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Bifurcation of periodic solutions near a collision of eigenvalues of opposite signature. (English) Zbl 0722.58032

When two purely imaginary eigenvalues of opposite Krein signature collide (in a Hamiltonian system), a small perturbation can drive these eigenvalues off the imaginary axis and create a linear instability. The best known example of this instability arises in the restricted three- body problem of celestial mechanics at Routh’s critical mass ratio. The author treats the collision of eigenvalues as a singularity. A variational form of the Lyapunov-Schmidt method together with a distinguished parameter \({\mathbb{Z}}_ 2\)-equivariant singularity theory (with frequency as the distinguished parameter) are used to determine the effect of the degeneracy on the branches of periodic solutions in a neighborhood.
The author recovers previous results of Meyer and Schmidt, Sokol’skij, and van der Meer in his formulation as codimension one singularities. The author’s results also include the effect of a codimension two singularity. Of special interest is the application of these techniques to a spinning double pendulum.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
70F07 Three-body problems
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