Regularisation for Planar vector fields

Recurrent switching rules for pairs of linear planar vector fields

Given a switched system formed by a pair of (not asymptotically stable) linear vector fields of R, we define a special class of hybrid feedback laws (called recurrent switching rules) whose structure depends on some conic regions with nonempty interior. We prove that if the system is asymptotically controllable and radially controllable, then it is robustly stabilized by any recurrent switching...

On the Closing Lemma for Planar Piecewise Smooth Vector Fields

A large number of papers deal with “Closing Lemmas” for Cr-vector fields (and Cr-diffeomorphisms). Here, we introduce this subject and formalize the terminology about nontrivially recurrent points and nonwandering points for the context of planar piecewise smooth vector fields. A global bifurcation analysis of a special family of piecewise smooth vector fields presenting a nonwandering set with...

Darboux Integrals for Schrödinger Planar Vector Fields via Darboux Transformations

In this paper we study the Darboux transformations of planar vector fields of Schrödinger type. Using the isogaloisian property of Darboux transformation we prove the “invariance” of the objects of the “Darboux theory of integrability”. In particular, we also show how the shape invariance property of the potential is important in order to preserve the structure of the transformed vector field. ...

Commuting Planar Polynomial Vector Fields for Conservative Newton Systems

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let f ∈...

Unique Normal Forms for Nilpotent Planar Vector Fields