A new algorithm for determining the monodromy of a planar differential system

https://doi.org/10.1016/j.amc.2014.03.129Get rights and content

Abstract

We give a new algorithmic criterium that determines wether an isolated degenerate singular point of a system of differential equations on the plane is monodromic. This criterium involves the conservative and dissipative parts associated to the edges and vertices of the Newton diagram of the vector field.

Introduction

We are interested in the behavior of the trajectories in a neighborhood of a singular point of the planar analytic differential systemẋ=F(x)and, in particular, we try to establish when a singular point (we can assume the origin to be the singular point) is surrounded by orbits of the system (monodromic singular point), i.e., each trajectory by lying on a vicinity of a monodromic singular point is either a spiral or an oval. Moreover, from the finiteness theorem for the number of limit cycles, a monodromic point of an analytic planar vector field can be only either a focus or a center, see Il’yashenko [17]. So, the monodromy problem is a previous step to solve the center problem of a vector field which is one of the open classical problems in the qualitative theory of planar differential systems, see [2], [12], [13], [14], [15], [16], [18], [20].

If the eigenvalues of the matrix of the linear part at the origin are conjugate complex or the matrix is nilpotent, the monodromy is a problem solved (Poincaré [20], Andreev [7]). However, there are only partial results when the quoted matrix is identically null (Medvedeva [19], Gasull et al. [12], Mañosa [18], García et al. [11]). All of them use the blow-up procedure introduced by Dumortier [10] which consists of performing a series of changes to desingularize the point.

Tang et al. [21] use a method of generalized normal sectors to determine orbits in exceptional directions near high degenerate equilibria.

Currently, the algorithms of [19] (in general) and [11] (for particular cases) are the criteria used in order to determine the monodromy of the origin.

Here we give an alternative algorithm which improves the Medvedeva algorithm in two aspects basically: it uses the expressions of the conservative and dissipative terms (reducing computational efforts) and it does not require to apply blow-up changes when the conservative quasi-homogeneous term of minor degree has trivial factors.

The paper is organized as follows. In the next section, we show the conservative–dissipative decomposition of a quasi-homogeneous vector field and recall some concept related to the Newton diagram of a vector field. In Section 3, we state the monodromy algorithm. In Section 4, we give the concept of characteristic orbit and show its relation with the monodromy problem. This section also contains some auxiliaries results in order to prove the correctness of the algorithm (Theorem 2). Last on, as an application of our algorithm, in Section 5, we obtain the systemsẋ=ay3+cxy2+gx2y+ex5,ẏ=dy3+hxy2+fx4y+bx7,with ab(g2+h2)<0, whose origin is a monodromic singular point (Theorem 13). Medvedeva [19] studied the case c=-d,g=0 and did not give all cases of monodromy.

Section snippets

Quasi-homogeneous vector field and Newton diagram

Next on, we give some definitions and concepts that we will use throughout the paper. Fixed a type t=(t1,t2) with t1 and t2 coprime natural numbers, which can be arbitrarily chosen, a function f of two variables is quasi-homogeneous of type t and degree k if f(εt1x,εt2y)=εkf(x,y). The vector space of quasi-homogeneous polynomials of type t and degree k will be denoted by Pkt. We will also consider the limit cases t=(1,0) and t=(0,1), being Pk(1,0)=xkR[[y]] and Pk(0,1)=ykR[[x]], where R[[y]]

Monodromy algorithm

We now give two concepts which play a main role in our algorithm.

Definition 1

Let hr+|t|Pr+|t|t and μrPrt polynomials associated to lowest-degree quasi-homogeneous term of type t of F.

We say that a polynomial of the form yt1-λxt2,λ0, is a strong factor associated to the type t if it satisfies one of the following properties:

  • (i)

    it is a factor of hr+|t| of odd multiplicity order,

  • (ii)

    it is a factor of hr+|t| of even multiplicity order (2m) and, either it is a no factor of μr with μr0 or is a factor of μr with

Characteristic orbits

Fixed a type t, it defines the generalized trigonometric functions, Cs(θ) and Sn(θ), as the unique solution of the initial value problemdxdθ=XH(x),x(0)=(1,0)T,where H is the Hamiltonian H(x,y)=x2t2+y2t1P2t1t2t. These functions are periodic and T will denote their minimal period. Moreover, they satisfy the equality Cs2t2(θ)+Sn2t1(θ)=1. For more details, see Dumortier [10].

We can already introduce the generalized polar coordinates, u and θ of the real plane (x,y)R2, asx=ut1Cs(θ),y=ut2Sn(θ),with

Application

We consider systemẋ=ay3+cxy2+gx2y+ex5ẏ=dy3+hxy2+fx4y+bx7,ab(g2+h2)<0.Its Newton diagram consists of two exterior vertices A=(0,4) associated to (ay3,0)T and E=(8,0) associated to vector field (0,bx7)T and an inner vertex C=(2,2) associated to (gx2y,hxy2)T.

We observe that all vertices have even coordinates and ab<0, therefore the system satisfies conditions (A) and (B) of Proposition 1. The Newton diagram has two edges, one associated to the type t=(1,1) and other of type s=(1,3). Fig. 2 shows

Acknowledgement

This work has been partially supported by Ministerio de Educación y Ciencia, Plan Nacional I+D+I co-financed with FEDER funds, in the frame of the projects MTM2007-64193 and MTM2010-20907-C02-02, and by Consejería de Educación y Ciencia de la Junta de Andalucía (FQM-276 and P08-FQM-03770).

Cited by (10)

  • Center problem for generic degenerate vector fields

    2022, Nonlinear Analysis, Theory, Methods and Applications
    Citation Excerpt :

    The so-called monodromy problem consists in characterize when a vector field has a well-defined return map in a neighborhood of an isolated singularity, that is, if the vector field has or not characteristic orbits passing through this isolated singularity, see [1–5].

  • Nilpotent centres via inverse integrating factors

    2016, European Journal of Applied Mathematics
View all citing articles on Scopus
View full text