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.

Planetary nebulae and determination of the bulge-disk boundary

In this paper, a sample of planetary nebulae in the Galaxy's inner-disk and bulge is used to find the galactocentric distance that optimally separates these two populations in terms of their abundances. Statistical distance scales were used to investigate the distribution of abundances across the disk–bulge interface, while a Kolmogorov–Smirnov test was used to find the distance at which the chemical properties of these regions separate optimally. The statistical analysis indicates that, on average, the inner population is characterized by lower abundances than the outer component. Additionally, for the α-element abundances, the inner population does not follow the disk's radial gradient toward the Galactic Center. Based on our results, we suggest a bulge–disk interface at 1.5 kpc, marking the transition between the bulge and the inner disk of the Galaxy as defined by the intermediate-mass population.

Projections of hypersurfaces in 'R POT.4' with boundary to planes

We study orthogonal projections of generic embedded hypersurfaces in 'R POT.4' with boundary to 2-spaces. Therefore, we classify simple map germs from 'R POT.3' to the plane of codimension less than or equal to 4 with the source containing a distinguished plane which is preserved by coordinate changes. We also go into some detail on their geometrical properties in order to recognize the cases of codimension less than or equal to 1.