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Double-Hopf Bifurcation

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

Abstract

The final bifurcation of codimension 2 is characterized by the intersection of 2 curves of Poincaré-Andronov-Hopf points on a two-dimensional surface of equilibria. As we shall see, the drift direction at the Hopf lines play an important role. In the case of a parameter-dependent fixed line of equilibria, drifts at both Hopf-lines can be opposite and spiraling orbits appear, see Sect. 12.1. In the generic case with a plane of equilibria without parameters, both drifts are transverse and generate a Lyapunov function. Only heteroclinic orbits arise. See Sect. 12.2.

Keywords

  • Final Bifurcation
  • Hopf Line
  • Spiraling Orbits
  • Drift Direction
  • Hopf Point

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. Vanderbauwhede, A.: Centre manifolds, normal forms and elementary bifurcations. In: Kirchgraber, U., Walther, H.O. (eds.) Dynamics Reported 2, pp. 89–169. Teubner & Wiley, Stuttgart (1989)

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© 2015 Springer International Publishing Switzerland

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Liebscher, S. (2015). Double-Hopf Bifurcation. In: Bifurcation without Parameters. Lecture Notes in Mathematics, vol 2117. Springer, Cham. https://doi.org/10.1007/978-3-319-10777-6_12

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