arXiv subjects
Publications and source records attributed to Klaus Jaenich.
This article is meant as a mathematical appendix or comment on [BT]. We first consider the notion of transcritical bifurcations of fixed points of general area-preserving maps, and then adress some questions related to [BT] on bifurcation in Poincaré maps of 2-dimensional Hamiltonian systems. [BT] M. Brack and K. Tanaka, arXiv:0705.0753
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 $δH=δ\cdot F(x,y,p_x,p_y)$ 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