Class Of Planar Vector Fields Research Articles

Periodic orbits in piecewise Riccati vector fields

In this paper a class of planar vector fields is characterized and their periodic orbits are studied. The class under study is formed by vector fields that can be written as Riccati ordinary differential equations. The characterization is stated in terms of some homogeneity conditions on the vector fields. The study of periodic orbits is done through Poincaré maps defined around monodromic points. In particular, some properties of the so-called Möbius transformations are considered. In the study, both smooth and discontinuous vector fields defined by sectors are studied.Theorem 2.2 characterizes the Riccati vector fields and Theorem 3.4 proves that the Poincaré maps of such vector fields are Möbius transformations. In Theorem 4.1 a sharp upper bound of two periodic orbits is stated. The polynomial case is considered in Theorems 4.3 and 4.4 for which existence, upper bounds, and hyperbolicity of periodic orbits are stated.

A class of planar vector fields with homogeneous singular points: Solvability and boundary value problems

This paper deals with the solvability of planar complex vector fields with homogeneous degeneracies. Hölder continuous solutions are obtained via a Cauchy type integral operator associated to the vector field. An associated boundary value problem of Riemann–Hilbert type is also considered.

The multiplicity of the equatorial limit cycle of a class of planar polynomial vector fields

In this paper, we have investigated the existence and the multiplicity of the equatorial limit cycle for a class of planar vector fields of degree 2 m + 1 , obtained the necessary and sufficient condition that the equator is a closed orbit and the multiplicity of the equatorial limit cycle of the planar polynomial vector fields is m + 1 . It is helpful to answer the second part of Hilbert’s 16th problem.

Qualitative analysis of the phase portrait for a class of planar vector fields via the comparison method

The phase portrait of the second-order differential equation x ̈ + ∑ l = 0 n f l ( x ) x ̇ l = 0 , is studied. Some results concerning existence, non-existence and uniqueness of limit cycles are presented. In particular, a generalization of the classical Massera uniqueness result is proved.

Uniqueness of limit cycles for a class of planar vector fields

We give sufficient conditions to ensure uniqueness of limit cycles for a class of planar vector fields. We also exhibit a class of examples with exactly one limit cycle.

Solvability near the characteristic set for a class of planar vector fields of infinite type

On etudie la resolubilite des equations associees a un champ de vecteurs complexe L dans R 2 a coefficients de classe C ∞ ou C ω . On suppose que L est partout elliptique, sauf le long d'une courbe simple etfermee Σ. Sur Σ, on suppose que L est de type infini et que L A L s'annule a un ordre constant. Les equations considerees sont de la forme Lu = pu + f, ou f satisfait des conditions de compatibilite. On prouve, en particulier, que lorsque l'ordre d'annulation de L L est > 1, l'equation Lu = f est resoluble dans la categorie C ∞ mais pas dans la categorie C ω .

Semiglobal solvability of a class of planar vector fields of infinite type

Semiglobal solvability of a class of planar vector fields of infinite type

Global properties of a class of planar vector fields of infinite type

This paper studies some global and semi global properties of infinite type, planar, C-valued real analytic vector fields that are invariant under the rotation group. Results are proved on the integrability, kernel, range and classification of such operators.