Planar Analytic Vector Fields Research Articles

Existence and nonexistence of Puiseux inverse integrating factors in analytic monodromic singularities

AbstractIn this work, we present some criteria about the existence and nonexistence of both Puiseux inverse integrating factors and Puiseux first integrals for planar analytic vector fields having a monodromic singularity. These functions are a wide generalization of their formal or algebraic counterpart in Cartesian coordinates . We prove that none of the functions and can be used to characterize degenerate centers although the existence of is a sufficient center condition.

Lyapunov coefficients for monodromic tangential singularities in Filippov vector fields

In planar analytic vector fields, a monodromic singularity can be distinguished between a focus or a center by means of the Lyapunov coefficients, which are given in terms of the power series coefficients of the first-return map defined around the singularity. In this paper, we are interested in an analogous problem for monodromic tangential singularities of piecewise analytic vector fields Z=(Z+,Z−). First, we prove that the first-return map, defined in a neighborhood of a monodromic tangential singularity, is analytic, which allows the definition of the Lyapunov coefficients. Then, as a consequence of a general property for pair of involutions, we obtain that the index of the first non-vanishing Lyapunov coefficient is always even. In addition, a general recursive formula together with a Mathematica algorithm for computing the Lyapunov coefficients is obtained. We also provide results regarding limit cycles bifurcating from monodromic tangential singularities. Several examples are analyzed.

ON THE CENTER CONDITIONS FOR ANALYTIC MONODROMIC DEGENERATE SINGULARITIES

In this paper, we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields and can be applied independently so that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allow to study monodromic degenerate singularities with characteristic directions.

Local behavior of planar analytic vector fields via integrability

We present an algorithm to study the local behavior of singular points of planar analytic vector fields having a first integral which is a quotient of analytic functions. The algorithm is based on the blow-up method. It emphasizes the curves passing through the singular points and avoids the computation of the desingularized systems. Vector fields having a rational first integral are a particular case.

A new approach to center conditions for simple analytic monodromic singularities

In this paper we present an alternative algorithm for computing Poincaré–Lyapunov constants of simple monodromic singularities of planar analytic vector fields based on the concept of inverse integrating factor. Simple monodromic singular points are those for which after performing the first (generalized) polar blow-up, there appear no singular points. In other words, the associated Poincaré return map is analytic. An improvement of the method determines a priori the minimum number of Poincaré–Lyapunov constants which must cancel to ensure that the monodromic singularity is in fact a center when the explicit Laurent series of an inverse integrating factor is known in (generalized) polar coordinates. Several examples show the usefulness of the method.

Automated computation of robust normal forms of planar analytic vector fields

We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighbourhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method.

Multiplicity of limit cycles and analytic m-solutions for planar differential systems

This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function f 0 ( x , y ) , defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of f 0 ( x , y ) being an m- solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if f 0 ( x , y ) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles.

The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré–Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincaré–Liapunov method.

Planar analytic vector fields with generalized rational first integrals

The main purpose of this paper is to characterize a germ of planar holomorphic vector field at an elementary singular point having a generalized rational first integral. Our results generalize a result due to Poincaré on a necessary condition of the existence of a rational first integral for planar polynomial systems. As two applications of our main result, we give the necessary and sufficient conditions on the existence of rational first integral for planar quadratic systems having either a weak nondegenerate singular point, or a degenerate elementary singular point.

Finiteness for critical periods of planar analytic vector fields

LIMBURGS UNIV CENTRUM,DEPT MATH,B-3610 DIEPENBEEK,BELGIUM.CHICONE, C, UNIV MISSOURI,DEPT MATH,COLUMBIA,MO 65211.