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Bogdanov-Takens Bifurcation

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2117)

Abstract

In the parameter space, curves of (classical) Poincaré–Andronov–Hopf bifurcations, saddle-node bifurcations and homoclinic orbits emerge.In this chapter, we discuss the intricate patterns of heteroclinic orbits which appear near the corresponding bifurcation without parameters.

Keywords

  • Unstable Manifold
  • Homoclinic Orbit
  • Heteroclinic Orbit
  • Horizontal Boundary
  • Melnikov Function

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.

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References

  1. Arnol’d, V.: Geometrical methods in the theory of ordinary differential equations. Grundl. Math. Wiss., vol. 250. Springer, New York (1983)

    Google Scholar 

  2. Bogdanov, R.: Bifurcation of the limit cycle of a family of plane vector fields. Trudy Semin. Im. I. G. Petrovskogo 2, 23–36 (1976)

    MathSciNet  Google Scholar 

  3. Bogdanov, R.: Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues. Sel. Mat. Sov. 1, 389–421 (1981)

    Google Scholar 

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

    CrossRef  MATH  Google Scholar 

  5. Fiedler, B., Liebscher, S.: Takens-Bogdanov bifurcations without parameters, and oscillatory shock profiles. In: Broer, H., Krauskopf, B., Vegter, G. (eds.) Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday, pp. 211–259. IOP, Bristol (2001)

    Google Scholar 

  6. Fiedler, B., Scheurle, J.: Discretization of Homoclinic Orbits and Invisible Chaos, Mem. AMS, vol. 570. Amer. Math. Soc., Providence (1996)

    Google Scholar 

  7. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Appl. Math. Sci., vol. 42. Springer, New York (1982)

    Google Scholar 

  8. Takens, F.: Forced oscillations and bifurcations (1973). The Utrecht preprint, reproduced in Broer, H., Krauskopf, B., Vegter, G. (eds.), Global Analysis of Dynamical Systems, Festschrift dedicated to Floris Takens for his 60th birthday. IOP, Bristol 2001, pp. 211–259

    Google Scholar 

  9. Takens, F.: Singularities of vector fields. Publ. Math. Inst. Hautes Etud. Sci. 43, 47–100 (1974)

    CrossRef  MathSciNet  Google Scholar 

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Liebscher, S. (2015). Bogdanov-Takens Bifurcation. In: Bifurcation without Parameters. Lecture Notes in Mathematics, vol 2117. Springer, Cham. https://doi.org/10.1007/978-3-319-10777-6_10

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