Polynomial Liénard Equations near Infinity

Big Deformations near Infinity

Unfolding Polynomial Maps at Infinity

Let f : Cn → C be a polynomial map. The polynomial describes a family of complex affine hypersurfaces f−1(c), c ∈ C. The family is locally trivial, so the hypersurfaces have constant topology, except at finitely many irregular fibers f−1(c) whose topology may differ from the generic or regular fiber of f . We would like to give a full description of the topology of this family in terms of easil...

The applied geometry of a general Liénard polynomial system

In this work, applying a canonical system with field rotation parameters and using geometric properties of the spirals filling the interior and exterior domains of limit cycles, we solve the limit cycle problem for a general Liénard polynomial systemwith an arbitrary (but finite) number of singular points. © 2012 Elsevier Ltd. All rights reserved.

Numerical calculations near spatial infinity

After describing in short some problems and methods regarding the smoothness of null infinity for isolated systems, I present numerical calculations in which both spatial and null infinity can be studied. The reduced conformal field equations based on the conformal Gauss gauge allow us in spherical symmetry to calculate numerically the entire Schwarzschild-Kruskal spacetime in a smooth way incl...

Symmetric Periodic orbits Near heteroclinic Loops at Infinity for a Class of Polynomial Vector Fields

For polynomial vector fields in R, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a ...