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Characterization of isochronous foci for
planar analytic differential systems ∗
Jaume Gin´
e & Maite Grau
Departament de Matem`atica. Universitat de Lleida.
Avda. Jaume II, 69. 25001 Lleida, SPAIN.
E–mails: gine@eps.udl.es and mtgrau@matematica.udl.es
Abstract
We consider the two-dimensional autonomous systems of differential equations
of the form:
˙x=λx −y+P(x, y),˙y=x+λy +Q(x, y),
where P(x, y) and Q(x, y) are analytic functions of order ≥2. These systems
have a focus at the origin if λ6= 0, and have either a center or a weak focus if
λ= 0. In this work we study necessary and sufficient conditions for the existence
of an isochronous critical point at the origin. Our result is original when applied
to weak foci and gives known results when applied to strong foci or to centers.
Keywords: nonlinear differential equations, isochronicity.
AMS classification: Primary 34C05; Secondary 34C23, 37G15.
1 Introduction
Let us consider an autonomous differential system:
˙x=λx −y+P(x, y),˙y=x+λy +Q(x, y),(1.1)
where P(x, y) and Q(x, y) are analytic functions in a neighborhood Uof the origin O
and of order greater or equal than two. We assume that Ois an isolated singular point
of (1.1). We denote by Xthe equivalent vector field:
X= (λx −y+P(x, y)) ∂
∂x + (x+λy +Q(x, y )) ∂
∂y .
An isolated singular point of (1.1) is said to be a focus if it has a neighborhood where
all the orbits spiral in forward or backward time. An isolated singular point of (1.1) is
said to be a center if it has a punctured neighborhood filled of periodic orbits. System
(1.1) has a strong focus at the origin if λ6= 0 and it has a weak focus or a center at
∗The authors are partially supported by a MCYT grant number BFM 2002-04236-C02-01. The first
author is partially supported by DURSI of Government of Catalonia “Distinci´o de la Generalitat de
Catalunya per a la promoci´o de la recerca universit`aria. The second author is also partially supported
by a FPU grant with reference AP2000-3585.
1
the origin if λ= 0.
A main problem is that of studying the existence and properties of periodic solu-
tions in a neighborhood of the origin of (1.1). In this field, different methods have
been used to study isolated periodic solutions, i.e. limit cycles, or non-isolated ones,
i.e. period annulus. The stability of the singular point Odoes not imply the stability
of the cycles close to the singular point. In fact, a non-isolated cycle is Liapunov stable
if and only if every neighboring cycle has the same period. This fact motivates the def-
inition of isochronicity. We give a precise definition of isochronicity in the forthcoming
paragraph. Isochronicity has been widely studied not only for its physical meaning and
for its role in stability theory, but also for its relationship with bifurcation problems
and to boundary value problems.
An essential tool to study the stability of the origin of system (1.1) is the Poincar´e
map, see [14, 15]. Let us consider a neighborhood Uof the origin and let Σ be a section
of system (1.1) through the origin, that is, a transversal curve through the origin for
the flow of system (1.1).
More precisely, we define a section through the origin as a simple arc without contact
with the origin Oas an endpoint. See the book of Andronov et al. [1], page 55, for a
precise definition of simple arc without contact. We also need some assumptions on its
regularity for technical reasons. Given a section through the origin Σ ⊂R2, we consider
a parameterization c:R→R2such that Σ = {c(σ)|σ∈R}and limσ→−∞ c(σ) = O.
We assume that c(σ) is analytic for all σ∈R.
For each point p∈Σ, the flow of system (1.1) through pwill cross Σ again at a
point P(p)∈Σ near p. The map p7→ P (p) is called the Poincar´e map. If we denote by
Φt(p) the flow of system (1.1) with the initial condition Φ0(p) = p, we can define the
Poincar´e map in the following way. Given p∈Σ, there is a unique analytic function
τ(p) such that Φτ(p)(p)∈Σ and Φt(p)6∈ Σ for any 0 < t < τ(p), see [14]. In these
terms, we have P(p)=Φτ(p)(p). We remark that both functions Pand τdepend on
the chosen section Σ. The function τ: Σ →R+is called the period function. As usual
R+denotes the set of positive real numbers. In this paper we study the existence of
a section Σ such that τ: Σ →R+is constant. When such a Σ exists, we say that the
origin Oof (1.1) is isochronous and that Σ is an isochronous section.
We will call a center an analytic system of the form (1.1) with λ= 0 and where
the origin Ois a center. Isochronicity has been widely studied for centers, see for
instance [8] and the references therein. We remark that the period function of a center
does not depend on the chosen section Σ. The main methods used in order to study
isochronicity of centers can be roughly classified in two categories, linearization and
commutation.
Finding a linearization for a center Xmeans finding a transformation φ:U → U
analytic in a neighborhood of the origin such that Dφ(O) = I, where I denotes the
2×2 identity matrix, such that the transformed system is a linear center, that is
φ∗(X) = −y ∂/∂x +x ∂/∂ y. If such a transformation exists, then all the orbits have the
same period, coinciding with the period of the linear center. So, a center is isochronous
if and only if a linearization can be found.
Finding a commutator for a center Xmeans finding a second vector field Yanalytic
in a neighborhood of the origin and of the form
Y= (x+A(x, y)) ∂
∂x + (y+B(x, y)) ∂
∂y ,(1.2)
2
with Aand Banalytic functions of order ≥2, such that the Lie bracket [X,Y] of the
center Xand Yidentically vanishes.
An isolated singular point of a real planar analytic autonomous system is called a
star node if the linear part of the vector field at the singular point has equal non-zero
eigenvalues and it is diagonalizable. Clearly, the origin is a star node for (1.2). By an
affine change of coordinates any vector field with a star node can be brought to the
form (1.2).
Given two analytic vector fields defined in an open set U,Xand Y, we say that
they are transversal at noncritical points when Xand Yhave isolated singular points,
they both have the same critical points in U, and if p∈ U is such that X(p)6= 0 then
the function given by the wedge product of Xand Yis not zero at p. From now on,
we always assume that Xand Yare analytic vector fields defined in a neighborhood
Uof the origin and transversal at non critical points.
We will always consider analytic vector fields although many of the stated results
apply also for vector fields with weaker differentiability restrictions. The results of
Sabatini [12] go on this direction. We define a smooth function as a function of class
C∞in a neighborhood Uof the origin O. Analogously, a smooth vector field is defined
by smooth functions.
The following result, proved in [2] (Theorem 2.4, page 140), characterizes centers
in terms of Lie brackets.
Theorem 1.1 [2] System (1.1) with λ= 0 has a center at the origin if, and only
if, there exists a smooth vector field Uof the form (1.2) and a smooth scalar function
ν(x, y)with ν(0,0) = 0 such that [X, U ] = νX.
The most important result on characterization of isochronous centers appears in [16,
10]. A further study can be found in, for instance, [2, 7] and the references therein. See
[3] for a constructive method of Uand νin special cases for polynomial vector fields.
The following theorem, which is stated and proved in [10] (Theorem 2, page 98),
gives the equivalence between commutation and isochronicity for centers.
Theorem 1.2 [10] Let Obe a center of system (1.1), with λ= 0. Then Ois isochro-
nous for system (1.1) if, and only if, there exists an analytic vector field Yof the form
(1.2), transversal to Xand such that [X,Y]≡0.
Another work on commuting systems is [11], where M. Sabatini discusses the local and
global behavior of the orbits of a pair of commuting systems and gives several illustra-
tive examples. A wide collection of commutators and linearizations can be found in [4].
When a center is isochronous, it is possible to construct an isochronous section Σ,
see for instance [12]. However, the existence of an isochronous section is not strictly
dependent on the existence of a center. A system can have a singular point of focus type
with an isochronous section. This implies the existence of a neighborhood covered with
solutions spiralling towards the singular point, all meeting Σ at equal time intervals.
Such a behavior may occur, for instance, in a pendulum with friction, or in an electric
circuit with dissipation, see also [12]. Our main result, Theorem 3.1, characterizes
when the origin of system (1.1) is isochronous, even when the origin is a center, a weak
focus or a strong focus. In this paper, we adapt the two different techniques usually
used for isochronous centers, in order to study isochronous foci.
In Section 2 we summarize the known results on isochronicity for foci. It is shown
that a strong focus of an analytic system is always isochronous. All the results described
in Section 2 only apply for systems of the form (1.1) with λ6= 0 or for centers.
3
Section 3 contains the main theorem of this work which characterizes isochronicity
for the origin of a system (1.1). Our result is original when applied to weak foci
and gives known results when applied to strong foci or to centers. We modify the
commutators’ method to study isochronous critical points. We prove that system (1.1)
has a transversal vector field Ysuch that the vector field [X,Y] is proportional to Y
if, and only if, system (1.1) has an isochronous critical point at the origin.
We give two examples of weak isochronous foci and we give an example of a family
of quadratic systems depending on a parameter w∈Rwhich never has an isochronous
critical point at the origin. When w= 0 the system is a center and when w6= 0 the
system is a weak focus (stable if w < 0 and unstable if w > 0). Hence, we show that
there is no isochronous section for any system of this family.
2 Summary of known results
We denote by Uany open neighborhood of the origin and by ρ:U → R+×Rthe change
to polar coordinates, that is, ρ(x, y )=(r, θ) with r=px2+y2and θ= arctan(y/x).
As usual, ρ∗is the push-forward defined by ρand ρ∗is the corresponding pull-back.
In order to give the definition of isochronous critical point, we consider the form of
(1.1) in polar coordinates, that is, ρ∗(X) = rf(r, θ)∂
∂r +g(r, θ)∂
∂θ , where fand gare
analytic functions in a neighborhood of ρ(O).
Definition 2.1 The critical point Oof (1.1) is said to be isochronous if there ex-
ists a local analytic change of variables φwith Dφ(O)=Iand such that ρ∗φ∗(X) =
rf (r, θ)∂
∂r +g(θ)∂
∂θ .
A system (1.1) with an isochronous critical point at the origin is more easily written
using the arc–length ϕ, defined by ϕ=Rθ
0dθ/g(θ), as new angular variable. In this
formulation we end up to the following definition.
Definition 2.2 The critical point Oof (1.1) is said to be isochronous if there ex-
ists a local analytic change of variables φwith Dφ(O)=Iand such that ρ∗φ∗(X) =
rf (r, θ)∂
∂r +k∂
∂θ ,k∈R,k6= 0.
The existence of an isochronous section is equivalent to the existence of the local ana-
lytic change of variables φ, as we will show in Theorem 3.1. We state the definition of
isochronous critical point by means of φsince this is its classical definition which let
us give the summary of known results.
Linear foci, (λx −y)∂/∂x + (x+λy)∂/∂y, are isochronous since their angular
speed is constant along rays through the origin. For a linear focus, every ray through
the origin is an isochronous section. We say that a vector field Xof the form (1.1) is
linearizable when there exists a local change of variables φwith Dφ(O) = Isuch that
φ∗(X) is a linear focus.
By the above definition, every analytic linearizable focus is isochronous. If φis
the linearizing transformation and Σ is a ray, then φ−1(Σ) is an isochronous section
of the analytic linearizable focus. Next theorem, which is a special case of classical
Poincar´e’s Theorem, shows that every strong focus of an analytic system is linearizable
and therefore isochronous. For a proof, see [5, 15].
Theorem 2.3 [15] Let us consider the planar real analytic system
˙x=αx −βy +g1(x, y),˙y=βx +αy +g2(x, y),(2.1)
4
with αβ 6= 0, and g1and g2are of second order in xand y. Then there exists a real
local analytic change of variables φ(x, y) = (u, v)with Dφ(O)=Iwhich transforms
system (2.1) into ˙u=αu −βv,˙v=βu +αv.
This result can also be stated for a system of the form (2.1) satisfying weaker differen-
tiability restrictions. Since we are only concerned with analytic vector fields, we state
the result only for the analytic case.
We have seen that every analytic linearizable focus is isochronous, but finding the
linearization, and hence the isochronous sections, is usually too difficult. Next theorem
proved in [12] shows that it is not necessary to find the explicit form of the linearization,
since the orbits of a suitable commutator are isochronous sections of X.
Theorem 2.4 [12] If the vector field Xgiven by (1.1) has a focus Oand a nontrivial
commutator Ywith a star node at O, then every orbit of Ycontained in a neighborhood
of Ois an isochronous section of X.
This result only applies when the vector field Xhas a strong focus at the origin or has
a center because if the vector field Xhas a weak focus at the origin with a nontrivial
commutator Ywith a star node at O, then by Theorem 1.1 the vector field has a center
at the origin. Next corollary proved in [12] shows that every system with a strong focus
and a nontrivial commutator has a commutator with a star node.
Corollary 2.5 [12] If the vector field Xhas eigenvalues with non-zero real part at
a focus Oand a nontrivial commutator Y, then it has infinitely many isochronous
sections.
In [9] and [12], different sufficient conditions for an analytic vector field to have an
isochronous weak focus at Oare given. In [12], the particular case of a differential
system equivalent to a Li´enard equation is taken into account.
3 Characterization of isochronous critical points
The following theorem characterizes when the origin Oof a system (1.1) has a section
Σ such that the period function τ: Σ →R+is constant, that is, it does not depend
on the point p∈Σ considered. We will see that if such section exists, then there are
an infinite number of them. In particular next theorem characterizes the existence of
isochronous critical points.
Theorem 3.1 Let us consider an analytic system (1.1). The following statements are
equivalent:
(i) There exists an analytic change of variables φ:U → U , where Uis a neighbor-
hood of the origin, with Dφ(O)=I, such that the transformed system reads for
ρ∗φ∗(X) = rf (r, θ)∂
∂r +g(θ)∂
∂θ .
(ii) There exists an analytic vector field Ydefined in a neighborhood of the origin of
the form
Y= (x+A(x, y)) ∂
∂x + (y+B(x, y)) ∂
∂y ,(3.1)
with Aand Banalytic functions of order ≥2, such that [X,Y] = µ(x, y)Y,
where µ(x, y)is a scalar function with µ(0,0) = 0.
(iii) There exists a section Σsuch that the period function τ: Σ →R+is constant.
5
Proof of Theorem 3.1.In order to prove the equivalence of the three statements,
it suffices to show (i)⇒(ii), (ii)⇒(iii) and (iii)⇒(i). We also include the proof of
(ii)⇒(i) and (iii)⇒(ii), for completeness.
In the subsequent, we will denote by a subindex a partial derivative, for instance,
if f(r, θ) is a function of (r, θ), ∂f
∂r is replaced by fr.
(i)⇒(ii) We define
Y=φ∗µx∂
∂x +y∂
∂y ¶,
and we have that Yhas the form described since φis an analytic change such that
Dφ(O) = I. Moreover,
[X,Y] = ·φ∗φ∗X, φ∗µx∂
∂x +y∂
∂y ¶¸ =φ∗ρ∗µ·rf (r, θ)∂
∂r +g(θ)∂
∂θ , r ∂
∂r ¸¶
=φ∗ρ∗µ−r2fr(r, θ)∂
∂r ¶.
We denote by µ(x, y) = φ∗ρ∗(−rfr(r, θ)). It is obvious that is an analytic scalar
function with µ(0,0) = 0. We have
[X,Y] = µ(x, y)φ∗ρ∗µr∂
∂r ¶=µ(x, y )φ∗µx∂
∂x +y∂
∂y ¶=µ(x, y )Y.
(ii)⇒(i) From normal form theory, see [5, 15], we have that there exists an analytic
change of variables φ, defined in a neighborhood Uof the origin and with Dφ(O) = I,
such that φ∗(Y) = x∂
∂x +y∂
∂y .
Since [X,Y] = µY, we have [φ∗(X), φ∗(Y)] = φ∗(µ)φ∗(Y). We introduce the fol-
lowing notation ˜µ(r, θ) := ρ∗φ∗(µ(x, y)) and ρ∗φ∗(X) := rf (r, θ)∂
∂r +g(r, θ)∂
∂θ . Hence,
·rf (r, θ)∂
∂r +g(r, θ)∂
∂θ , r ∂
∂r ¸= ˜µ(r, θ)r∂
∂r .
We compute the Lie bracket and we have the following equality
−r2fr(r, θ)∂
∂r −rgr(r, θ)∂
∂θ = ˜µ(r, θ)r∂
∂r ,
which implies gr(r, θ)≡0 and, therefore, g(r, θ) = g(θ). We remark that since the ori-
gin of the system defined by Xis a monodromic critical point, we have that g(θ)>0
or g(θ)<0 for all θ∈R. Moreover, as before, we may consider the arc–length
ϕ=Rθ
0dθ/g(θ). This integral is well defined and it gives a change of variable since
g(θ) has a definite sign for all θ∈R. Then, after this change, the angular speed of the
corresponding system is constant.
(ii)⇒(iii) This statement is a clear corollary of Theorem 2.4. However, a geo-
metric outline of its proof is easy enough to be given here.
Let, for any p∈ U, be Φt(p) the flow of Xand Ψs(p) that of Y, with the initial
condition Φ0(p) = Ψ0(p) = p. Without lack of generality, we can assume that Ois
the unique singular point for Xand Yin U. Let p, q ∈ U ,p, q 6=O. By classical Lie
6
theory, we have that the relation [X,Y] = µ(x, y)Yimplies that if Σ = {Ψs(p)|s∈R}
is a solution of Y, then for any t∈R, Φt(Σ) is another solution for Y.
It is clear that Σ is a transversal section for X. Let τ, Pbe the corresponding period
function and Poincar´e map defined on it. We will show that any two points p, q ∈Σ
have the same period function. We have that P(p) = Φτ(p)(p). The time τ(p) leaves Σ
invariant: Φτ(p)(Σ) ⊆Σ. Let q∈Σ, then there exists s∈Rsuch that q= Ψs(p). The
minimal time to meet Σ again, that is τ(q), must coincide with τ(p) since the time
τ(p) brings the solution Σ into itself. Then τ(p) = τ(q).
(iii)⇒(ii) We consider Φt(p) the flow of system Xdefined in the neighborhood
Uof the origin and with the initial condition Φ0(p) = p.
Given a section through the origin, Σ ⊂R2, we consider its parameterization by
its arc-parameter σ, that is, there exists a map c:R→Σ such that Σ = {c(σ)|σ∈
R}. We can assume without loss of generality that limσ→−∞ c(σ) = Oand that
limσ→−∞ c0(σ)6= (0,0). As usual, c0(σ) denotes the derivative of the parameterization
of the curve c:σ7→ c(σ) at the value σ. We define the following set of transformations
Ψ : R× U → U in the following way, see Figure 1.
If p∈Σ, that is p=c(σ0) for a certain σ0∈R, and s∈Rthen Ψs(p) := c(σ0+s).
If p6∈ Σ, there exists t0(p)∈Rsuch that Φt0(p)(p)∈Σ, that is, there exists σ0∈R
such that c(σ0) = Φt0(p)(p). Assume that t0(p)>0 is the lowest positive real with this
property. For any s∈Rwe define Ψs(p) = Φ−t0(p)(c(σ0+s)).
In the subsequent, for any p∈ U we denote by t0(p) as the lowest positive real
such that Φt0(p)(p)∈Σ. It is clear that t0:U → [0, T ) where T > 0 is the period
defined by the section Σ. We denote by σ0(p)∈Rthe value of the parameter such
that Φt0(p)(p) = c(σ0(p)).
Figure 1: Definition of Ψs(p).
We are going to prove that the set of transformations defined by Ψsis a one–
parameter Lie group of point transformations. We need to show the following state-
ments:
(a) For all s∈R, Ψs:U → U is bijective.
(b) Ψ0is the identity map.
(c) For any s1, s2∈R, Ψs1◦Ψs2= Ψs1+s2.
(d) Ψ ∈ Cω(R)× Cω(U).
7
(a) Fixed s∈R, let us consider any p∈ U and we have Ψs(p)=Φ−t0(p)(c(σ0(p) +
s)). Let p1, p2∈ U . If Ψs(p1) = Ψs(p2), let qbe this point q= Ψs(pi). Then, the points
Φt0(p1)(q) = c(σ0(p1) + s) and Φt0(p2)(q) = c(σ(p2) + s) belong to Σ. Both, t0(p1) and
t0(p2) are defined as the minimum positive time with this property so, t0(p1) = t0(p2).
Therefore, c(σ0(p2) + s) = c(σ0(p1) + s) and this gives σ0(p1) = σ0(p2) which implies
p1=p2. Then, Ψsis injective.
Let us see that it is exhaustive. Given q∈ U let p= Φ−t0(q)(c(σ0(q)−s)). Then,
t0(p) = t0(q), σ0(p) = σ0(q)−sand Ψs(p)=Φ−t0(q)(c(σ0(q))) = q. The fact that the
section Σ is isochronous ensures the well-definition of this p.
(b) Given p∈ U we have that Ψ0(p) = Φ−t0(p)(c(σ0(p))) where c(σ0(p)) = Φt0(p)(p).
Then, clearly, Ψ0(p) = p.
(c) Given p∈ U, it is clear that t0(Φ−t0(p)(c(σ0+s1))) = t0(Φ−t0(p)(c(σ0+s1+
s2))) = t0(p). We have Ψs1◦Ψs2(p)=Ψs1(Φ−t0(p)(c(σ0+s1))) = Φ−t0(p)(c(σ0+s1+
s2)) = Ψs1+s2(p).
(d) The regularity of Ψ is clear due to the regularity of Φ and c.
Once we have that Ψ is a one–parameter Lie group of point transformations, we
apply the first fundamental theorem of Lie, see [13], and we have that there exists
an analytic vector field Ywhose flow coincides with Ψs(p). Moreover, Yis given
by ∂Ψs
∂s (p)|s=0 . By the definition Ψs(p) = Φt0(p)(c(σ0(p) + s)) we have that Y(p) =
TΦt0(p)(c(σ0(p) + s)) ·c0(σ0(p) + s)|s=0 = TΦt0(p)(c(σ0(p))) ·c0(σ0(p)), where TΦt(q)
denotes the jacobian matrix of the analytic change of variables Φtat the point qand,
as before, c0(σ) denotes the derivative of the parameterization of the curve c:σ7→ c(σ)
at the value σ.
Moreover, by construction, Yhas a star node at the origin. This is clear by the
fact that each of its orbits Φt(Σ), t∈[0, T ), has a different tangent at the origin.
Let Y=ξ(x, y)∂
∂x +η(x, y)∂
∂y . Since Yhas a star node at the origin, by a classical
result stated in [17], page 63, we have that ξ(x, y) = xh(x, y) + h.o.t. and η(x, y) =
yh(x, y) + h.o.t., where h(x, y ) is a homogeneous polynomial and h.o.t. denotes higher
order terms. Therefore, in order to see that Yis of the form (1.2), we only need to
show that the divergence of the vector field Y, that is divY, is different from zero at
the origin, where divY(x, y) = ∂ξ
∂x (x, y) + ∂ η
∂y (x, y). The divergence of the vector field
Yis related to the inverse integrating factor of Y. The inverse integrating factor of
Yis given by V(x, y)=(λx −y+P(x, y))η(x, y)−(x+λy +Q(x, y))ξ(x, y) which is
defined in the neighborhood Uof the origin. An easy computation shows that
V(Ψs(x0, y0)) = V(x0, y0) exp ½Zs
0
divY(Ψu(x0, y0))du¾(3.2)
for any (x0, y0)∈ U. It is clear that V(0,0) = 0. Let p0:= (x0, y0)∈ U − {(0,0)}and
assume that V(p0) = 0. This implies that the vectors Y(p0) and X(p0) are parallel.
By the definition of Y, we have that
Y(p0) = TΦ−t0(p0)(c(σ0(p0))) ·c0(σ0(p0)) = TΦ−t0(p0)(Φt0(p0)(p0)) · Y(Φt0(p0)(p0)).
We denote by q0= Φt0(p0)(p0) and we have TΦt0(p0)(q0)· Y(p0) = Y(q0). Since Φ is
the flow of X, we have TΦt0(p0)(q0)· X (p0) = X(q0).Therefore, if Y(p0) and X(p0) are
parallel, then Y(q0) and X(q0) are parallel. However, q0∈Σ and the vector Y(q0) is
tangent to Σ at q0, so the parallelism between Y(q0) and X(q0) is a contradiction with
Σ being a transversal section for X. Therefore, we conclude that V(x0, y0)6= 0 for any
(x0, y0)∈ U − {(0,0)}.
8
By using this fact, we prove that divY(0,0) 6= 0. Let us consider (x0, y0)∈
U − {(0,0)}and we have that lims→−∞ Ψs(x0, y0) = (0,0). By continuity and the
identity (3.2), we have that the integral I(x0, y0) := R0
−∞ divY(Ψu(x0, y0))du di-
verges. I(x0, y0) is continuous, so I(0,0) also diverges. Hence, if divY(0,0) = 0,
then I(0,0) = R0
−∞ divY(Ψu(0,0))du =R0
−∞ divY(0,0)du = 0, in contradiction with
being divergent. Therefore, divY(0,0) 6= 0.
Moreover, by definition it is clear that the flow of Xtakes solutions of Yto solu-
tions of Y. Another classical result on Lie symmetries gives that Xis a Lie symmetry
for Yand therefore, there exists an analytic scalar function µ:U → Rsuch that
[X,Y] = µY. Moreover, µ(0,0) = 0 since both functions defining the vector field
[X,Y] have order two at the origin and the vector field Yhas order one at the origin.
(i)⇒(iii) The ray ˜
Σ = {(x, 0) |x > 0}is an isochronous section for the sys-
tem φ∗(X), where ρ∗φ∗(X) = rf (r, θ)∂
∂r +g(θ)∂
∂θ , since ˜τ:˜
Σ→R+is given by
˜τ(x) = R2π
0dθ/g(θ), which is constant for every x∈˜
Σ. Then, Σ := φ−1(˜
Σ) is an
isochronous section for system (1.1) and the period function is given by τ:= φ∗(˜τ). ¥
Using Theorem 1.1 and Theorem 3.1, we reencounter the following result which
characterizes isochronous centers and which is stated and proved in [2] (Theorem 2.3,
page 140).
Theorem 3.2 System (1.1) with λ= 0 has an isochronous center at the origin if, and
only if, there exists a smooth vector field Zof the form (1.2) such that [X,Z] = 0.
Proof. Assume that system (1.1), with λ= 0, has an isochronous center at the
origin. Then by Theorem 1.1 there exists a smooth vector field Uof the form (1.2)
and a smooth function ν, with ν(0,0) = 0 such that [X,U] = νX. Moreover, by
Theorem 3.1 there exists an analytic vector field Yof the form (1.2) and an analytic
functions µ, with µ(0,0) = 0, satisfying [X,Y] = µY. Since Xand Yare transversal in
a neighborhood of the origin, they define a basis in this neighborhood and therefore,
there exist two smooth functions α, β such that U=αX+βY. Since both Uand Y
have the form (1.2), we have that β= 1 + β1where β1is a smooth function of order
≥1. We compute
[X,U]=[X, α X+βY] = X(α)X+α[X,X] + X(β)Y+β[X,Y]
=X(α)X+ (X(β) + βµ)Y.
Since [X,Y] = νX, we deduce X(β) = −µβ.
We define Z=βYwhich is a smooth vector field with the form (1.2) since β= 1+β1
where β1is a smooth function of order ≥1. Then, [X,Z] = [X, βY] = β[X,Y] +
X(β)Y=βµY − µβY= 0. ¥
We want to remark a difference between this Theorem 3.2 and Theorem 1.2. In The-
orem 1.2 the origin is required to be a center and the result characterizes its isochronic-
ity. In Theorem 3.2, the origin is only a linear center (that the equilibrium is a center
is not required), thanks to Theorem 1.1.
The methods developed in this work can be used to classify isochronous critical
points for polynomial systems. We give some examples of systems of the form (1.1)
with an isochronous critical point at the origin. The determination of the origin being
a focus is straightforward by computing Liapunov constants, see for instance [9]. When
9
the origin is a center, a first integral defined on a neighborhood of it is provided. We
also give an example of a family of quadratic systems depending on a real parameter
w6= 0 which never has an isochronous critical point at the origin. When w= 0, the
system has a center, and when w6= 0 the system has a weak focus at the origin.
Example 1. The following system has an isochronous critical point at the origin.
˙x=−y+λ2x3+λ3x2y+λ4xy2,
˙y=x+λ2x2y+λ3xy2+λ4y3,(3.3)
with λ2, λ3, λ4∈R. In polar coordinates, this system reads for
˙r=r3
2(λ2+λ4+ (λ2−λ4) cos(2θ) + λ3sin(2θ)),˙
θ= 1.
Then, by definition, the origin is an isochronous critical point. A first integral for
system (3.3) is given by
H(x, y) = x2+y2
1−λ3x2+ (λ2−λ4)xy + (λ2+λ4)(x2+y2) arctan( y
x).
When λ2+λ46= 0, the origin is a focus and when λ2+λ4= 0, the origin is a cen-
ter. Let us consider Xthe corresponding vector field and Y=x∂
∂x +y∂
∂y . We have
[X,Y] = −2(λ2x2+λ3xy +λ4y2)Y.
Example 2. The following system has an isochronous focus at the origin.
˙x=−y−2xy +xy2−2y3+µ2(x3−xy2) + µ3x2y−y4+
µ2(x2y2+y4)−µ2xy4−µ3y5−µ2y6,
˙y=x+y2+y3+µ2(x2y−y3) + µ3xy2+ 2µ2xy3+µ3y4+µ2y5,
(3.4)
where µiare arbitrary real constants for i= 2,3. This system has no constant angular
speed. An easy computation shows that the first Liapunov constant equals −1/2, so
the origin of (3.4) is a stable weak focus. We use Theorem 3.1 to ensure the property
of isochronicity.
Let us consider Xthe corresponding vector field and Ythe following analytic vector
field
Y= (x−y2)∂
∂x +y∂
∂y .
The Lie bracket [X,Y] gives [X,Y] = −2(y2+µ2(x2−y2+2xy2+y4) + µ3(xy +y3)) Y.
Therefore, the hypothesis of Theorem 3.1 are satisfied and the origin of system (3.4)
is an isochronous focus.
Example 3. The following family of quadratic systems depending on the parameter
w∈R
˙x=−y, ˙y=x−4wxy + 2y2,(3.5)
never has an isochronous critical point at the origin.
It can be shown that wis the first Liapunov constant for this family of quadratic
systems. Hence, when w > 0 the origin is an unstable weak focus and when w < 0 the
origin is a stable weak focus. When w= 0, we have that H(x, y) = (4x+ 8y2−1)e4x
defines a first integral which is analytic in a neighborhood of the origin. So, the origin
is a center for w= 0.
10
We will try to construct a vector field Yand a function µsatisfying Theorem 3.1 and
we will get a contradiction. Assume that there exists a vector field Ywith a star node at
the origin such that the Lie bracket between the vector field Xwdefined by (3.5) and Yis
equal to µ(x, y)Yfor a certain scalar analytic function µ(x, y) with µ(0,0) = 0. We can
write Y= (x+Pi>1Ai(x, y)) ∂/∂x + (y+Pi>1Bi(x, y)) ∂/∂y, where Ai(x, y ), Bi(x, y)
are homogeneous polynomials of degree iand µ(x, y) = Pi>0mi(x, y ) where mi(x, y)
is a homogeneous polynomial of degree i.
Equating the terms of order 2 in the equation [Xw,Y] = µYwe get the two following
equations:
−y∂A2
∂x +x∂A2
∂y +B2=m1x,
−y∂B2
∂x +x∂B2
∂y + 4xwy −2y2−A2=m1y.
The solution of these two equations is A2(x, y) = ax2+bxy −(2/3)y2,B2(x, y) =
(4w/3)x2+axy +by2and m1(x, y ) = (b+ (4w/3))x−((4/3) + a)y, where a, b are any
two real numbers.
Equating the terms of order 3 in the equation [Xw,Y] = µY, we get the two
following equations:
−y∂A3
∂x +x∂A3
∂y + (2y2−4wxy)∂A2
∂y +B3=m2x+m1A2,
−y∂B3
∂x +x∂B3
∂y + (2y2−4wxy)∂B2
∂y + 4wyA2−A3−4(y−wx)B2=m2y+m1B2.
Let us write m2(x, y) = Pi+j=2 mij xiyj,A3(x, y) = Pi+j=3 aijxiyjand B3(x, y ) =
Pi+j=3 bij xiyj. We consider the vector of unknowns
v={m20, m11 , m02, a30 , a21, a12 , a03, b30 , b21, b12 , b03}
and we can write the previous two equations as a linear system of eight equations in
these eleven unknowns : M v =k. The matrix Mis
M=
−10001001000
0−1 0 −3 0 2 0 0 1 0 0
0 0 −1 0 −2 0 3 0 0 1 0
0 0 0 0 0 −1 0 0 0 0 1
0 0 0 −1 0 0 0 0 1 0 0
−1 0 0 0 −1 0 0 −3 0 2 0
0−1 0 0 0 −1 0 0 −2 0 3
0 0 −1 0 0 0 −1 0 0 −1 0
which can be seen that it is of rank 7. The vector kis
k=½a
3(3b+ 4w),1
3(−4a−3a2+ 3b2+ 16bw),1
9(−36b−9ab −56w),
2
9(16 + 3a),4
9(3b−8w)w, 1
9(9ab + 32w−36aw),
1
3(2a−3a2+ 3b2+ 4bw),1
3(−4b−3ab + 8w)¾.
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The matrix (M|k) has rank 8 as the determinant of one of its 8 ×8 minors equals
1 + w2. So, the linear system does not satisfy the compatibility condition and, hence,
no such Ynor µcan exist.
Acknowledgements
The authors would like to thank Prof. I.A. Garc´ıa from Univ. de Lleida and Prof. M.
Sabatini from Univ. degli Studi di Trento for several useful conversations and remarks
about this work.
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