SINGULAR POINTS AND LIE ROTATED VECTOR FIELDS

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Vol. 24, No. 3 (2000) 179–185 S016117120001005X

SINGULAR POINTS AND LIE ROTATED VECTOR FIELDS

JIE WANG and CHEN CHEN (Received 2 April 1999)

Abstract.This paper gives the definition of Lie rotated vector fields in the plane and the conditions of movement of singular points on Lie rotated vector fields with variable parameters.

Keywords and phrases. Lie rotated vector ﬁelds, Lie bracket, one parameter group, singular points.

2000 Mathematics Subject Classiﬁcation. Primary 34C05.

1. Introduction. Many engineering problems are usually run into a class of non- linear equations that contain variable parameters. In order to study whole orbits or whole phase diagrams of vector fields that contain parameters, it is a complicated and interesting problem how the whole orbit or whole phase diagram change as parameter is changed. It is extremely complicated for general containing parameter vector fields to change in the plane, but for some special containing parameter rotated vector fields, their change has regular rule as parameter is changed. These are many results in this respects [3, 4, 5, 6, 7].

In Section 2, we present the basic definitions of Lie rotated vector fields. We define Lie rotated vector fields using one parameter group approach. In accordance with the strict definition of rotated vector field, the singular points ofX(µ)must be kept fixed, but in this paper, the singular points ofX(µ)can be moved as parameterµis changed. In Section 3, we discuss the motion of singular points on Lie rotated vector fields. In the section, we require the singular points of X(µ) to be strictly moved as parameterµ is changed, and permit the moved singular points to disappear or decompose, which do not coincide with the singular points of original vector field. We give some conditions and properties corresponding to the vector fieldY. In this paper, we give some examples to illustrate the concept and notion of Lie rotated vector fields.

2. Lie rotated vector ﬁelds. We consider vector ﬁelds on the planex=(x1,x2)

∈R²,

X=

X1(x),X2(x)

, Y=

Y1(x),Y2(x)

. (2.1)

For the vector ﬁelds (2.1), we deﬁne

X∧Y=X1Y2−X2Y1, X,Y =X1Y1+X2Y2. (2.2) If X and Y are vector ﬁelds, then [X,Y ]is a vector ﬁeld which is operated by Lie

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bracket, i.e.,

[X,Y ]= Z1,Z2

, (2.3)

whereZ1andZ2are expressed as,

Z1= X,∇Y1−Y ,∇X1, Z2= X,∇Y2−Y ,∇X2, (2.4) respectively, where∇is gradient operator.

Let the plane vector fieldsX(µ)=(X1(x,µ),X2(x,µ))be defined by the following differential equations:

dx1

dt =X1(x,µ), dx2

dt =X2(x,µ), (2.5)

whereX1andX2are functions ofxand parameterµ∈I⊂R, and the singular points are isolated.

Deﬁnition2.1. Let the plane vector ﬁeld X(µ) be determined by (2.5), where X1,X2∈C³(R²×I,R),I= {µ| |µ|< δ}is a real interval,δis a given positive number.

If vector fieldY exists which is defined by the following differential equations:

dx1

dt =Y1(x), dx2

dt =Y2(x), (2.6)

whereY1andY2∈C³(R²,R). At all ordinary points ofX(0), such that the following relation holds

L(0)^def= X(0)∧

X_µ(0)+[X(0),Y ]

>0(<0), (2.7) whereX_µ(0)is the derivative of the vector ﬁeldX(µ)atµ=0, then X(µ), µ∈I, is called Lie rotated vector ﬁelds.

Remark2.2. If the vector fieldX(µ)is defined onD×I, whereD⊂R², such that X(0)satisfies relation (2.7) at all ordinary points ofX(0)onD, thenX(µ), µ∈I, is called Lie rotated vector fields onD.

Lemma2.3. Letψ^s be a one parameter transform group which is produced byC¹ vector fieldY,s∈R, and letXbeC¹vector field. Ifsis fixed, andϕp(t)is an integral curve ofXthrough the pointp,ϕp(0)=p, thenψ^s◦ϕp(t)is an integral curve ofψ^s∗X through the pointψ^s(p). IfX|p=0, then(ψ^s_∗X)|ψ^s(p)=0.

Proof. The proof follows from [1] and [2]. In fact, ifϕp(t)is an integral curve of Xthrough the pointp, then

ψ^s◦ϕp(t)

t=0=ψ^s(p) (2.8)

and

ψ^s◦ϕp(t)

∗t· d

dt_t

=ψ^s_∗_ϕp(t)◦ϕp(t)∗t· d

dt_t

=ψ^s_∗_ϕp(t)·Xϕp(t)= ψ_∗^sX

ψ^s◦ϕp(t). (2.9)

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It follows thatψ^s◦ϕp(t)is an integral curve ofψ^s∗Xthrough the pointψ^s(p).

Next, due to

ψ^s_∗X|q=Dψ^s

ψ^−s(q)

·X

ψ^−s(q)

, q∈R². (2.10)

Setq=ψ^s(p), note that we already suppose X|p=0, again note that ψ^s is a one parameter transform which is produced byY, then

ψ∗^sX|ψ^s(p)=Dψ^s(p)·X(p)=Dψ^s(p)·X|p=0, (2.11) i.e.,ψ^s(p)is a singular point ofψ_∗^sX.

Lemma2.4. Letψ^s be a one parameter transform group which is produced byC¹ vector fieldY,s∈R, fixs, then the index of isolated singularity ofC¹vector fieldXis not changed under theψ^s transform.

Proof. In fact, by the condition of the lemma, it is known thatψ^sis a diﬀerentiable homeomorphism, then the lemma follows from [8, Theorem 4.2].

Next, ifX(µ) is a Lie rotated vector field, then Y is a corresponding vector field which satisfies (2.7), andψ^s is a one parameter transform group which is produced byY,s∈R.

Lemma2.5. LetX(µ)be a Lie rotated vector ﬁeld, for allε >0, there existδ=δ(ε), such that when|µ|< δ,ψ_∗^sX(µ)constitutes a rotated vector ﬁeld.

Proof. Let the singular points ofψ^s_∗X(µ), µ=0, on the planeR²bepµ1,...,pµ_k

and the singular points ofX(0)on the planeR²arep1,...,pm,∀ε >0, 0< ε1, let Sε(pµ_i)orSε(pj) (1≤i≤k, 1≤j≤m)be open neighborhood pµ_i (1≤i≤k)and pj(1≤j≤m), and radiusε, such thatSε(p)∩Sε(q)= ∅, wherepandq∈ {pµ_i}∪{pj} (1≤i≤k, 1≤j≤m),p=q. Letψ^s be a one parameter transform group which is produced byC¹vector ﬁeldY,s∈R. By the limit deﬁnition of Lie bracket, we have

ψ_∗^sX(µ)=ψ⁰_∗X(µ)+ s 1!

d

dt_s=0ψ^s_∗X(µ)+s² 2!

d²

dt²_s=0ψ^s_∗X(µ)+···

=X(µ)+s

1![X(µ),Y ]+s²

2![[X(µ),Y ],Y ]+···.

(2.12)

Next, we notice thatX(µ)can be unfolded as X(µ)=X(0)+µ

1!X_µ(0)+µ²

2!X_µ(0)+···, (2.13) since

[X(µ),Y ]=[X(0),Y ]+µ 1!

X_µ(0),Y +µ² 2!

X_µ(0),Y +···. (2.14)

Lets=µ, it follows from (2.12), (2.13), and (2.14) that ψ^µ∗X(µ)=X(0)+µ

X_µ(0)+[X(0),Y ] +1

2µ²

X_µ(0)+2

X_µ(0),Y +[[X(0),Y ],Y ]

+···. (2.15)

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At the ordinary points of R²\_k

i=1Sε(pµ_i)_m

j=1Sε(pj)

, for given ε >0, we sooner or later can ﬁndδ1=δ1(ε) >0, such that when|µ|< δ1, we have

ψ^µ∗X(µ)∧ ∂

∂µ

ψ^µ∗X(µ)

=X(0)∧

X_µ(0)+[X(0),Y ] +µX(0)∧

X_µ(0)+2

X_µ(0),Y +[[X(0),Y ],Y ] +···

=L(0)+O(µ) >0(<0)

(2.16) and letϑ(µ)be the crossing angle ofψ^µ∗X(µ)and the x1axis, for givenε >0, we sooner or later can ﬁndδ2=δ2(ε), such that when|µ|< δ2, at the ordinary points of R²\_k

i=1Sε(pµi)_m

j=1Sε(pj)

(ψ^µ∗X(µ)isX(0)whenµ=0,ϑ(0)is the crossing angle ofX(0)and thex1axis), so

0<|ϑ(µ)−ϑ(0)|< π. (2.17) Takeδ=min{δ1,δ2}, then when|µ|< δ,ψ^µ∗X(µ)constitutes a rotated vector ﬁeld.

Remark2.6. In accordance with the strict definition of rotated vector field, the singular points must be kept fixed, but the singular points ofψ^µ∗X(µ)in Lemma 2.5 can be moved as parameterµ is changed. In the unmistakable circumstance, when

|µ|< δ, we callψ∗^µX(µ)a rotated vector.

In the above lemma,δneeds not be a quite small positive number, i.e., 0< δ1 need not be set up. For the sake of distinctness, we cite an example to illustrate this equation.

Example2.7. LetX(µ)=(x2,−x1+µx2), if we takeY=(−x2/2,0), then at all the ordinary points ofX(0), we have

X(0)= 1 2

x²₁+x₂²>0, (2.18) that is,X(µ)is a Lie rotated vector ﬁeld.

Now we consider the range ofµ, because ψ^µ∗X(µ)=

1 2µx1+

1−1

4µ²

x2,−x1+1 2µx2

(2.19) so

ψ^µ∗X(µ)∧ ∂

∂µ

ψ^µ∗X(µ)

=1 2

x₁²+x₂²

−1

2µx1x2+1

8µ²x₂². (2.20) Formula (2.16) is compared with formula (2.20), we can ﬁnd thatO(µ)in formula (2.16) is replaced byO(µ)in formula (2.20),

O(µ)= −1

2µx1x2+1

8µ²x²₂ (2.21)

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yet go a step further calculating, we have ψ^µ∗X(µ)∧ ∂

∂µ

ψ^µ∗X(µ)

=1 2x²₂+1

8

µx2−2x12 (2.22)

which is larger than zero at the ordinary points ofX(0)and ψ^µ∗X(µ) for allµ∈R, but the range ofµ that satisﬁes formula (2.17) is|µ|<4, thus we takeδ=4, when

|µ|< δ=4,ψ^µ∗X(µ)constitutes rotated vector ﬁeld.

3. The motion of singular points. LetX(µ)be a Lie rotated vector ﬁeld, we require the singular points ofX(µ)to be strictly moved as parameterµis changed, and permit the singular points that have been moved disappear or decompose, but require the singular points that have been decomposed to be at most limited in number, which do not coincide with the singular points of the original vector ﬁeld.

Ifpis a singular point ofX(µ), we nameJµ(p)for index of singular pointpofX(µ), under the same circumstances,J0(p0)for index of singular pointp0ofX(µ),Jµ∗(q) for index of singular pointqofψ^µ∗X(µ) (µ=0).

Theorem3.1. LetX(µ)be a Lie rotated vector ﬁeld,X(0)|p0=0, and letY|p0=0. If the singular pointp0ofX(0)disappears or decomposes aspi(1≤i≤k)inX(µ) (µ= 0), thenJ0(p0)=0, andJµ(pi)=0(µ=0, 1≤i≤k).

Proof. First of all, we prove thatJ0(p0)=0. In fact, because ofX(µ)|p0=0(µ=0), utilize Lemma 2.3 and conditionY|p0=0, we know thatψ^µ∗X(µ)|p0=0(µ=0), it follows from Lemma 2.5, for givenδ >0, when|µ|< δ,ψ^µ∗X(µ)constitutes a rotated vector ﬁeld. Takeη >0 as quite small positive number, such that Sη(p0)does not contain the singular points ofψ^µ∗X(µ) (µ=0), and only contains the isolate singular pointp0 ofX(0). It is easy to know thatJµ∗(p0)=0 about ∂Sη(p0). By (2.17) of Lemma 2.5, it follows thatJ0(p0)=0 when|µ|< δ.

Using the same method, we prove Jµ∗(ψ^µ(pi))= 0 (µ = 0, 1 ≤i≤ k) and by Lemma 2.4, we ﬁndJµ(pi)=0(µ=0,1≤i≤k).

Corollary3.2. LetX(µ) be a Lie rotated vector ﬁeld,X(0)|p0 =0, ifY|p0=0, and moved the singular pointspi=ψ^−µ(p0) (µ=0, 1≤i≤k), thenJ0(p0)=0and Jµ(pi)=0(µ=0,1≤i≤k).

Proof. SinceX(0)|p0 =0, let the singular point of X(µ) (µ=0)disappears or decomposes intop1,...,pk points which do not coincide with singular point p0 of X(0), i.e.,X(µ)|p_i =0(1≤i≤k), yet because ofX(µ)|p₀=0(µ=0)and Y|p₀=0.

By Lemma 2.3, we haveψ^µ∗X(µ)|ψ^µ(p0)=0 andψ^µ∗X(µ)|ψ^µ(p_i)=0, but by condition ψ^µ(pi)=p0, we know thatψ^µ∗X(µ)|p0=0, as in the proof of Theorem 3.1, we can prove thatJ0(p0)=0 andJµ(pi)=0(µ=0,1≤i≤k).

Corollary3.3. LetX(µ) be a Lie rotated vector ﬁeld,X(0)|p0 =0, ifY|p0=0, but for some i0 (1≤i0≤ k), set up ψ^µ(pi0)=p0 (µ =0), then J0(p0)=jµ(pi0), Jµ(pi)=0(µ=0,1≤i≤kandi=i0).

Example3.4. LetX(µ)=

x₂²,−x1+µ

, and let Y=

3x1−αx2,2x2

(3.1)

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when|µ|< δ, we takeα >0 andα1, on the range ofD= {(x1,x2)|x2< α⁻¹} ⊂R², at all ordinary points∈DofX(0), set up

L(0)=αx₁²+x₂²−αx²₃>0, (3.2) that is,X(µ)constitutes a Lie rotated vector ﬁeld onD, the singular points ofX(µ)are strictly moved as parameterµis changed. We note thatY|p0=0,p0=(0,0)is singular point ofX(0), by Theorem 3.1, we can ﬁnd thatJ0(p0)=0 andJµ(pi)=0(µ=0), wherepi=(µ,0).

Theorem3.5. LetX(µ)be a Lie rotated vector ﬁeld,X(0)|p0=0,p0is elementary.

(1)IfY|p₀=0, thenp0cannot be moved as parameterµis changed.

(2)IfY|p0=0, thenp0can be moved as parameterµis changed, and the moved point is the singular pointψ^−µ(p0)ofX(µ) (µ=0).

Proof. (1) We noteJ0(p0)= ±1=0, it is proved immediately from Theorem 3.1.

(2) First of all, we prove thatp0 is indeed moved asµ is changed, suppose that it is not real, i.e.,p0 is not moved asµ is changed, then that X(µ)|p0 =0 (µ=0), by Lemma 2.3, we know thatψ∗^µX(µ)|ψ^µ(p₀)=0. Becausep0is isolate singular point ofX(0), we take ¯δ >0 and ample smallη >0, it follows thatψ^µ(p0)∈Sη(p0). When 0<|µ|<δ < δ, then for¯ ∂Sη(p0), we haveJµ∗(p0)=0 (sinceψ^µ∗X(µ)|p0=0), where µ=0. ButJ0(p0)= ±1=0, this is a contradiction from Lemma 2.5. Thus we have provedp0is indeed moved asµis changed, and by Corollaries 3.2 and 3.3, it follows thatp0is moved as the singular pointψ^−µ(p0)ofX(µ) (µ=0)whenµ is changed.

Lemma3.6. LetX(µ)be a Lie rotated vector ﬁeld,X(0)|p0=0, and there is an elliptic region at the singular pointp0.

(1)IfY|p0=0, then the singular pointp0cannot be moved when parameterµ=0.

(2)IfY|p0=0, then when parameterµ=0, singular pointp0 is moved, andp0be moved as singular pointψ^−µ(p0)ofX(µ).

Proof. (1) We already know thatY|p0, suppose the original equation is not real, then when µ=0, singular pointp0 is moved, thus we letp0 moved as the singular point pµ ofX(µ), X(µ)|pµ =0, pµ=p0, µ=0. From Lemma 2.3, we know that ψ∗^µX(µ)|ψ^µ(pµ)=0, and byY|p0=0, we know thatψ^µ(pµ)=p0(µ=0). LetΩbe an elliptic region at the singular pointp0ofX(0), for arbitrary ﬁxedµ (0<|µ|< δ), it is sure to have some elliptic trajectoryr ofX(0), which does not contain the point ofψ^µ(pµ)onr and inr. Sincer has single direction, and there is no singular point ofψ^µ∗X(µ)onrand inr. By Lemma 2.5, we can know that positive half trajectory or negative half trajectory ofψ^µ∗X(µ)which pass through the pointpwill wander about without a home to go to, wherepis any point which passes the inner region ofr, this is a contradiction.

(2) Now we knowY|p0=0, yet use reduction to absurdity. Suppose, whenµ=0, singular pointp0is not moved, i.e., establishX(µ)|p0=0, namely we haveψ^µ∗X(µ)|ψ^µ(pµ)= 0, andψ^µ(p0)=p0. The method of the proof is completely alike as part (1), we can prove it is a contradiction. Thus let µ =0, singular point p0 is moved as singular point pµ (p0 =pµ) of X(µ), i.e., ψ^µ∗X(µ)|ψ^µ(pµ) =0 (µ= 0). If ψ^µ(pµ)= p0, the

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method of the proof is alike as in part (1), yet it is a contradiction, thus only establish ψ^µ(pµ)=p0, orpµ=ψ^−µ(p0).

Lemma3.7. LetX(µ)be a Lie rotated vector ﬁeld,X(0)|p₀=0, and whenµ=0,p0

is moved as the singular pointpµ(pµ=p0)ofX(µ)asµis changed. IfY|p0=0, then for singular pointpµ(orp0), at least there are a positive half trajectory and a negative half trajectory ofX(µ)(orX(0)) to get into it.

Proof. We only prove the circumstance of pointpµ(the proof is completely alike as the circumstance of pointp0).

From Lemma 3.6, we know that there is no elliptic region which links with the singular pointp0ofX(0), the same do the singular pointpµofX(µ), and from Theorem 3.1, we know that the index ofpµofX(µ)is zero. Takepµ as circular center, make the circumference of a circlelwith radius rather small, and let that hyperbolic region of pointpµwhich intersects with the circumference of a circlelhash. By the Bendixson’s formula in§6of Chapter 3 of [8], we can immediately ﬁndh=2.

From Lemmas 3.6and 3.7, we have the following theorem.

Theorem3.8. LetX(µ)be a Lie rotated vector ﬁeld,X(0)|p₀=0, and letY|p₀=0, then some singulars while can be moved as parameterµis changed inX(µ)only contain two hyperbolic regions and their index is zero.

Acknowledgement. The authors express grateful thanks to Professor Yaoxian Wang for his help and direction in the work.

References

[1] W. M. Boothby,An Introduction to Diﬀerentiable Manifolds and Riemannian Geometry, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London, 1975. Zbl 333.53001.

[2] S. S. Chen and W. H. Chen,Lectures in Diﬀerential Geometry, Beijing Univ. Press, Beijing, 1983.

[3] Y. Y. Chen,Limit cycles and quasirotational vector ﬁelds, Acta Math. Sinica32(1989), no. 6, 786–792. MR 91c:58113.

[4] G. F. D. Duﬀ,Limit-cycles and rotated vector ﬁelds, Ann. of Math. (2)57(1953), 15–31.

MR 14,751c. Zbl 050.09103.

[5] Z. E. Ma,The motion of singular closed orbit rotated vector ﬁelds, J. Xi’an Jiaotong Univ.4 (1978), 49–65.

[6] L. M. Perko,Rotated vector ﬁelds and the global behavior of limit cycles for a class of qua- dratic systems in the plane, J. Diﬀerential Equations18(1975), 63–86. MR 51#10752.

Zbl 297.34024.

[7] Y. Q. Ye, S. L. Cai, L. S. Chen, K. C. Huang, D. J. Luo, Z. E. Ma, E. N. Wang, M. S. Wang, and X. A. Yang,Theory of Limit Cycles, American Mathematical Society, Providence, R.I., 1986. MR 88e:58080.

[8] Z. F. Zhang, T. R. Ding, W. Z. Huang, and Z. X. Dong,The Qualitative Theory of Diﬀerential Equation, Science Publishing House, Beijing, 1985.

Jie Wang and Chen Chen: School of Electric Power, Shanghai Jiaotong University, Shanghai,200030, China