Quasi Normal Sectors and Orbits in Regular Critical Directions of Planar System?

Shimin LI Yulin ZHAO

1 Introduction

One of the classic problems in the qualitative theory of the differential system is to characterize the local phase portraits near an isolated singular point.Consider Ckplanar differential system

where the dot denotes derivatives with respect to the variable t,k denotes a positive integer,+∞or w,Cwstands for an analytic function,and the origin O(0,0)is the isolated singular point such that X(0,0)=Y(0,0)=0.

Concerning the simple singular point(those both eigenvalues of the Jacobian matrix at the singular point are different from zero),the Hartman-Grobman theorem completely solved them except when the singularity is monodromic,that is,the solution of the differential system turns around the singular point.The semi-simple singular points(where one of the eigenvalues equal to zero)are also classified(see for instance[3]).

Regarding the degenerate singular points(where both eigenvalues of the Jacobian matrix at the point equal to zero),the situation is much more difficult.The Andreev theorem(see[2])classifies the nilpotent singular points,whose associated Jacobian matrix is not identically zero,except the monodromic case.If the Jacobian matrix is identically null,the problem is open.In this case,the only possibility is studying each degenerate singular point case by case.

In polar coordinates x=rcosθ,y=rsinθ,the system(1.1)becomes

Following Frommer[7],a direction of θ= θ0at the origin is called a critical direction of system(1.2)if there exists a sequence of pointssuch that0,tanαn→ 0 as n → ∞,where αnis the angle turning anti-clockwise from the directionto the vector field at Pn.

There are several methods to study the number of orbits tending to the singular point in the critical direction.The first method is the so-called Z-sector(see[8]).The small neighborhood U(O,r)near the origin is divided into a finite number of sectors by the branches of Z(x,y)=,these sectors are called Z-sectors.Within a Z-sector,has a fixed sign and contains a finitely number of critical directions.

The second method is to analysis normal sectors(normal domain)(see[10,14]).A circular sector S:is called a normal sector if(i)neither θ1nor θ2is a critical direction,(ii)withinbetween θ1and θ2there has a finite(possible zero)number of critical directions.Normal sectors usually construct as a small neighborhood of a critical direction,so in each normal sector there has at most one critical direction.

The third method is generalized normal sectors(see[12–13]),which is developed from normal sectors,do not restrict edges of the sectors to the radial lines but even allow orbits,horizontal isoclines and vertical isoclines to be edges.

A Z-sector possibly has more than one critical direction,so we can not determine which critical direction an orbit in the sector will be tangent to.Moreover,sometimes Z-branches of Z(x,y)=0 are hardly solved,so it is difficult to find all Z-sector branches.For a normal sector,sometimes it is not always constructible about an critical direction.Construction of generalized normal sector is a technical task,generally drawing horizontal isoclines and vertical isoclines is helpful for analysis.We note that the common point in the above three methods is to apply the classic polar coordinate to analysis.

In this paper we apply the method of quasi normal sectors,which is a generalization of normal sectors,to determine orbits in the critical direction.

The organization of this paper is as follows:In Section 2,we construct the method of quasi normal sector and statement some preliminary results.Section 3 contains the main results.In Section 4,we give two concrete examples to show that our method maybe more effective than the above three methods in some cases.In the last section,we introduce the method of Newton polyhedron.

2 Quasi Normal Sector

Given p,q,d∈N,we say that a function F(x,y)is called(p,q)-quasi-homogeneous function of weight degree d if F(λpx,λqy)= λdF(x,y).A vector field(P(x,y),Q(x,y))is called(p,q)-quasi-homogeneous of weight degree d if P(x,y)and Q(x,y)are(p,q)-quasi-homogeneous functions of weight degree d+p?1 and d+q?1 respectively.

For a given Ckdifferential system(1.1),if k is sufficiently large,we can always choose conveniently a pair of positive integers(p,q)by the Newton polyhedron(see[4]),hence the system(1.1)can be written as the following differential system:

Definition 2.1 Ifthen we sayis a regular critical direction.

Proposition 2.1 From(2.3)–(2.4),the following statements hold:

(i)If m

(ii)If m>n and Yn(0,1)6=0,then are regular critical direction.

(iii)Assume m=n.

If p is even and q is odd,then θkis a regular critical direction if and only if?θkis a regular critical direction.

If p is odd and q is odd,then θkis a regular critical direction if and only ifregular critical direction.

If p is odd and q is even,then θkis a regular critical direction if and only ifregular critical direction.

Proof The statements(i)–(ii)are obvious from the Definition of regular critical direction.

Now we prove the statement(iii).Suppose that p is even and q is odd,then we get that

Figure 1 Three types of QNS.

The following four lemmas are generalization of the results in[14],and the method of proof these lemmas is similar.

3 Main Results

For convenience,in this section we say the orbit(x(t),y(t))of(2.1)tends to the origin if(x(t),y(t))→(0,0)as t→+∞ or t→?∞.

We want to determine the number of orbits tending to the origin O(0,0)in regular critical direction.According to the number of real roots of equation G(θ)=0,we can distinguish to the following three cases.

3.1 G(θ)=0 has no real root

3.2 G(θ)≡ 0,the number of regular critical direction is in finite

In this case,it is obvious that m=n.Assume that Φ(x,y)and Ψ(x,y)are analytic functions with respect to variables x and y near the origin O(0,0).

Note that if p is odd,then the positive x directional blow-up provides the information of the negative x directional blow-up.We only need employ the x directional blow-up x=up∈R,

In the following we only consider the positive x directional blow-up,the negative case is similar.

Case(a)

Note that(0,vk)is a nonsingular point of system(3.1)+,then there exists one and only one orbit cutting across the v-axis at(0,vk)in the(u,v)-plane(see Figure 2.1).

Figure 2 Xm(1,vk)6=0.

Case(b)and

In this case,consider the following system

and Ym(1,v)is a polynomial in variable v with degree less than or equal toTherefore Xm(1,v)is a polynomial in variable v with degree less than or equal to

Figure 3 Xm(1,vk)=0,The orbit remains in the same half-plane u≥0.

In this case,(0,vk)is a singular point of system(3.1)+,the behavior of solution near such a singular point may belong to any of the types discussed in the above cases.Since(2.1)is an analytic system,the desingularization theorem(see[6])ensures that after a finite number of such blow-up,it is possible to solve this problem.

To study the behavior of orbits near the critical directions,we employ the quasi-homogeneous directional blow-up in the positive(resp.negative)y direction x=becomes

Figure 4 The orbit cuts across the v-axis.

In the following we also only consider the positive y directional blow-up.

Case(d)Ym(0,+1)6=0.

There is a unique orbit of system(3.3)+through the origin O(0,0)and cuts cross the eu-axis on the(see Figure 5.1).

Case(e)

In this case,consider the following system

We know that(0,0)is a nonsingular point of system(3.4)and the unique orbit through(0,0)is tangent to theFigures 6.1 and 7.1 denote that the orbit remains in the same half-planeand cuts across the

Case(f)Ym(0,+1)=0 and

In this case,O(0,0)is a singular point of(3.3)+, finite number of such blow-up will solve this problem.

From the above analysis,we can obtain the following theorems.

Theorem 3.2 Consider the analytic differential system(2.1)with G(θ)≡ 0,and assume that p and q are odd.

(i)If,then there is a unique orbit of(2.1)tending to the origin in the direction θkas well as a unique orbit tending to the origin in the direction(see Figure 2.2),where

Figure 5 Ym(0,1)6=0.

Tn

(ii)Let

If the orbit remains in the same half-plane near(0,vk),say in the half-plane u≥0(resp.u ≤ 0),then there are exactly two orbits tending to the origin in the directionone on each side of the curve and there is no orbit of(2.1)tending to the origin in the direction(see Figure 3.2).

If the orbit cuts across the u-axis at(0,vk),then there is a unique orbit of(2.1)tending to the origin in the direction θkas well as a unique orbit tending to the origin in the direction

(iii)If Ym(0,1)6=0,then there is a unique orbit of(2.1)tending to the origin in the directionas well as a unique orbittending to the origin in the direction(see Figure 5.2).

(iv)Let

If the orbit remains in the same half-planethen there exist two orbit of(2.1)tending to the origin in the direction,one on each side of the y-axis and there is no orbit of(2.1)tending to the origin in the direction(see Figure 6.2).

If the orbit cuts across theat(0,0),then there is a unique orbit of(2.1)tending to the origin in the directionas well as a unique orbit tending to the origin in the direction?(see Figure 7.2).

Figure 6Orbits remains in the same half-plane ev ≥ 0.

(v)Ifthen successive application of the above transformation can solve this problem.

Proof Because p and q are odd,we only need employ the x directional blow-upR,y=uqv and y directional blow-up

(i)In this case,there exist orbits of system(2.1)tending to the origin along the curve.For every generalized polar direction θkin the(x,y)-plane such that Tn θk=,there is a unique orbit of(2.1)tending to the origin in the direction θkas well as a unique orbit tending to the origin in the direction

(ii)In the first subcase,if u≥0(resp.u≤0),thenSo there are exactly two orbits tending to the origin in the directioneach side of the curve and there is no orbit of(2.1)tending to the origin in the direction(resp.θk).The second subcase is similar to case(i).

(iii)If Ym(0,1)6=0,then it is clearly that the orbit of system(3.3)cuts across theThe statement follows from

(iv)In the first subcase,ifthen(resp.So there are exactly two orbits tending to the origin in the directionone on each side of the y-axis and there is no orbit of(2.1)tending to the origin in the direction(resp.The second subcase is similar to case(iii).

Figure 7.Orbits cuts across the eu-axis.

(v)It is obvious from cases(c)and(f).

Theorem 3.3 Consider the analytic differential system(2.1)with G(θ)≡ 0,and assume that p and q are even.

(i)If Xm(1,vk)6=0,then there is a unique orbit of(2.1)tending to the origin in the direction θkas well as a unique orbit tending to the origin in the direction

(ii)Let

If the orbit remains in the same-half-plane near(0,vk),say in the half-plane u≥0(resp.u ≤ 0),then there are exactly two orbits tending to the origin in the directionone on each side of the curve and there is no orbit of(2.1)tending to the origin in the direction(see Figure 3.3).

If the orbit cuts across the u-axis at(0,vk),then there is a unique orbit of(2.1)tending to the origin O(0,0)in the direction θkas well as a unique orbit tending to the origin in the direction

(iii)If,then there exist two orbits of(2.1)tending to the origin in the directionand there is no orbit tending to the origin in the direction,(see Figure 5.3).

(iv)If,then there exist two orbit of(2.1)tending to the origin in the directionand there is no orbit of(2.1)tending to the origin in the direction(see Figures 6.3 and 7.3).

(v)If ,then there exist two orbit of(2.1)tending to the origin in the directionand there is no orbit of(2.1)tending to the origin in thedirection

(vi)If successive application of the above transformation can solve this problem.

Proof Because p is odd,we only need to employ the x directional blow-up.Statements(i)and(ii)follow from the proof of Theorem3.2 and the statement(vi)is obvious.

If employ the quasi-homogeneous directional blow-up in the positive y direction,thenso the statements(iii)and(iv)are proved.

If ,employ the quasi-homogeneous directional blow-up in the negative y direction,then,so there exist two orbits going into the origin in the directionThe statement(v)is proved.

Theorem 3.4 Consider the analytic differential system(2.1)with G(θ)≡ 0,and assume that p and q are odd.

(i)Ifthen there is a unique orbit of(2.1)tending to the origin in the direction θk(resp.)as well as a unique orbit tending to the origin in the direction)(see Figure 2.4).

(ii)Let

If the orbit remains in the same half-plane near(0,vk),say in the half-plane u≥0(resp.u ≤ 0),then there are exactly two orbits tending to the origin in the direction sgn(vk)θk(resp.?sgn(vk)θk),one on each side of the curve and there is no orbit of(2.1)tending to the origin in the direction ?sgn(vk)θk(resp.sgn(vk)θk)(see Figure 3.4),where sgn(vk)denotes the sign function of vk.

If orbit cuts across the u-axis at(0,vk),then there is a unique orbit of(2.1)tending to the origin in the direction θkas well as a unique orbit tending to the origin in the direction ?θk(see Figure 4.4).

(iii)Let

If the orbit remains in the same-half-plane near(0,vk),say in the half-plane u≥0(resp.u ≤ 0),then there are exactly two orbits tending to the origin in the direction(resp.,one on each side of the curve and there is no orbit of(2.1)tending to the origin in the direction

If the orbit cuts across the u-axis at(0,vk),then there is a unique orbit of(2.1)tending to the origin in the directionas well as a unique orbit tending to the origin in the direction

(iv)If,there is a unique orbit of(2.1)tending to the origin in the directionas well as

(v)Let

If the orbit remains in the same half-planethen there exist two orbit of(2.1)tending to the origin in the direction,one on each side of the y-axis and there is no orbit of(2.1)tending to the origin in the direction(see Figure 6.4).

If the orbit cuts across theat(0,0),then there is a unique orbit of(2.1)tending to the origin in the directionas well as a unique orbit tending to the origin in the direction(see Figure 7.4).

(vi)If,then successive application of the above transformation can solve this problem.

Proof Because q is odd,we only need to employ the y directional quasi-homogeneous blow-up,so the statements(iv),(v)and(vi)are obvious from the proof of Theorem 3.2.

Ifemploy the quasi-homogeneous directional blow-up in the positive x direction,then x ≥ 0 and

For the first subcase,suppose,then x≥0,y≥0(resp.x≥0,y≤0).There are exactly two orbits of(2.1)tending to the origin in the directionand there is no orbit tending to the origin in the direction,so the statement(ii)is proved.The second subcase is similar.

If,employ the quasi-homogeneous directional blow-up in the negative x direction,the proof of statement(iii)is similar.

The statement(i)is obvious from the above analysis.

3.3 G(θ)=0 have finite real roots

Remark 3.1 If the differential system(2.1)is an analytic system near the origin,then the condition of Proposition 3.2 and the condition(3.8)of Proposition 3.3 are naturally satisfied.

Furthermore,if l is odd,then the following statements hold.

4 Applications

Figure 8 In Figure 8.1 the dashed curve isIn Figure 8.2,the dashed curve is

Note that if we use the normal sector method for the system(4.1),then G(θ)= ?sin4(θ),H(θ)=cos(θ)sin3(θ).We can not determine the number of orbits tending to the origin in the direction θ=0 and θ= π.

Example 4.2 Consider the differential system

where both p and q are positive integers and q≥2,a>0,b>0.The above differential system is considered in[13].We show that our method of QNS is more effective than the method of generalized normal sector in this case.

Since the system(4.2)has concrete background as stated in[13],we only consider the first quadrant(x≥0,y≥0).

For the system(4.2),Apply the generalized polar coordinate,we have

5 Appendix:Method of Newton Polyhedron

In this appendix we show how to obtain the parameters p,q and d by Newton polyhedron,see for instance[1].

Consider the following system origin of system(5.1)is degenerate,then(0,0)6∈N.We define the Newton polyhedron as the convex hull of N+R2+in the(i,j)-plane.We call γkthe segments of this polyhedron.If one of these segments is completely in the half-plane i≤ 0(resp.j ≤ 0)we call it γ0(resp.γn+1).The rest of the segments are calledfrom left to right,and they have at least one endpoint in the first quadrant of the(i,j)-plane.For k=1,···,n,the segment γksatisfies the equation of the straight linefor some coprime αWe choose the suitable(p,q,d)from the set,k=1,···,n}provided by

If(i,j)∈N,then eitheris a monomial of X(x,y),oris a monomial of Y(x,y).We call dk=pi+qj the quasi-degree of type(p,q)of these monomials.The monomials.of quasi-degree dkof type(p,q)are grouped in a polynomial.Hence the vector field χ=(X,Y)can be decomposed into its quasi-homogeneous components of type(p,q)whereandThis is a different way of writing system(5.1).

AcknowledgementThe authors would like to thank the reviewers for their valuable comments and suggestions,which improve the presentation of this paper a lot.

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