Homoclinic bifurcation at resonant eigenvalues

  • Shui -Nee Chow
  • Bo Deng
  • Bernold Fiedler


We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.

Key words

homoclinic orbit period doubling pathfollowing global bifurcation resonance 

AMS classification

34C15 34C35 58F14 


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  1. Abraham, R., and Marsden, J. (1978).Foundation of Mechanics, Benjamin/Cummings, Reading, Mass.Google Scholar
  2. Abraham, R., and Robbin, J. (1967).Transversal Mappings and Flows, Benjamin, Amsterdam.Google Scholar
  3. Afraimovich, V. S., and Shilnikov, L. P. (1974). On attainable transitions from Morse-Smale systems to systems with many periodic points.Math. USSR Izvest. 8, 1235–1270.Google Scholar
  4. Alexander, J. C., and Antman, S. S. (1981). Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems.Arch. Rat. Mech. Anal. 76, 339–354.Google Scholar
  5. Alligood, K. T., Mallet-Paret, J., and Yorke, J. A. (1983). An index for the global continuation of relatively isolated sets of periodic orbits. In J. Palis (ed.),Geometric Dynamics, Springer-Verlag, New York, 1–21.Google Scholar
  6. Amick, C. J., and Kirchgässner, K. (1988). A theory of solitary water-waves in the presence of surface tension. Preprint.Google Scholar
  7. Angenent, S., and Fiedler, B. (1988). The dynamics of rotating waves in scalar reaction diffusion equations.Trans. AMS 307, 545–568.Google Scholar
  8. Armbruster, D., Guckenheimer, J., and Holmes, P. (1988). Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry.Physica 29D, 257–282.Google Scholar
  9. Arneodo, A., Coullet, P., and Tresser, C. (1981). A possible new mechanism for the onset of turbulence.Phys. Lett. 81A, 197–201.Google Scholar
  10. Arnol'd, V. I. (1972). Lectures on bifurcations and versal systems.Russ. Math. Surv. 27, 54–123.Google Scholar
  11. Belitskii, G. R. (1973). Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class.Fund. Anal. Appl. 7, 268–277.Google Scholar
  12. Belyakov, L. A. (1980). Bifurcation in a system with homoclinic saddle curve.Mat. Zam. 28, 911–922.Google Scholar
  13. Belyakov, L. A. (1984). Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero.Mat. Zam. 36, 838–843.Google Scholar
  14. Berger, M. S. (1977).Nonlinearity and Functional Analysis, Academic Press, New York.Google Scholar
  15. Bogdanov, R. I. (1976). Bifurcation of the limit cycle of a family of plane vector fields (Russ.).Trudy Sem. I. G. Petrovskogo 2, 23–36; (Engl.) (1981).Sel. Mal. Sov. 1, 373–387.Google Scholar
  16. Bogdanov, R. I. (1981). Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues (Russ.).Trudy Sem. I. G. Petrovskogo 2, 37–65; (Engl.)Sel. Mat. Sov.1, 389-421.Google Scholar
  17. Bogdanov, R. I. (1985). Invariants of singular points in the plane (Russ.).Uspekhi Mat. Nauk 40, 199–200.Google Scholar
  18. Brunovský, P., and Fiedler, B. (1986). Numbers of zeros on invariant manifolds in reactiondiffusion equations.Nonlin. Anal. TMA 10, 179–193.Google Scholar
  19. Brunovský, P., and Fiedler, B. (1988). Connecting orbits in scalar reaction diffusion equations. In U. Kirchgraber and H.-O. Walther (eds.),Dynamics Reported 1, 57/2-89.Google Scholar
  20. Brunovský, P., and Fiedler, B. (1989). Connecting orbits in scalar reaction diffusion equations II: The complete solution.J. Diff. Eg. 81, 106–135.Google Scholar
  21. Bykov, V. V. (1980). Bifurcations of dynamical systems close to systems with a separatrix contour containing a saddle-focus (Russ.). In E. A. Leontovich-Andronova (ed.),Methods of the Qualitative Theory of Differential Equations, Gor'kov. Gos. Univ., Gorki, 44–72.Google Scholar
  22. Chow, S.-N., and Deng, B. (1989). Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite-dimensional systems.Trans. AMS 312, 539–587.Google Scholar
  23. Chow, S.-N., Deng, B., and Terman, D. (1986). The bifurcation of a homoclinic orbit from two heteroclinic orbit-a topological approach. Preprint.Google Scholar
  24. Chow, S.-N., Deng, B., and Terman, D. (1990). The bifurcation of homoclinic and periodic orbits from two heteroclinic orbits.SIAM J. Math. Anal. (in press).Google Scholar
  25. Chow, S.-N., and Hale, J. K. (1982).Methods of Bifurcation Theory, Grundl. math. Wiss. 251, Springer-Verlag, New York.Google Scholar
  26. Chow, S.-N., and Lin, X.-B. (1988). Bifurcation of a homoclinic orbit with a saddle-node equilibrium. Preprint.Google Scholar
  27. Chow, S.-N., Mallet-Paret, J., and Yorke, J. A. (1983). A periodic orbit index which is a bifurcation invariant. In J. Palis (ed.),Geometric Dynamics, Springer-Verlag, New York, pp. 109–131.Google Scholar
  28. Coullet, P., Gambaudo, J.-M., and Tresser, C. (1984). Une nouvelle bifurcation de codimension 2: le collage de cycles.C.R. Acad. Sci. Paris 299, 253–256.Google Scholar
  29. de Hoog, P. and Weiss, R. (1979). The numerical solution of boundary value problems with an essential singularity.SIAM J. Numer. Anal. 16, 637–669.Google Scholar
  30. Deng, B. (1989a). The Šil'nikov problem, exponential expansion, strong λ-lemma, C1-linearization, and homoclinic bifurcation.J. Diff. Eq. 79, 189–231.Google Scholar
  31. Deng, B. (1989b). Exponential expansion with Šil'nikov's saddle-focus.J. Diff. Eq. 82, 156–173.Google Scholar
  32. Deng, B. (1990a). The bifurcation of countable connections from a twisted heteroclinic loop.SIAM J. Math. Anal. (in press).Google Scholar
  33. Deng, B. (1990b). Homoclinic bifurcations with nonhyperbolic equilibria.SIAM J. Math. Anal. (in press).Google Scholar
  34. Deuflhard, P., Fiedler, B., and Kunkel, P. (1987). Efficient numerical pathfollowing beyond critical points.SIAM J. Numer. Anal. 24, 912–927.Google Scholar
  35. Doedel, E. J., and Kernevez, J. P. (1985). Software for continuation problems in ordinary differential equations with applications. CALTECH.Google Scholar
  36. Evans, J., Fenichel, N., and Feroe, J. A. (1982). Double impulse solutions in nerve axon equations.SIAM J. Appl. Math. 42, 219–234.Google Scholar
  37. Feroe, J. A. (1982). Existence and stability of multiple impulse solutions of a nerve axon equation.SIAM J. Appl. Math. 42, 235–246.Google Scholar
  38. Fiedler, B. (1985). An index for global Hopf bifurcation in parabolic systems.J. reine angew. Math. 359, 1–36.Google Scholar
  39. Fiedler, B. (1986). Global Hopf bifurcation of two-parameter flows.Arch. Rat. Mech. Anal. 94, 59–81.Google Scholar
  40. Fiedler, B., and Kunkel, P. (1987a). A quick multiparameter test for periodic solutions. In T. Küpper, R. Seydel, and H. Troger (eds.),Bifurcation: Analysis, Algorithms, Applications, ISNM 79, Birkhäuser, Basel, pp. 61–70.Google Scholar
  41. Fiedler, B., and Kunkel, P. (1987b). Multistability, scaling, and oscillations. In J. Warnatz and W. Jäger (eds.),Complex Chemical Reaction Systems, Springer Ser. Chem. Phys. 47, Berlin, pp. 169–180.Google Scholar
  42. Fiedler, B., and Mallet-Paret, J. (1989). Connections between Morse sets for delay-differential equations.J. reine angew. Math. 397, 23–41.Google Scholar
  43. Fischer, G. (1984). Zentrumsmannigfaltigkeiten bei elliptischen Differentialfgleichungen.Math. Nachr. 115, 137–157.Google Scholar
  44. FitzHugh, R. (1969). Mathematical models of excitation and propagation in nerves. In H. P. Schwan (ed.),Biological Engineering, McGraw-Hill, New York, pp. 1–85.Google Scholar
  45. Gambaudo, J.-M., Glendinning, P., and Tresser, C. (1984). Collage de cycles et suites de Farey.C.R. Acad. Sci. Paris 299, 711–714.Google Scholar
  46. Glendinning, P. (1984). Bifurcation near homoclinic orbits with symmetry.Phys. Lett. 103A, 163–166.Google Scholar
  47. Glendinning, P. (1987). Travelling wave solutions near isolated double-pulse solitary waves of nerve axon equations.Phys. Lett. 121A, 411–413.Google Scholar
  48. Glendinning, P., and Sparrow, C. (1986). T-points: A codimension two heteroclinic bifurcation.J. Stat. Phys. 43, 479–488.Google Scholar
  49. Guckenheimer, J., and Holmes, P. (1983).Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci.42, Springer-Verlag, New York.Google Scholar
  50. Guckenheimer, J., and Holmes, P. (1988). Structurally stable heteroclinic cycles.Math. Proc. Camb. Phil. Soc. 103, 189–192.Google Scholar
  51. Hale, J. K., and Lin, X.-B. (1986). Heteroclinic orbits for retarded functional differential equations.J. Diff. Eq. 65, 175–202.Google Scholar
  52. Hastings, S. P. (1982). Single and multiple pulse waves for the FitzHugh-Nagumo equations.SIAM J. Appl. Math. 42, 247–260.Google Scholar
  53. Hirsch, M. W., Pugh, C. C., and Shub, M. (1977).Invariant Manifolds, Lect. Notes Math. 583, Springer-Verlag, Berlin.Google Scholar
  54. Hofer, H., and Toland, J. (1984). Homoclinic, heteroclinic, and periodic orbits for a class of indefinite Hamiltonian systems.Math. Ann. 268, 387–403.Google Scholar
  55. Holmes, P. (1980). A strange family of three-dimensional vector fields near a degenerate singularity.J. Diff. Eq. 37, 382–403.Google Scholar
  56. Hurewicz, W., and Wallman, H. (1948).Dimension Theory, Princeton University Press.Google Scholar
  57. Kielhöfer, H., and Lauterbach, R. (1983). On the principle of reduced stability.J. Funct. Anal. 53, 99–111.Google Scholar
  58. Kirchgässner, K. (1982). Wave-solutions of reversible systems and applications.J. Diff. Eq. 45, 113–127.Google Scholar
  59. Kirchgässner, K. (1983). Homoclinic bifurcation of perturbed reversible systems. InEquadiff 82, H. W. Knobloch and K. Schmitt (eds.), Lect. Notes Math. 1017, Springer-Verlag, Heidelberg, pp. 328–363.Google Scholar
  60. Kirchgässner, K. (1988). Nonlinearly resonant surface waves and homoclinic bifurcation. Preprint.Google Scholar
  61. Kokubu, H. (1987). On a codimension 2 bifurcation of homoclinic orbits.Proc. Jap. Acad. 63A, 298–301.Google Scholar
  62. Kokubu, H. (1988). Homoclinic and heteroclinic bifurcations of vector fields.Jap. J. Appl. Math. 5, 455–501.Google Scholar
  63. Kubiček, M., and Marek, M. (1983).Computational Methods in Bifurcation Theory and Dissipative Structures, Springer-Verlag, New York.Google Scholar
  64. Kupka, I. (1963). Contribution à la théorie des champs génériques.Contrib. Diff. Eq. 2, 457–484; (1964)3, 411-420.Google Scholar
  65. Kuramoto, Y., and Koga, S. (1982). Anomalous period-doubling bifurcations leading to chemical turbulence.Phys. Lett. 92A, 1–4.Google Scholar
  66. Lentini, M., and Keller, H. B. (1980). Boundary value problems on semi-infinite intervals and their numerical solution.SIAM J. Numer. Anal. 17, 577–604.Google Scholar
  67. Leontovich, E. (1951). On the generation of limit cycles from separatrices (Russ.).Dokl. Akad. Nauk 78, 641–644.Google Scholar
  68. Lin, X.-B. (1986). Exponential dichotomies and homoclinic orbits in functional differential equations.J. Diff. Eq. 63, 227–254.Google Scholar
  69. Lukyanov, V. I. (1982). Bifurcations of dynamical systems with a saddle-point separatrix loop.Diff. Eq. 18, 1049–1059.Google Scholar
  70. Lyubimov, D. V., and Zaks, M. A. (1983). Two mechanisms of the transition to chaos in finite-dimensional models of convection.Physica 9D, 52–64.Google Scholar
  71. Mallet-Paret, J., and Yorke, J. A. (1980). Two types of Hopf bifurcation points: Sources and sinks of families of periodic orbits. In R. H. G. Helleman (ed.),Nonlinear Dynamics, Proc. N.Y. Acad. Sci.357, pp. 300–304.Google Scholar
  72. Mallet-Paret, J., and Yorke, J. A. (1982). Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation.J. Diff. Eq. 43, 419–450.Google Scholar
  73. A. Mielke. A reduction principle for nonautonomous systems in infinite-dimensional spaces.J. Diff. Eq. 65, 68–88.Google Scholar
  74. Mielke, A. (1986). Steady flows of inviscid fluids under localized perturbations.J. Diff. Eq. 65, 89–116.Google Scholar
  75. Moser, J. (1973).Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, N.J.Google Scholar
  76. Nozdracheva, V. P. (1982). Bifurcation of a noncoarse separatrix loop.Diff. Eq. 18, 1098–1104.Google Scholar
  77. Ovsyannikov, I. M., and Shil'nikov, L. P. (1987). On systems with a saddle-focus homoclinic curve.Math. USSR Sbornik 58, 557–574.Google Scholar
  78. Palmer, K. J. (1984). Exponential dichotomies and transversal homoclinic points.J. Diff. Eq. 55, 225–256.Google Scholar
  79. Rabinowitz, P. H. (1971). Some global results for nonlinear eigenvalue problems.J. Funct. Anal. 7, 487–513.Google Scholar
  80. Reyn, J. W. (1980). Generation of limit cycles from separatrix polygons in the phase plane. In R. Martini (ed.),Geometrical Approaches to Differential Equations, Lect. Notes Math. 810, Springer-Verlag, Berlin, pp. 264–289.Google Scholar
  81. Rinzel, J., and Terman. D. (1982). Propagation phenomena in a bistable reaction-diffusion system.SIAM J. Appl. Math. 42, 1111–1137.Google Scholar
  82. Robinson, C. (1988). Differentiability of the stable foliation for the model Lorenz equations. Preprint.Google Scholar
  83. Rodriguez, J. A. (1986). Bifurcations to homoclinic connections of the focus-saddle type.Arch. Rat. Mech. Anal. 93, 81–90.Google Scholar
  84. Sanders, J. A., and Cushman, R. (1986). Limit cycles in the Josephson equation.SIAM J. Math. Anal. 17, 495–511.Google Scholar
  85. Schecter, S. (1987a). The saddle-node separatrix-loop bifurcation.SIAM J. Math. Anal. 18, 1142–1156.Google Scholar
  86. Schecter, S. (1987b). Melnikov's method at a saddle-node and the dynamics of the forced Josephson junction.SIAM J. Math. Anal. 18, 1699–1715.Google Scholar
  87. Sell, G. R. (1984). Obstacles to linearization.Diff. Eq. 20, 341–345.Google Scholar
  88. Sell, G. R. (1985). Smooth linearization near a fixed point.Am. J. Math. 107, 1035–1091.Google Scholar
  89. Seydel, R. (1988).From Equilibrium to Chaos, Elsevier, New York.Google Scholar
  90. Shil'nikov, L. P. (1962). Some cases of generation of periodic motions in an n-dimensional space.Soviet Math. Dokl. 3, 394–397.Google Scholar
  91. Shil'nikov, L. P. (1965). A case of the existence of a countable number of periodic motions.Soviet Math. Dokl. 6, 163–166.Google Scholar
  92. Shil'nikov, L. P. (1966). On the generation of a periodic motion from a trajectory which leaves and re-enters a saddle state of equilibrium.Soviet Math. Dokl. 7, 1155–1158.Google Scholar
  93. Shil'nikov, L. P. (1967). The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus.Soviet Math. Dokl. 8, 54–57.Google Scholar
  94. Shil'nikov, L. P. (1968). On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type.Math. USSR Sbornik 6, 427–437.Google Scholar
  95. Shil'nikov, L. P. (1969). On a new type of bifurcation of multidimensional dynamical systems.Soviet Math. Dokl. 10, 1368–1371.Google Scholar
  96. Shil'nikov, L. P. (1970). A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type.Math. USSR Sbornik 10, 91–102.Google Scholar
  97. Shub, M. (1987).Global Stability of Dynamical Systems, Springer-Verlag, New York.Google Scholar
  98. Smale, S. (1963). Stable manifolds for differential equations and diffeomorphisms.Ann. Sc. Norm. Sup. Pisa 17, 97–116.Google Scholar
  99. Sparrow, C. (1982).The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Appl. Math. Sci. 41, Springer-Verlag, New York.Google Scholar
  100. Sternberg, S. (1957). Local contractions and a theorem of Poincaré.Am. J. Math. 79, 809–824.Google Scholar
  101. Sternberg, S. (1958). On the structure of local homeomorphisms of Euclidean n-space.Am. J. Math. 80, 623–631.Google Scholar
  102. Takens, F. (1974). Singularities of vector fields.Publ. IHES 43, 47–100.Google Scholar
  103. Tresser, C. (1984). About some theorems by L. P. Shil'nikov.Ann. Inst. H.Poincaré 40, 441–461.Google Scholar
  104. Vanderbauwhede, A. (1989). Center manifolds, normal forms and elementary bifurcations. U. Kirchgraber and H.-O. Walther (eds.),Dynamics Reported 2, 89–169.Google Scholar
  105. Walther, H.-O. (1981). Homoclinic solution and chaos in x(t)=f(x(t−1)).Nonlin. Anal. TMA 5, 775–788.Google Scholar
  106. Walther. H.-O. (1985). Bifurcation from a heteroclinic solution in differential delay equations.Trans. AMS 290, 213–233.Google Scholar
  107. Walther, H.-O. (1986). Bifurcation from a saddle connection in functional differential equations: An approach with inclination lemmas.Dissertationes Mathematicae, to appear.Google Scholar
  108. Walther, H.-O. (1987a). Inclination lemmas with dominated convergence.J. Appl. Math. Phys. (ZAMP) 32, 327–337.Google Scholar
  109. Walther, H.-O. (1987b). Homoclinic and periodic solutions of scalar differential delay equations. Preprint.Google Scholar
  110. Walther, H.-O. (1989). Hyperbolic periodic solutions, heteroclinic connections and transversal homoclinic points in autonomous differential delay equations.Memoirs of the AMS 402, Providence, R.I.Google Scholar
  111. Yanagida, E. (1987). Branching of double pulse solutions from single pulse solutions in nerve axon equations.J. Diff. Eq. 66, 243–262.Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • Shui -Nee Chow
    • 1
  • Bo Deng
    • 2
  • Bernold Fiedler
    • 3
  1. 1.Center for Dynamical Systems and Nonlinear StudiesGeorgia Institute of TechnologyAtlanta
  2. 2.Department of Mathematics and StatisticsUniversity of NebraskaLincoln
  3. 3.Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany

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