Bifurcation of straight-line librations

arXiv0710.3466MaRDI QIDQ6207176

Publication date: 18 October 2007

Abstract: We study a class of 2-dimensional Hamiltonian systems $H(x,y,p_x,p_y)=frac12(p_x^2+p_y^2) +V(x,y)$ in which the plane $x$=$p_x$=0 is invariant under the Hamiltonian flow, so that straight-line librations along the y axis exist, and we also consider perturbations $delta H=deltacdot F(x,y,p_x,p_y)$ that preserve these librations. We describe a procedure for the analytical calculation of partial derivatives of the Poincar'e 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. J"anich, arXiv.org/abs/0710.3464 [2] M. Brack and K. Tanaka, arXiv:0705.0753

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