Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems

  • Yuri A. Kuznetsov
Part of the Applied Mathematical Sciences book series (AMS, volume 112)

Abstract

The list of possible bifurcations in multidimensional systems is not exhausted by those studied in the previous chapters. Actually, even the complete list of all generic one-parameter bifurcations is unknown. In this chapter we study several unrelated bifurcations that occur in one-parameter continuous-time dynamical systems
$$\dot x = f(x,\alpha ),x \in {R^n},\alpha \in {R^1},$$
(7.1)
where f is a smooth function of (x, α). We start by considering global bifurcations of orbits that are homoclinic to nonhyperbolic equilibria. As we shall see, under certain conditions they imply the appearance of complex dynamics. We also briefly touch some other bifurcations generating “strange” behavior, including homoclinic tangency and the “blue-sky” catastrophe. Then we discuss bifurcations occurring on invariant tori. These bifurcations are responsible for such phenomena as frequency and phase locking. Finally, we give a brief introduction to the theory of bifurcations in symmetric systems, which are those systems that are invariant with respect to the representation of a certain symmetry group. After giving some general results on bifurcations in such systems, we restrict our attention to bifurcations of equilibria and cycles in the presence of the simplest symmetry group ℤ2, composed of only two elements.

Keywords

Hopf Bifurcation Homoclinic Orbit Rotation Number Center Manifold Invariant Torus 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical notes

  1. Andronov, A.A. & Pontryagin, L.S. [ 1937 ], ‘Systèmes grossières’, C.R. (Dokl.) Acad. Sci. URSS (N.S.) 14, 247–251.Google Scholar
  2. Shil’nikov, L.P. [1963], ‘Some cases of generation of periodic motions from singular trajectories’, Mat. Sbornik 61 443–466. In Russian.Google Scholar
  3. Shil’nikov, L.P. [ 1966 ], ‘On the generation of a periodic motion from a trajectory which leaves and re-enters a saddle-saddle state of equilibrium’, Soviet Math. Dokl. 7, 1155–1158.MATHGoogle Scholar
  4. Shil’nikov, L.P. [ 1969 ], ‘On a new type of bifurcation of multidimensional dynamical systems’, Sov. Math. Dokl. 10, 1368–1371.MATHGoogle Scholar
  5. Ilyashenko, Yu. & Li, Weigu. [ 1999 ], Nonlocal Bifurcations, American Mathematical Society, Providence, RI.MATHGoogle Scholar
  6. Champneys, A., Härterich, J. & Sandstede, B. [ 1996 ], ‘A non-transverse homoclinic orbit to a saddle-node equilibrium’, Ergodic Theory Dynamical Systems 16, 431–450.MATHCrossRefGoogle Scholar
  7. Lukyanov, V.I. [ 1982 ], ‘Bifurcations of dynamical systems with a saddle-point separatrix loop’, Differential Equations 18, 1049–1059.MathSciNetGoogle Scholar
  8. Chow, S.-N. & Lin, X.-B. [ 1990 ], ‘Bifurcation of a homoclinic orbit with a saddle-node equilibrium’, Differential Integral Equations 3, 435–466.MathSciNetMATHGoogle Scholar
  9. Deng, B. [ 1990 ], ‘Homoclinic bifurcations with nonhyperbolic equilibria’, SIAM J. Math. Anal. 21, 693–720.MathSciNetMATHCrossRefGoogle Scholar
  10. Gavrilov, N.K. & Shilnikov, L.P. [ 1972 ], ‘On three-dimensional systems close to systems with a structurally unstable homoclinic curve: I’, Math. USSR-Sb. 17, 467485.Google Scholar
  11. Gavrilov, N.K. & Shilnikov, L.P. [ 1973 ], ‘On three-dimensional systems close to systems with a structurally unstable homoclinic curve: II’, Math. USSR-Sb. 19, 139–156.CrossRefGoogle Scholar
  12. Gruzdev, V.G. & Neimark, Ju.I. [1975], A symbolic description of motion in the neighborhood of a not structurally stable homoclinic structure and of its change in transition to close systems, in ‘System Dynamics, Vol. 8’, Gorkii State University, Gorkii, pp. 13–33. In Russian.Google Scholar
  13. Newhouse, S., Palis, J. & Takens, F. [1983], ‘Bifurcations and stability of families of diffeomorphisms’, Inst. Hautes Etudes Sci. Publ. Math. 57, 5–71.Google Scholar
  14. Palis, J. & Takens, F. [ 1993 ], 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.Google Scholar
  15. Gonchenko, S.V. & Gonchenko, V.S. [ 2000 ], On Andronov-Hopf bifurcations of two-dimensional diffeomorphisms with homoclinic tangencies, Preprint 556, WIAAS, Berlin.Google Scholar
  16. Afraimovich, V.S. & Shil’nikov, L.P. [1972], ‘Singular trajectories of dynamical systems’, Uspekhi. Mat. Nauk 27 189–190. In Russian.Google Scholar
  17. Afraimovich, V.S. & Shil’nikov, L.P. [ 1974 ], ‘The attainable transitions from MorseSmale systems to systems with several periodic motions’, Math. USSR-Izv. 8, 1235–1270.CrossRefGoogle Scholar
  18. Afraimovich, V.S. & Shil’nikov, L.P. [1982], ‘Bifurcation of codimension 1, leading to the appearance of a countable set of tori’, Dokl. Akad. Nauk SSSR 262 777–780. In Russian.Google Scholar
  19. Ilyashenko, Yu. & Li, Weigu. [ 1999 ], Nonlocal Bifurcations, American Mathematical Society, Providence, RI.MATHGoogle Scholar
  20. Palis, J. & Pugh, C. [1975], Fifty problems in dynamical systems, in ‘Dynamical Systems (Warwick, 1974)’, Vol. 468 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, pp. 345–353.Google Scholar
  21. Medvedev, V.S. [1980], ‘A new type of bifurcation on manifolds’, Mat. Sbornik 113 487–492. In Russian.Google Scholar
  22. Turaev, D.V. & Shil’nikov, L.P. [ 1995 ], ‘Blue sky catastrophes’, Dokl. Math. 51, 404407.Google Scholar
  23. Gavrilov, N.K. & Shilnikov, A.L. [ 2000 ], Example of a blue sky catastrophe, in ‘Methods of Qualitative Theory of Differential Equations and Related Topics’, Vol. 200 of Amer. Math. Soc. Transi. Ser. 2, Amer. Math. Soc., Providence, RI, pp. 99–105.Google Scholar
  24. Arnol’d, V.I. [ 1983 ], Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York.CrossRefGoogle Scholar
  25. Fenichel, N. [ 1971 ], ‘Persistence and smoothness of invariant manifolds for flows’, Indiana Univ. Math. J. 21, 193–226.MathSciNetMATHCrossRefGoogle Scholar
  26. Broer, H., Simó, C. & Tatjer, J. [ 1998 ], ‘Towards global models near homoclinic tangencies of dissipative diffeomorphisms’, Nonlinearity 11, 667–770.MathSciNetMATHCrossRefGoogle Scholar
  27. Aronson, D., Chory, M., Hall, G. & McGehee, R. [ 1982 ], ‘Bifurcations from an invariant circle for two-parameter families of maps on the plane: A computer assisted study’, Comm. Math. Phys. 83, 303–354.MathSciNetMATHCrossRefGoogle Scholar
  28. Golubitsky, M. & Schaeffer, D. [ 1985 ], Singularities and Groups in Bifurcation Theory I, Springer-Verlag, New York.MATHGoogle Scholar
  29. Ruelle, D. [ 1973 ], ‘Bifurcation in the presence of a symmetry group’, Arch. Rational Mech. Anal. 51, 136–152.MathSciNetMATHCrossRefGoogle Scholar
  30. Fiedler, B. [ 1988 ], Global Bifurcations of Periodic Solutions with Symmetry, Vol. 1309 of Lecture Notes in Mathematics, Springer-Verlag, Berlin.Google Scholar
  31. Nikolaev, E.V. [ 1994 ], Periodic motions in systems with a finite symmetry group, Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region.Google Scholar
  32. Nikolaev, E. [ 1995 ], ‘Bifurcations of limit cycles of differential equations that admit involutory symmetry’, Mat. Sb. 186, 143–160.MathSciNetGoogle Scholar
  33. Nikolaev, E.V. [ 1992 ], On bifurcations of closed orbits in the presence of involutory symmetry, Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region.Google Scholar
  34. Nikolaev, E.V. & Shnol, E.E. [ 1998a ], ‘Bifurcations of cycles in systems of differential equations with a finite symmetry group. I’, J. Dynam. Control Systems 4, 315–341.MathSciNetMATHCrossRefGoogle Scholar
  35. Nikolaev, E.V. & Shnol, E.E. [ 1998b ], ‘Bifurcations of cycles in systems of differential equations with a finite symmetry group. II’, J. Dynam. Control Systems 4, 343–363.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Yuri A. Kuznetsov
    • 1
    • 2
  1. 1.Department of MathematicsUtrecht UniversityUtrechtThe Netherlands
  2. 2.Institute of Mathematical Problems of BiologyRussian Academy of SciencesPushchino, Moscow RegionRussia

Personalised recommendations