Realization of non-holonomic constraints and singular perturbation theory for plane dumbbells

On singular non-holonomic geometry

The aim of the paper is to prove that every smooth generalized vector subbundle D of a vector bundle E is the image of a smooth endomorphism Φ on the fibers of E that induces authomorphisms of fibers of D and is called here a natural anchor. Two new constructions of finite sets of smooth generators of the fibers of D are obtained, using any finite set of generators of the module Γ(E). Natural a...

Geometric Singular Perturbation Theory for Non-smooth Dynamical Systems

In this article we deal with singularly perturbed Filippov systems Zε:

Singular perturbation theory

When we apply the steady-state approximation (SSA) in chemical kinetics, we typically argue that some of the intermediates are highly reactive, so that they are removed as fast as they are made. We then set the corresponding rates of change to zero. What we are saying is not that these rates are identically zero, of course, but that they are much smaller than the other rates of reaction. The st...

Geometric Singular Perturbation Theory

On Hodge Theory of Singular Plane Curves

The dimensions of the graded quotients of the cohomology of a plane curve complement U = P \ C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H(U,C).