Bifurcation of straight-line librations

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We study a class of 2-dimensional Hamiltonian systems H(x,y,pₓ,py)=12(pₓ²+py²) +V(x,y) in which the plane x=pₓ=0 is invariant under the Hamiltonian flow, so that straight-line librations along the y axis exist, and we also consider perturbations δH=δ· F(x,y,pₓ,py) that preserve these librations. We describe a procedure for the analytical calculation of partial derivatives of the Poincaré map. These partial derivatives can be used to predict the bifurcation behavior of the libration, in particular to distinguish between transcritical and fork-like bifurcations, as was mathematically investigated in [1] and numerically studied in [2]. [1] K. Jänich, arXiv.org/abs/0710.3464 [2] M. Brack and K. Tanaka, arXiv:0705.0753