TYPE-ZERO SADDLE-NODE BIFURCATIONS AND STABILITY REGION ESTIMATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS

A complete characterization of the stability boundary of a class of nonlinear dynamical systems that admit energy functions is developed in this paper. This characterization generalizes the existing results by allowing the type-zero saddle-node nonhyperbolic equilibrium points on the stability boundary. Conceptual algorithms to obtain optimal estimates of the stability region (basin of attraction) in the form of level sets of a given family of energy functions are derived. The behavior of the stability region and the corresponding estimates are investigated for parameter variation in the neighborhood of a type-zero saddle-node bifurcation value.

Homogeneous metallicities and radial velocities for Galactic globular clusters

Well determined radial velocities and abundances are essential for analyzing the properties of the globular cluster system of the Milky Way. However more than 50% of these clusters have no spectroscopic measure of their metallicity. In this context, this work provides new radial velocities and abundances for twenty Milky Way globular clusters which lack or have poorly known values for these quantities. The radial velocities and abundances are derived from spectra obtained at the Ca II triplet using the FORS2 imager and spectrograph at the VLT, calibrated with spectra of red giants in a number of clusters with well determined abundances. For about half of the clusters in our sample we present significant revisions of the existing velocities or abundances, or both. We also confirm the existence of a sizable abundance spread in the globular cluster M 54, which lies at the center of the Sagittarius dwarf galaxy. In addition evidence is provided for the existence of a small intrinsic internal abundance spread (sigma[Fe/H](int) approximate to 0.11-0.14 dex, similar to that of M 54) in the luminous distant globular cluster NGC 5824. This cluster thus joins the small number of Galactic globular clusters known to possess internal metallicity ([Fe/H]) spreads.

Stability boundary characterization of nonlinear autonomous dynamical systems in the presence of saddle-node equilibrium points

A dynamical characterization of the stability boundary for a fairly large class of nonlinear autonomous dynamical systems is developed in this paper. This characterization generalizes the existing results by allowing the existence of saddle-node equilibrium points on the stability boundary. The stability boundary of an asymptotically stable equilibrium point is shown to consist of the stable manifolds of the hyperbolic equilibrium points on the stability boundary and the stable, stable center and center manifolds of the saddle-node equilibrium points on the stability boundary.