Bifurcation of straight-line librations

Abstract

We study a class of 2-dimensional Hamiltonian systems $H(x,y,p_x,p_y)=\frac{1}{2}(p_x^2+p_y^2)+V(x,y)$ 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 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

References in corpus (2)

• Transcritical bifurcations in non-integrable Hamiltonian systems
• Mathematical remarks on transcritical bifurcation in Hamiltonian systems

Cited by in corpus (2)

• Transcritical bifurcations in non-integrable Hamiltonian systems
• Mathematical remarks on transcritical bifurcation in Hamiltonian systems