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Application: Fluid Flow in a Planar Channel, Spatial Dynamics with Reversible Bogdanov-Takens Bifurcation

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

Abstract

In [1, 2] the Kolmogorov problem of viscous incompressible planar fluid flow under external spatially periodic forcing has been studied. Kirchgässner reduction has been used to find time-independent bounded solutions at the onset of instability of the system when the Reynolds number increases. We regard bounded solutions as evolutions in the unbounded direction of a cross-sectional profile, and find a six-dimensional center manifold. Three conserved quantities yield a reduction to a three-dimensional reversible system with a line of equilibria. When we take the Reynolds number into account, a Bogdanov-Takens point along a one-parameter family of lines of equilibria appears, see Chap. 10. Additional reversibilities, however, change the resulting dynamics.

Keywords

  • Homoclinic Orbit
  • Elliptic Case
  • Cusp Point
  • Additional Reversibility
  • Periodic Bubble

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

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Liebscher, S. (2015). Application: Fluid Flow in a Planar Channel, Spatial Dynamics with Reversible 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_14

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