouse, S., Palis, J., Takens, F.: Bifurcations and stability of families of diffeomorphisms. Publ. Math. Inst. Hautes Etud. Sci. 57, 5–71 (1983)

    MathSciNet  MATH  Google Scholar 

  231. Nicholson, A.J.: An outline of the dynamics of animal populations. Aust. J. Zool. 2, 9–65 (1954)

    Google Scholar 

  232. Nussbaum, R.D.: Periodic solutions of some nonlinear functional differential equations. Ann. Math. Pura Appl. 101, 263–338 (1974)

    MathSciNet  MATH  Google Scholar 

  233. Nussbaum, R.D.: A global bifurcation theory with application to functional differential equations. J. Funct. Anal. 19, 319–338 (1975)

    MathSciNet  MATH  Google Scholar 

  234. Nussbaum, R.D.: Global bifurcation of periodic solutions of some autonomous functional differential equations. J. Math. Anal. Appl. 55, 699–725 (1976)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  236. Nussbaum, R.D.: A global Hopf bifurcation theorem of functional differential systems. Trans. Am. Math. Soc. 238, 139–164 (1978)

    MathSciNet  MATH  Google Scholar 

  237. Nussbaum, R.D.: Circulant matrices and differential-delay equations. J. Differ. Equat. 60, 201–217 (1985)

    MathSciNet  MATH  Google Scholar 

  238. Olien, L., Bélair, J.: Bifurcations, stability, and monotonicity properties of a delayed neural network model. Phys. D 102, 349–363 (1997)

    MathSciNet  MATH  Google Scholar 

  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. Palais, R.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)

    MathSciNet  MATH  Google Scholar 

  241. Palis, J., de Melo, W.: Geometric Theory of Dynamical Systems. Springer, New York (1982)

    MATH  Google Scholar 

  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. 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. Peixoto, M.M.: Structural stability on two-dimensional manifolds. Topology 1, 101–120 (1962)

    MathSciNet  MATH  Google Scholar 

  245. Pliss, V.: Principal reduction in the theory of stability of motion. Izv. Akad. Nauk. SSSR Math. Ser. 28, 1297–1324 (1964) (in Russian)

    MathSciNet  MATH  Google Scholar 

  246. Poincaré, H.: Sur les propriétés des fonctions définies par les équations aux différences partielles. Thése. Gauthier-Villars, Paris (1879)

    Google Scholar 

  247. Poincaré, H.: Mémoire sur les courbes définis par une equation différentielle IV. J. Math. Pures Appl. 1, 167–244 (1885)

    MATH  Google Scholar 

  248. Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste, vol. I. Cauthier-Villars, Paris (1892)

    Google Scholar 

  249. Pontryagin, L.: On the dynamical systems close to Hamiltonian systems. J. Exp. Theor. Phys. 4, 234–238 (1934) (in Russian)

    Google Scholar 

  250. Poore, A.B.: On the theory and application of the Hopf-Friedrichs bifurcation theory. Arch. Ration. Mech. Anal. 60, 371–393 (1976)

    MathSciNet  MATH  Google Scholar 

  251. Rabinowitz, P.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)

    MathSciNet  MATH  Google Scholar 

  252. Ruan, S., Filfil, R.F.: Dynamics of a two-neuron system with discrete and distributed delays. Phys. D 191, 323–342 (2004)

    MathSciNet  MATH  Google Scholar 

  253. Ruan, S., Wei, J.: Periodic solutions of planar systems with two delays. Proc. Math. Roy. Soc. Edinb. 129, 1017–1032 (1999)

    MathSciNet  MATH  Google Scholar 

  254. Rudin, W.: Functional Analysis. McGraw-Hill Science, New York (1991)

    MATH  Google Scholar 

  255. Ruelle, D.: Bifurcations in the presence of a symmetry group. Arch. Ration. Mech. Anal. 51, 136–152 (1973)

    MathSciNet  MATH  Google Scholar 

  256. Ruelle, D., Takens, F.: On the nature of turbulence. Comm. Math. Phys. 20, 167–192, and 23, 343–344 (1971)

    Google Scholar 

  257. Rustichini, A.: Hopf bifurcation for functional differential equations of mixed type. J. Dynam. Differ. Equat. 1, 145–177 (1989)

    MathSciNet  MATH  Google Scholar 

  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. 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. Sattinger, D.H.: Bifurcation of periodic solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 41, 66–80 (1971)

    MathSciNet  MATH  Google Scholar 

  261. Sattinger, D.H.: Bifurcation and symmetry breaking in applied mathematics. Bull. Am. Math. Soc. 3, 779–819 (1980)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  263. Shil’nikov, L.P.: On a Poincaré-Birkhoff problem. Math. USSR-Sb. 3, 353–371 (1967)

    Google Scholar 

  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. Sieberg, H.W.: Some historical remarks concerning degree theory. Am. Math. Mon. 88, 125–139 (1981)

    Google Scholar 

  266. Sijbrand, J.: Properties of center manifolds. Trans. Am. Math. Soc. 289, 431–469 (1985)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  270. Staffans, O.J.: Hopf bifurcation for functional and functional differential equations with infinite delay. J. Differ. Equat. 70, 114–151 (1987)

    MathSciNet  MATH  Google Scholar 

  271. Stech, H.: Hopf bifurcation calculations for functional differential equations. J. Math. Anal. Appl. 1109, 472–491 (1985)

    MathSciNet  Google Scholar 

  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. Takens, F.: Normal forms for certain singularities of vector fields. Ann. Inst. Fourier (Grenoble) 23, 163–195 (1973)

    Google Scholar 

  274. Takens, F.: Singularities of vector fields. Publ. Math. IHES 43, 47–100 (1974)

    MathSciNet  Google Scholar 

  275. Thom, R.: Topological models in biology. Topology 8, 313–335 (1969)

    MathSciNet  MATH  Google Scholar 

  276. Thom, R.: Stabilité structurelle et morphogénése. Benjamin, New York (1972)

    Google Scholar 

  277. Tsiligiannis, C.A., Lyberatos, G.: Normal forms, resonance and bifurcation analysis via the Carleman linearization. J. Math. Anal. Appl. 139, 123–138 (1989)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  279. Turaev, D., Shil’nikov, L.: Blue sky catastrophes. Dokl. Math. 51, 404–407 (1995)

    MATH  Google Scholar 

  280. Ushiki, S.: Normal forms for singularities of vector fields. Jpn. J. Appl. Math. 1, 1–34 (1984)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    MATH  Google Scholar 

  283. Vanderbauwhede, A.: Symmetry and bifurcation near families of solutions. J. Diff. Equat. 36, 173–178 (1980)

    MathSciNet  MATH  Google Scholar 

  284. Vanderbauwhede, A.: Local Bifurcation and Symmetry. Research Notes in Mathematics, vol. 75. Pitman, London (1982)

    Google Scholar 

  285. Vanderbauwhede, A.: Center manifolds, normal forms and elementary bifurcations. Dynamics Reported, vol. 2. Wiley, New York (1989)

    Google Scholar 

  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. Vidossich, G.: On the structure of periodic solutions of differential equations. J. Differ. Equat. 21, 263–278 (1976)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  289. Walther, H.-O.: Bifurcation from periodic solutions in functional differential equations. Math. Z. 182, 269–290 (1983)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  292. Weedermann, M.: Normal forms for neutral functional differential equations. Field. Inst. Comm. 29, 361–368 (2001)

    MathSciNet  Google Scholar 

  293. Weedermann, M.: Hopf bifurcation calculations for scalar delay differential equations. Nonlinearity 19, 2091–2102 (2006)

    MathSciNet  MATH  Google Scholar 

  294. Wei, J., Li, M.Y.: Hopf bifurcation analysis in a delayed Nicholson blowflies equation. Nonlinear Anal. 60, 1351–1367 (2005)

    MathSciNet  MATH  Google Scholar 

  295. Wei, J., Ruan, S.: Stability and bifurcation in a neural network model with two delays. Phys. D 130, 255–272 (1999)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  297. Wiggins, S.: Normally Hyperbolic Invariant Manifolds in Dynamical Systems. Springer, New York (1994)

    MATH  Google Scholar 

  298. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  300. Wright, E.M.: A nonlinear differential difference equation. J. Reine Angew. Math. 194, 66–87 (1955)

    MathSciNet  MATH  Google Scholar 

  301. Wu, J.: Global continua of periodic solutions to some differential equations of neutral type. Tôhoku Math J. 45, 67–88 (1993)

    MATH  Google Scholar 

  302. Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)

    MATH  Google Scholar 

  303. Wu, J.: Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998)

    MATH  Google Scholar 

  304. Wu, J.: Introduction to Neural Dynamics and Signal Transmission Delay. Walter de Gruyter, Berlin (2001)

    MATH  Google Scholar 

  305. Wu, J., Xia, H.: Self-sustained oscillations in a ring array of coupled lossless transmission lines. J. Differ. Equat. 124 247–278 (1996)

    MathSciNet  MATH  Google Scholar 

  306. Wu, J., Xia, H.: Rotating waves in neutral partial functional-differential equations. J. Dynam. Differ. Equat. 11, 209–238 (1999)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, China, People’s Republic

    Shangjiang Guo

  2. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada

    Jianhong Wu

Authors
  1. Shangjiang Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianhong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Guo, S., Wu, J. (2013). Lyapunov–Schmidt Reduction. In: Bifurcation Theory of Functional Differential Equations. Applied Mathematical Sciences, vol 184. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6992-6_5

Download citation

Publish with us

Policies and ethics

Search

Navigation