We obtain center conditions for a $O$-symmetric system of degree 5 for which
the origin is a uniformly isochronous singular point. In the revised paper some
misprints are corrected in the reference list.Comment: 9 pages, 0 figures, LaTeX 2.0

Evgenii P. Volokitin

International Journal of Mathematics and Mathematical Sciences

English

arXiv

Crossref

IJMMS 29:11 (2002) 625–632
PII. S0161171202012802
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
CENTER CONDITIONS FOR A SIMPLE CLASS
OF QUINTIC SYSTEMS
EVGENII P. VOLOKITIN
Received 13 April 2001 and in revised form 25 September 2001
We obtain center conditions for an O-symmetric system of degree 5 for which the origin
is a uniformly isochronous singular point.
2000 Mathematics Subject Classiﬁcation: 34C05, 34C25.
1. Consider a planar diﬀerential system
x˙ =y+xRn−1(x,y), y˙ =−x+yRn−1(x,y), (1.1)
where Rn−1(x,y) is a polynomial in x,y of degree n−1.
System (1.1) has a unique singular point O(0,0) whose linear part is of center type.
Orbits of (1.1) move around the origin with a constant angular velocity and the
origin is a uniformly isochronous singular point.
In [3], the following problem was proposed.
Problem 1.1. Identify (1.1) of odd degree that are O-symmetric (not necessarily
quasi-homogeneous) having O as a (uniformly isochronous) center.
We solve this problem for n = 5 and derive necessary and suﬃcient center condi-
tions for the system
x˙ =y+x(ax2+bxy+cy2+dx4+ex3y+fx2y2+gxy3+hy4),
y˙ =−x+y(ax2+bxy+cy2+dx4+ex3y+fx2y2+gxy3+hy4),
a,b,c,d,e,f ,g,h∈R.
(1.2)
Theorem 1.2. The origin is a center of (1.2) if and only if one of the following sets
of conditions is satisﬁed:
a= b = c = 0, f =−3(d+h); (1.3a)
a= c = d= f = h= 0; (1.3b)
a≠ 0, c =−a, f = 3b(ae−bd)(
2a2
) ,
g =
(
2a2bd+(2a2−b2)(bd−ae))(
2a3
) ,
h=
(−2a2d+b(bd−ae))(
2a2
) .
(1.3c)
626 EVGENII P. VOLOKITIN
Proof
Necessity. To describe the behaviour of trajectories of (1.2) near the origin, we
construct the comparison function (see [6])
F(x,y)=
(
x2+y2)
2
+f3(x,y)+f4(x,y)+··· , (1.4)
where fk is a homogeneous polynomial of degree k whose derivative is
dF
dt
=D1
(
x4+y4)+D2(x6+y6)+D3(x8+y8)+··· . (1.5)
The number of the ﬁrst coeﬃcient Di other than zero deﬁnes the multiplicity of a
complex focus and the sign of this coeﬃcient deﬁnes stability of a focus; if Di = 0 for
all i, the origin is a center of (1.2). We refer to coeﬃcientsDi as the Poincaré-Lyapunov
constants.
To ﬁnd the Poincaré-Lyapunov constants of a system x˙ = p(x,y), y˙ = q(x,y) with
a linear center, we used computer algebra and wrote a Mathematica code that rests
on the Poincaré algorithm in [6]; (see [9] for more details)
PLconst[n_] :=
Module[{dF, ff, fF, x, y, pP, qQ, dD},
fF[2] := (xˆ2+yˆ2)/2;
fF[i_] := Sum[ff[i-j, j]*xˆ(i-j)*yˆj, {j, 0, i}];
pP[1] := y;
pP[i_] := Sum[p[i-j, j]*xˆ(i-j)*yˆj, {j, 0, i}];
qQ[1] := -x;
qQ[i_] := Sum[q[i-j, j]*xˆ(i-j)*yˆj, {j, 0, i}];
dF[k_] := (Sum[D[fF[i], x]*pP[k+1-i], {i, 2, k}]+
Sum[D[fF[i], y]*qQ[k+1-i], {i, 2, k}])//Expand;
Do[
Solve[Table[Coefficient[dF[k], xˆ(k-j) yˆj],
{j, 0, k}]
==Table[0, {k+1}],
Table[ff[k-j, j], {j, 0, k}]
]/.Rule->Set;
Solve[Table[Coefficient[dF[k+1], xˆ(k+1-j)*yˆj],
{j, 0, k+1}]
==Flatten[{dD[k], Table[0, {k}], dD[k]}]
&&ff[0, k+1]==0,
Flatten[{Table[ff[k+1-j, j],{j, 0, k+1}], dD[k]}]
]/.Rule->Set,
{k, 3, 2n+1, 2}];
Table[Numerator[Together[dD[k]]],{k, 3, 2n+1, 2}]
]
The procedure PLconst[n] returns a list {D1, . . . ,Dn} of the Poincaré-Lyapunov
constants if we deﬁne the coeﬃcients pij,qij (2 ≤ i+j ≤ 2n+1) in the Taylor series
expansion of the functions p(x,y) and q(x,y) beforehand.
CENTER CONDITIONS FOR A SIMPLE CLASS OF QUINTIC SYSTEMS 627
Using this procedure, we found the ﬁrst four Poincaré-Lyapunov constants of (1.2).
D1 = 2(a+c),
D2 =−4ab−4bc+3d+f +3h,
D3 = 2
(−85a3+15ab2−67a2c+15b2c+61ac2+43c3−24bd−34ae
−22ce−12bf −50ag−38cg−48bh),
D4 = 44600a3b+2736ab3+84696a2bc+2736b3c+47688abc2+7592bc3
−37120a2d−1782b2d−32552acd−2704c2d+2364abe+1284bce
−2673de−6120a2f −234b2f −3384acf +792c2f −891ef
+6876abg+5076bcg−3807dg−1269fg+4720a2h+1098b2h
+31448ach+19456c2h−2673eh−3807gh.
(1.6)
It is easy to verify that the equalities Di = 0; i = 1,2,3,4, are equivalent to the
following relations:
a+c = 0, 3d+f +3h= 0,
3ce−bf +3cg−6bh= 0, 2c2f −3bcg+3b2h= 0. (1.7)
If a = 0 then our simultaneous polynomial equations have two sets of solutions
indicated in (1.3a) and (1.3b). If a≠ 0 then, in view of the condition c =−a, we see that
the other three equations constitute a nondegenerate linear system for determining
the variables f , g, h. The solution is given by (1.3c).
The necessity part of the theorem is proved.
Sufficiency
Case 1. System (1.2) now takes the form
x˙ =y+x(dx4+ex3y+fx2y2+gxy3+hy4)≡y+xp4(x,y),
y˙ =−x+y(dx4+ex3y+fx2y2+gxy3+hy4)≡ x+yp4(x,y). (1.8)
This is a quasi-homogeneous system of degree 5 whose coeﬃcients satisfy the
equality f = −3(d+h), that is, the necessary and suﬃcient center condition in the
case we study (see [2]).
Case 2. System (1.2) now takes the form
x˙ =y+x2y(b+ex2+gy2),
y˙ =−x+xy2(b+ex2+gy2). (1.9)
The planar diﬀerential system
x˙ = p(x,y), y˙ = q(x,y) (1.10)
is said to be reversible (in the sense of Z˙ola¸dek), if its orbits are symmetric with respect
to a line passing through the origin.
System (1.10) is reversible if there is a linear transformation S : R2 → R2, sending
a point (x,y) to the point (x′,y ′) symmetric to (x,y) with respect to the line αx+
βy = 0 and satisfying the condition S(p(x,y),q(x,y))=−(p(S(x,y)),q(S(x,y))).
628 EVGENII P. VOLOKITIN
A more general condition of reversibility is as follows:
2αβ
(
p(x,y)p
(
x′,y ′
)−q(x,y)q(x′,y ′))
+(β2−α2)(p(x,y)q(x′,y ′)+p(x′,y ′)q(x,y))= 0. (1.11)
It is well known that if (1.10) is reversible and has a linear center at the origin then
the origin is a center of this system (cf. [6]).
Obviously, system (1.9) is reversible because its trajectories are symmetric with
respect to both coordinate axes. So, the origin is a center for (1.9).
Case 3. System (1.2) now takes the form
(
2a3
)
x˙ = (2a3)y+x(ax2+bxy−ay2)
×(2a3+2a2dx2−2abdxy+2a2exy+2a2dy2−b2dy2+abey2),(
2a3
)
y˙ =−(2a3)x+y(ax2+bxy−ay2)
×(2a3+2a2dx2−2abdxy+2a2exy+2a2dy2−b2dy2+abey2).
(1.12)
It turns out that system (1.12) is reversible. Its trajectories are symmetric with respect
to each of the two perpendicular lines deﬁned by the equation ax2+bxy−ay2 = 0.
The appropriate linear transformation S is given by each of the two matrices
S1,2 =±
(
4a2+b2)−1/2
(−b 2a
2a b
)
. (1.13)
This fact is conﬁrmed by straight calculations. We used Mathematica here.
With the coordinate change x  x cosϕ+y sinϕ, y  −x sinϕ+y cosϕ, where
the angle ϕ is deﬁned from the condition atan2ϕ+b tanϕ−a = 0, system (1.12)
becomes as follows:
x˙ =y+x2y(b1+e1x2+g1y2),
y˙ =−x+xy2(b1+e1x2+g1y2). (1.14)
Hence the origin is a center for (1.2) in this case once again.
The theorem is proved.
2. It is known that isochronism of a center of a planar polynomial system is equiva-
lent to the existence of an analytic transversal system commuting with a given system
in a neighborhood of a center [7]; observe that an arbitrary polynomial system with
isochronous center does not necessarily commute with a polynomial system [4, 8].
It is proved in [1] that if the systems
x˙ = p(x,y), y˙ = q(x,y),
x˙ = r(x,y), y˙ = s(x,y) (2.1)
commute, then µ(x,y)= 1/(p(x,y)s(x,y)−q(x,y)r(x,y)) is an integrating factor
of both systems.
CENTER CONDITIONS FOR A SIMPLE CLASS OF QUINTIC SYSTEMS 629
Thereby, if both commuting systems are polynomial thenwe can ﬁnd the integrating
Darboux factor for the given system and integrate the latter, (about the method of
Darboux and the relevant deﬁnitions see, for example, [5]).
We now state the following fact that will be useful later.
Considering (2.1), assume that
p(x,y)=y+xR(x,y), q(x,y)=−x+yR(x,y),
r(x,y)= xQ(x,y), s(x,y)=yQ(x,y), (2.2)
where R(x,y),Q(x,y) are polynomials in x,y . Then the algebraic curves x2+y2 = 0,
Q(x,y)= 0 are invariants for each of these systems.
Indeed, it is immediately obvious that x2+y2 = 0 is an invariant of both systems
with the cofactor 2R(x,y) and 2Q(x,y), respectively. The curve Q(x,y) = 0 is an
invariant of the second system with the cofactor xQx(x,y)+yQy(x,y).
Because our systems commute, the Lie bracket of vector ﬁelds (p,q) and (r ,s)
vanishes and we have
px(x,y)r(x,y)+py(x,y)s(x,y)−rx(x,y)p(x,y)−ry(x,y)q(x,y)= 0,
xQ(x,y)
(
R(x,y)+xRx(x,y)
)+yQ(x,y)(1+xRy(x,y))
−p(x,y)(Q(x,y)+xQx(x,y))−xq(x,y)Qy(x,y)= 0,
(2.3)
or
x
(
Qx(x,y)p(x,y)+Qy(x,y)q(x,y)
)
= (R(x,y)+xRx(x,y))xQ(x,y)+(1+xRy(x,y))yQ(x,y)−Q(x,y)p(x,y)
= (R(x,y)+xRx(x,y))xQ(x,y)+(1+xRy(x,y))yQ(x,y)
−Q(x,y)(y+xR(x,y))
= x(xRx(x,y)+yRy(x,y))Q(x,y).
(2.4)
We see that the curve Q(x,y) = 0 is an invariant with the cofactor xRx(x,y) +
yRy(x,y).
In this case, µ(x,y)= 1/(Q(x,y)(x2+y2)) is an integrating Darboux factor.
3. In each of the three cases, we have found a nontrivial polynomial system com-
muting with the respective system.
In Case 1 such a system is
x˙ = x(1+ex4−4dx3y+4hxy3−gy4)≡ x(1+q4(x,y)),
y˙ =y(1+ex4−4dx3y+4hxy3−gy4)≡y(1+q4(x,y)). (3.1)
The function
µ(x,y)= 1(
x2+y2)(1+q4(x,y)) (3.2)
630 EVGENII P. VOLOKITIN
is the integrating Darboux factor of (1.8) and the function
H(x,y)=
(
x2+y2)2
1+q4(x,y) (3.3)
is the ﬁrst rational integral of (1.8).
The algebraic curves x2+y2 = 0 and 1+q4(x,y)= 0 are invariant curves for (1.8).
According to [2], system (1.1) has a center of type Bk, 1 ≤ k ≤ n−1, whose bound-
ary is a ﬁnite union of k unbounded open trajectories. Using (3.3), in Case 1 we can
describe this boundary explicitly:
= 1(
c0−q4(cosϕ,sinϕ)
)1/4 , (3.4)
where c0 =max[0,2π] q4(cosϕ,sinϕ), x = cosϕ, y = sinϕ.
A straight analysis of this expression allows us to conclude that in our case a center
may be of type B2 or B4 only.
In Case 2, (1.9) commutes with the system
x˙ = (e−g)x+x(ex2+gy2)(b+ex2+gy2),
y˙ = (e−g)y+y(ex2+gy2)(b+ex2+gy2). (3.5)
This permits us to ﬁnd an integrating Darboux factor
µ(x,y)= 1(
x2+y2)(e−g+(ex2+gy2)(b+ex2+gy2)) . (3.6)
The algebraic curves x2+y2 = 0, e−g+ (ex2+gy2)(b+ ex2+gy2) = 0 are the
invariant ones for (1.9).
If b = 0, then (1.9) is a system of the form (1.8) for which the condition f =−3(d+h)
is obviously fulﬁlled. Then its ﬁrst integral is
H(x,y)=
(
x2+y2)2
1+ex4−gy4 . (3.7)
If b ≠ 0 then we may suppose that b = 1. The general case reduces to this by the
change of variables x → x/√b, y → y/√b for b > 0 or x → y/√−b, y → x/√−b,
t→−t for b < 0.
Then our system takes the form
x˙ =y+x2y(1+ex2+gy2)≡X1(x,y),
y˙ =−x+xy2(1+ex2+gy2)≡ Y1(x,y). (3.8)
The function µ1(x,y), which is equal to µ(x,y) from (3.6) for b = 1, is an integrating
factor of (3.8). The ﬁrst integral H1(x,y) of (3.8) associated to the integrating factor
µ1(x,y) can be computed via the integral
H1(x,y)=
∫
µ1(x,y)Y1(x,y)dx+m(y) (3.9)
imposing the condition ∂H1(x,y)/∂y =−µ1(x,y)X1(x,y).
CENTER CONDITIONS FOR A SIMPLE CLASS OF QUINTIC SYSTEMS 631
B2 B4
Figure 3.1
We have
H1(x,y)= 1
2(e−g)
(
1
2
ln
(
e−g+(ex2+gy2)(1+ex2+gy2))
− ln(x2+y2)+ 1√
4(e−g)−1 arctan
1+2ex2+2gy2√
4(e−g)−1
)
.
(3.10)
We used Mathematica here.
Then the function H(x,y) = exp(−4(e−g)H1(x,y)) is the ﬁrst integral of (3.8)
also. It has the form
H(x,y)=
(
x2+y2)2
e−g+(ex2+gy2)(1+ex2+gy2)
×exp
(
− 2√
4(e−g)−1 arctan
1+2ex2+2gy2√
4(e−g)−1
)
.
(3.11)
Since (3.8) has a unique ﬁnite singular point at the origin, the phase portraits are
obtained by studying the points at inﬁnity. A standard inspection of the location and
types of such points on the equator of the Poincaré sphere allows us to conclude that
(3.8) has phase portraits of two types only: a center is of type B2 when eg ≥ 0 or of
type B4 when eg < 0.
The relevant phase portraits are presented in Figure 3.1. These portraits are fairly
typical (cf. [5]) and we do not supply explanations for them.
In Case 3, a commuting system and integrating Darboux factor and ﬁrst integral
may be found on considering that (1.12) is equivalent to (3.8).
Observe that for d = e = 0, (1.12) is a quasi-homogeneous O-symmetric cubic sys-
tem of the form
x˙ =y+x(ax2+bxy−ay2),
y˙ =−x+y(ax2+bxy−ay2). (3.12)
It commutes with the system
x˙ = x+x(bx2−2axy),
y˙ =y+y(bx2−2axy), (3.13)
632 EVGENII P. VOLOKITIN
and has the ﬁrst integral
H(x,y)= x
2+y2
1+bx2−2axy . (3.14)
Summarizing, we conclude that Figure 3.1 presents all possible phase portraits of
(1.2) having the origin as a center.
Acknowledgments. The author thanks the referees for their helpful remarks
and suggestions. This work was supported by Russian Foundation for Basic Research.
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Evgenii P. Volokitin: Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia
E-mail address: volok@math.nsc.ru
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Center conditions for a simple class of quintic systems

We obtained center conditions for an O-symmetric system of degree
5 for which the origin is a uniformly isochronous singular point

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Center conditions for a simple class of quintic systems

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