Bifurcation of straight-line librations

TL;DR

This paper analyzes bifurcations of straight-line librations in 2D Hamiltonian systems, providing an analytical method to predict bifurcation types and behaviors under perturbations.

Contribution

It introduces a procedure for calculating derivatives of the Poincaré map to predict bifurcation behavior of librations in Hamiltonian systems.

Findings

Analytical method for Poincaré map derivatives

Prediction of bifurcation types (transcritical, fork-like)

Application to perturbed Hamiltonian systems

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 $\delta H=\delta\cdot F(x,y,p_x,p_y)$ that preserve these librations. We describe a procedure for the analytical calculation of partial derivatives of the 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