#  The criticality of reversible quadratic centers
at the outer boundary of its period annulus 00footnotetext: 2010 AMS Subject
Classification: 34C07; 34C20; 34C23. 00footnotetext: Key words and phrases:
criticality, period function, asymptotic expansion, critical periodic orbit,
Hilbert’s 16th Problem. 00footnotetext: This work has been partially funded by
the Ministry of Science, Innovation and Universities of Spain through the
grants PGC2018-095998-B-I00 and MTM2017-86795-C3-2-P and by the Agency for
Management of University and Research Grants of Catalonia through the grants
2017SGR1725 and 2017SGR1617.

D. Marín and J. Villadelprat
Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona,
08193 Cerdanyola del Vallès (Barcelona), Spain
Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra,
08193 Cerdanyola del Vallès (Barcelona), Spain
Departament d’Enginyeria Informàtica i Matemàtiques, ETSE,
Universitat Rovira i Virgili, 43007 Tarragona, Spain

###### Abstract

This paper deals with the period function of the reversible quadratic centers

$X_{{{\nu}}}=-y(1-x)\partial_{x}+(x+Dx^{2}+Fy^{2})\partial_{y},$

where ${{\nu}}=(D,F)\in\mathbb{R}^{2}.$ Compactifying the vector field to
$\mathbb{S}^{2}$, the boundary of the period annulus has two connected
components, the center itself and a polycycle. We call them the inner and
outer boundary of the period annulus, respectively. We are interested in the
bifurcation of critical periodic orbits from the polycycle $\Pi_{{\nu}}$ at
the outer boundary. A critical period is an isolated critical point of the
period function. The criticality of the period function at the outer boundary
is the maximal number of critical periodic orbits of $X_{{\nu}}$ that tend to
$\Pi_{{{\nu}}_{0}}$ in the Hausdorff sense as ${{\nu}}\to{{\nu}}_{0}.$ This
notion is akin to the cyclicity in Hilbert’s 16th Problem. Our main result
(Theorem A) shows that the criticality at the outer boundary is at most 2 for
all ${{\nu}}=(D,F)\in\mathbb{R}^{2}$ outside the segments
$\\{-1\\}\times[0,1]$ and $\\{0\\}\times[0,2]$. With regard to the bifurcation
from the inner boundary, Chicone and Jacobs proved in their seminal paper on
the issue that the upper bound is 2 for all ${{\nu}}\in\mathbb{R}^{2}.$ In
this paper the techniques are different because, while the period function
extends analytically to the center, it has no smooth extension to the
polycycle. We show that the period function has an asymptotic expansion near
the polycycle with the remainder being uniformly flat with respect to
${{\nu}}$ and where the principal part is given in a monomial scale containing
a deformation of the logarithm, the so-called Écalle-Roussarie compensator.
More precisely, Theorem A follows by obtaining the asymptotic expansion to
fourth order and computing its coefficients, which are not polynomial in
${{\nu}}$ but transcendental. Theorem A covers two of the four quadratic
isochrones, which are the most delicate parameters to study because its period
function is constant. The criticality at the inner boundary in the isochronous
case is bounded by the number of generators of the ideal of all the period
constants but there is no such approach for the criticality at the outer
boundary. A crucial point to study it in the isochronous case is that the
flatness of the remainder in the asymptotic expansion is preserved after the
derivation with respect to parameters. We think that this constitutes a
novelty that is of particular interest also in the study of similar problems
for limit cycles in the context of Hilbert’s 16th Problem. Theorem A also
reinforces the validity of a long standing conjecture by Chicone claiming that
the quadratic centers have at most two critical periodic orbits. A less
ambitious goal is to prove the existence of a uniform upper bound for the
number of critical periodic orbits in the family of quadratic centers. By a
compactness argument this would follow if one can prove that the criticality
of the period function at the outer boundary of any quadratic center is
finite. Theorem A leaves us very close to this existential result.

###### Contents

  1. 1 Introduction and main results
  2. 2 Criticality vs bifurcation
  3. 3 Asymptotic expansion of the period function
    1. 3.1 Study of $\\{D=-1/2,F\in(1/2,1)\\}$ and $\\{F=1/2,D\in(-1,0)\\}$
    2. 3.2 Study of $\\{F=2,D\in(-2,0)\\}$ and $\\{D=\mathcal{G}(F):F\in(1,3/2)\\}$
    3. 3.3 Study of $\\{F=1,D\in(-1,0)\\}$
  4. 4 Distinguished cases
  5. 5 Proof of Theorem A
  6. A Coefficient formulas
    1. A.1 Previous results about the Dulac time
    2. A.2 Proof of Proposition 3.2
    3. A.3 Proof of Proposition 3.3
  7. B Beta and hypergeometric functions
  8. C A technical result for the proof of Proposition 3.3

##  1 Introduction and main results

A singular point $p\in\mathbb{R}^{2}$ of a planar differential system

$\left\\{\\!\begin{array}[]{l}\dot{x}=f(x,y),\\\\[2.0pt]
\dot{y}=g(x,y),\end{array}\right.$

is a _center_ if it has a punctured neighbourhood that consists entirely of
periodic orbits surrounding $p$. The _period annulus_ of the center is the
largest punctured neighbourhood with this property and we denote it by
$\mathscr{P}$. The period annulus is an open subset of $\mathbb{R}^{2}$ that
may be unbounded. For this reason we embed $\mathscr{P}$ in $\mathbb{RP}^{2}$
and, abusing notation, we denote the boundary of the resulting set by
$\partial\mathscr{P}$. Clearly the center $p$ belongs to $\partial\mathscr{P}$
and in what follows we call it the _inner boundary_ of the period annulus. We
also define the _outer boundary_ of the period annulus to be
$\Pi\\!:=\partial\mathscr{P}\setminus\\{p\\}$, which is a nonempty compact
subset of $\mathbb{RP}^{2}.$ The subject of our study is the _period function_
of the center, that assigns to each periodic orbit in $\mathscr{P}$ its
period. Since the period function is defined on the set of periodic orbits in
$\mathscr{P},$ in order to study its qualitative properties we need to
parametrize this set. This can be done by taking a transverse section to the
vector field $X=f(x,y)\partial_{x}+g(x,y)\partial_{y}$ on $\mathscr{P}$, for
instance an orbit of the orthogonal vector field $X^{\bot}$. To fix ideas let
us suppose that $\\{\gamma_{s}\\}_{s\in(0,1)}$ is such a parametrization where
$s\approx 0$ corresponds to the periodic orbits near $p$ and $s\approx 1$ to
the ones near $\Pi$. Then the map ${P}\\!:{(0,1)}\longrightarrow{(0,+\infty)}$
defined by $P(s)\\!:=\\!\\{\mbox{period of $\gamma_{s}$}\\}$ provides the
qualitative properties of the period function that we are concerned with and
one can readily show by using the Implicit Function Theorem that it is as
smooth as $X$. It is also well-known that if $X$ is analytic and the center
$p$ is non-degenerate then $P$ extends analytically to $s=0.$ Let us advance
that, on the contrary, $P$ does not extend smoothly to $s=1.$ The _critical
periods_ are the isolated critical points of $P$, i.e. $\hat{s}\in(0,1)$ such
that $P^{\prime}(\hat{s})=0$ and $P^{\prime}(s)\neq 0$ if
$0<|s-\hat{s}|<\varepsilon.$ In this case, more geometrically, we shall say
that $\gamma_{\hat{s}}$ is a _critical periodic orbit_ of $X$. One can easily
see that the property of being a critical periodic orbit does not depend on
the particular parametrization of the set of periodic orbits used, see Remark
2. The study of the critical periodic orbits is another issue arising from the
famous Hilbert’s 16th Problem and it has strong parallelisms with the research
on limit cycles, from both the conceptual and technical point of views. In
this regard we can mention for instance that the isochronicity problem (i.e.,
to decide whether a center has a constant period function) is the counterpart
of the center-focus problem. The renowned conjecture claiming that a quadratic
differential system can have at most four limit cycles has also an analogue in
the context of the period function and it was posed by C. Chicone [5]. More
specifically this conjecture asserts that if a quadratic center has some
critical periodic orbit then by an affine transformation and a constant
rescaling of time it can be brought to Loud normal form

$\left\\{\\!\begin{array}[]{l}\dot{x}=-y+Bxy,\\\\[2.0pt]
\dot{y}=x+Dx^{2}+Fy^{2},\end{array}\right.$ (1)

and that this center has at most two critical periodic orbits for any
$(B,D,F)\in\mathbb{R}^{3}$. In fact there is much analytic evidence that this
conjecture is true (see [7, 40, 41] for instance).

The problems that we are interested in take place when the vector field $X$
depends on parameters. To fix notation, let $U$ be an open subset of
$\mathbb{R}^{N}$ and consider a family of planar vector fields
$\\{X_{\mu},\mu\in U\\}$ such that each $X_{\mu}$ has a center $p_{\mu}$ with
period annulus $\mathscr{P}_{\mu}$. Let us denote the period function of the
center $p_{\mu}$ by $P(\,\cdot\,;\mu)$ and observe that, given some
$\mu_{0}\in U$, the number of critical periodic orbits of $X_{\mu}$ can vary
as we perturb $\mu\approx\mu_{0}.$ Under some regularity assumptions on the
dependence of $\mathscr{P}_{\mu}$ with respect to $\mu$ it can be proved (see
Lemma 2.12) that the emergence/disappearance of critical periodic orbits can
only occur from three different places:

  1. $(a)$

Bifurcations at the inner boundary of the period annulus (i.e., the center
$p_{\mu}$).

  2. $(b)$

Bifurcations at the outer boundary of the period annulus (i.e., the polycycle
$\Pi_{\mu}$).

  3. $(c)$

Bifurcations at the interior of the period annulus $\mathscr{P}_{\mu}.$

Chicone and Jacobs give in their seminal paper [6] a complete description of
the bifurcations from the inner boundary for the whole family of quadratic
centers. In this case the parameter $\mu$ are the coefficients of the vector
field and since the center is non-degenerate $P(s;\mu)$ extends analytically
to $s=0$, so that one can consider its Taylor series
$P(s;\mu)=\sum_{i=0}^{\infty}a_{i}(\mu)s^{i}$ at $s=0$, whose coefficients
$a_{i}$ belong to the polynomial ring $\mathbb{R}[\mu]$. On account of this
the result about the bifurcations from the isochronous centers (see [6,
Theorem 2.2]), which are the most difficult ones to study, follows by
analyzing the ideal $(a_{1},a_{2},\ldots)$ of all Taylor coefficients exactly
as N. Bautin does in [4] to study the bifurcations of limit cycles from the
quadratic centers. In the present paper we resume our study of the
bifurcations from the outer boundary that we initiated in [22, 23]. Let us
recall that the differential system $(\ref{BDF})$ has no critical periodic
orbits if $B=0,$ see [10, Theorem 1]. By means of a rescaling the case $B\neq
0$ can be brought to $B=1$, i.e.,

$X_{{{\nu}}}\\!:=-y(1-x)\partial_{x}+(x+Dx^{2}+Fy^{2})\partial_{y}\text{ with
${{\nu}}\\!:=(D,F).$}$ (2)

Here we already adopt the parameter notation that we shall use throughout the
paper, which is devoted to the bifurcation of critical periodic orbits from
the outer boundary in the family $\\{X_{{\nu}},{{\nu}}\in\mathbb{R}^{2}\\}.$
Since each vector field $X_{{\nu}}$ is polynomial we can consider its Poincaré
compactification $p(X_{{\nu}})$, see [3, §5], which is an analytic vector
field on the sphere $\mathbb{S}^{2}$ topologically equivalent to $X_{{\nu}}.$
The outer boundary $\Pi_{{\nu}}$ becomes then a polycycle of $p(X_{{\nu}})$
that can be studied using local charts of $\mathbb{S}^{2}$, but even so the
period function $P(s;{{\nu}})$ cannot be smoothly extended to $s=1.$ For the
family under consideration we show that $P(s;{{\nu}})$ has an asymptotic
expansion at $s=1$ with the remainder being uniformly flat with respect to
${{\nu}}$ and where the principal part is given in a monomial scale containing
a deformation of the logarithm, the so-called Écalle-Roussarie compensator.
Our main theorem follows by obtaining the asymptotic expansion to fourth order
and computing its coefficients, which are not polynomial in ${{\nu}}$ but
transcendental (more concretely, they are hypergeometric functions). To this
end we strongly rely on the tools that we develop in our recent papers [29,
30, 31]. The results that we obtain in the present paper can be viewed
conceptually as the analogue for the outer boundary of the work carried out by
Chicone and Jacobs in [6] on the bifurcation of critical periodic orbits from
the inner boundary of the quadratic centers. That being said, the proofs of
the results on the outer boundary are technically tougher than the ones on the
inner boundary because $\Pi_{{\nu}}$ is a polycycle and the period function
$P(s;{{\nu}})$ cannot be analytically extended there. By way of example, to
determine the parameters that vanish simultaneously two coefficients in the
asymptotic expansion at $s=1$ takes 5 pages of computations dealing with a
hypergeometric function (see Appendix C), whereas the same problem for the
Taylor series at $s=0$ can be solved readily by taking resultants because the
coefficients are polynomials.

In this paper we use the notion of _criticality_ of the period function at the
outer boundary which, roughly speaking, is the number of critical periodic
orbits that can emerge or disappear from $\Pi_{{\nu}}$ as we perturb ${{\nu}}$
slightly. It is defined in exactly the same way as the notion of _cyclicity_
of a limit periodic set, which is used to study the bifurcation of limit
cycles in the context of Hilbert’s 16th Problem, see [36] for instance. Before
giving its precise definition, and the statement of our main contribution, we
enumerate the previous results about the bifurcation of critical periodic
orbits from the outer boundary $\Pi_{{\nu}}$ for the family
$\\{X_{{\nu}},{{\nu}}\in\mathbb{R}^{2}\\}.$ In this regard we stress that
these results are given according to the dichotomy between local regular value
and local bifurcation value (of the period function at the outer boundary)
that we introduce in our early paper [23]. This notion (see Definition 2)
enables to obtain a structure theorem for the bifurcation diagram of the
period function in its full domain (see Lemma 2.12), but it has the
inconvenience of not being so quantitative and geometric as the criticality.
In order to simplify the exposition for the moment we can think that
${{\nu}}_{0}\in\mathbb{R}^{2}$ is a local regular value if and only if the
criticality of the period function at $\Pi_{{{\nu}}_{0}}$ is zero (i.e., no
critical periodic orbit bifurcates from $\Pi_{{{\nu}}_{0}}$ as we perturb
${{\nu}}\approx{{\nu}}_{0}).$ That said, let $\Gamma_{U}$ be the union of
dotted straight lines in Figure 1, whatever its colour is. Consider also the
thick curve $\Gamma_{B}.$ (Here the subscripts $B$ and $U$ stand for
bifurcation and unspecified respectively.)

[l(0mm),r(0mm),t(0mm),b(5mm)]dib2b(1)[l]13,76.5;$F=-D$ [l]67,85;$F$
[l]83,17;$D$ [l]67,34;$1/2$ [l]67,46;$1$ [l]67,57;$3/2$ [l]67,69;$2$
[l]48,17;$-1/2$ [l]36,17;$-1$ [l]20,60;$\Gamma_{B}$ [l]35,41;$\Gamma_{U}$
[l]45.0,50.5;$D=\mathcal{G}(F)$

Figure 1: The thick (closed) curve $\Gamma_{B}$ consists of local bifurcation
values of the period function at the outer boundary according to [23], where
the curve that joins $\left(-\frac{3}{2},\frac{3}{2}\right)$ and
$\left(-\frac{1}{2},1\right)$ is the graphic of an analytic function
$D=\mathcal{G}(F)$, see Remark 3.2. The dotted lines $\Gamma_{U}$ correspond
to parameters that remained unspecified in that paper and we colour the
subsequent improvements obtained in [19, 24, 25, 27, 28, 38, 39]. The
parameters outside $\Gamma_{B}\cup\Gamma_{U}$ are local regular values by the
result in [23].

Then according to [23, Theorem A] the open set
$\mathbb{R}^{2}\setminus\left(\Gamma_{B}\cup\Gamma_{U}\right)$ corresponds to
local regular values and $\Gamma_{B}$ consists of local bifurcation values (of
the period function at the outer boundary). In that paper we also conjecture
that any parameter in $\Gamma_{U}$ is regular, except for the segment
$\\{0\\}\\!\times\\!\left[0,\frac{1}{2}\right]$ in the vertical axis, that
should consist of bifurcation values. Since the formulation of this conjecture
there has been some progress in the study of the parameters in $\Gamma_{U}$:

  * •

From the results in [19, 39] it follows that the parameters in blue are
regular. In these papers the authors determine a region $M$ in the parameter
plane for which the corresponding center has a globally monotonous period
function (i.e., it has no critical periodic orbits). The parameters that we
draw in blue are inside the interior of $M$, which prevents the bifurcation of
critical periodic orbits.

  * •

Along the straight line $F=-D$ there is a breaking of a heteroclinic
connection between two hyperbolic saddles at the outer boundary. From the
results in [24] it follows that the parameters in red are regular.

  * •

Along the two segments in green it occurs a saddle-node bifurcation at the
outer boundary of the period annulus. An asymptotic expansion of the Dulac
time of this type of unfolding is obtained in [25] and as an application it is
proved that the parameters in the segment $(-1,0)\times\\{1\\}$, with the
exception of $(-\frac{1}{2},1)$, are local regular values. A subsequent
refinement of this approach shows in [27] that the segment
$(-1,0)\times\\{0\\}$ also consists of local regular values.

  * •

By [28, Theorem B] the parameters in brown, more precisely the segment
$\\{0\\}\\!\times\\!\left[\frac{1}{4},\frac{1}{2}\right]$, are local
bifurcation values of the period function at the outer boundary.

  * •

Along the segment $(-1,0)\times\\{\frac{1}{2}\\}$ there is a resonant saddle
at $\Pi_{{\nu}}$ and the parameters in yellow are local regular values at the
outer boundary of the period annulus according to [38, Corollary B].

As we already explained, these results are addressed to solve the dichotomy
between local regular value and local bifurcation value (of the period
function at the outer boundary). Beyond this dichotomy a challenging problem
is the computation of the exact number of critical periodic orbits that can
bifurcate from the outer boundary, which constitutes the counterpart of the
result by Chicone and Jacobs [6] about the bifurcation from the inner
boundary. The following is the precise definition of the number that we aim to
compute for the quadratic centers, where $d_{H}$ stands for the Hausdorff
distance between compact sets of $\mathbb{RP}^{2}.$

  * Definition 1.1.

Consider a $\mathscr{C}^{\infty}$ family $\\{X_{\mu},\mu\in U\\}$ of planar
vector fields with a center and fix some $\mu_{0}\in U$. Suppose that the
outer boundary of the period annulus varies continuously at $\mu_{0}\in U$,
meaning that $d_{H}(\Pi_{\mu},\Pi_{\mu_{0}})$ tends to zero as
$\mu\to\mu_{0}.$ Then, setting

$N(\delta,\varepsilon)=\sup\left\\{\text{\\# critical periodic orbits $\gamma$
of $X_{\mu}$ in $\mathscr{P}_{\mu}$ with
$d_{H}(\gamma,\Pi_{\mu_{0}})\leqslant\varepsilon$ and
$\|\mu-\mu_{0}\|\leqslant\delta$}\right\\},$

the _criticality_ of $(\Pi_{\mu_{0}},X_{\mu_{0}})$ w.r.t. the deformation
$X_{\mu}$ is
$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{0}},X_{\mu_{0}}),X_{\mu}\bigr{)}\\!:=\inf_{\delta,\varepsilon}N(\delta,\varepsilon).$
$\square$

We stress that in this definition the vector field $X_{\mu}$ is not required
to be polynomial but $\mathscr{C}^{\infty}$. This is so because in order to
define the outer boundary $\Pi_{\mu}$ of the period annulus
$\mathscr{P}_{\mu}$ of $X_{\mu}$ we do not compactify the vector field but
only the set $\mathscr{P}_{\mu}$ and to this end there is no need that
$X_{\mu}$ is polynomial. Certainly
$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{0}},X_{\mu_{0}}),X_{\mu}\bigr{)}$ may be
infinite but, if it is not, then it gives the maximal number of critical
periodic orbits of $X_{\mu}$ that tend to $\Pi_{\mu_{0}}$ in the Hausdorff
sense as $\mu\to\mu_{0}.$ Related with this issue we point out that the
contour of the period annulus $\mathscr{P}_{\mu_{0}}$ may change for
$\mu\approx\mu_{0}.$ The assumption that the period annulus varies
continuously ensures that this change does not occur abruptly. In this regard
note that $X_{\mu}=-y\partial_{x}+(x+\mu x^{3}+x^{5})\partial_{y}$, with
$\mu\in\mathbb{R},$ is a polynomial family of vector fields with a center at
the origin for which the outer boundary does not vary continuously at $\mu=2.$
This is so because the period annulus $\mathscr{P}_{\mu}$ is the whole plane
for $\mu<2$, whereas it is bounded for $\mu=2$ (see [20] for details). In this
example $\mathrm{Crit}\bigl{(}(\Pi_{\mu_{0}},X_{\mu_{0}}),X_{\mu}\bigr{)}$, as
introduced in Definition 1, does not give the number of critical periodic
orbits bifurcating from $\Pi_{\mu}$ as $\mu\to\mu_{0}.$ Let us mention that
this assumption is also required in [17, 18], where the authors obtain several
results addressed to bound the criticality at the outer boundary of families
of vector fields of potential type, i.e.,
$-y\partial_{x}+V^{\prime}(x)\partial_{y}$.

Let us remark at this point that if Chicone’s conjecture about the number of
critical periodic orbits of the quadratic centers is true then
$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}\leqslant
2$ for all ${{\nu}}\in\mathbb{R}^{2},$ see $(\ref{sist_loud})$. In this paper,
by applying our recent results from [29, 30, 31], we prove the following:

######  Theorem A.

Let $\\{X_{{{\nu}}},{{\nu}}\in\mathbb{R}^{2}\\}$ be the family of quadratic
vector fields given in $(\ref{sist_loud})$ and consider the period function of
the center at the origin. Then the following assertions hold:

  1. $(a)$

$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}=0$
for
${{\nu}}_{0}\notin\Gamma_{B}\cup\big{\\{}D=-1,F\in[0,1]\big{\\}}\cup\big{\\{}D=0,F\in[0,\frac{1}{2}]\big{\\}}$.

  2. $(b)$

$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}=1$
for
${{\nu}}_{0}\in\Gamma_{B}\setminus\left(\left\\{D=0\right\\}\cup\left\\{(-2,2),(\mathcal{G}(\frac{4}{3}),\frac{4}{3})\right\\}\right)$.

  3. $(c)$

$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}\geqslant
1$ for ${{\nu}}_{0}\in\big{\\{}D=0,F\in[\frac{1}{4},2]\big{\\}}$.

  4. $(d)$

$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}=2$
for
${{\nu}}_{0}\in\big{\\{}(-2,2),(\mathcal{G}(\frac{4}{3}),\frac{4}{3})\big{\\}}$.

  5. $(e)$

There is a $\mathscr{C}^{1}$ curve arriving at
$\nu=(\mathcal{G}(\frac{4}{3}),\frac{4}{3})$ tangent to $\Gamma_{B}$ and there
is a $\mathscr{C}^{0}$ curve with a exponential flat contact with $\\{F=2\\}$
at $\nu=(-2,2),$ consisting both of local bifurcation values of the period
function at the interior.

There are some papers containing results related with assertion $(b)$ in
Theorem A to be referred. Thus, by [33, Theorem A],
$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}=1$
for any ${{\nu}}_{0}=(\mathcal{G}(F_{0}),F_{0})$ with
$F_{0}\in(\frac{4}{3},\frac{3}{2})$. This is a piece of the curve that joins
$\left(-\frac{3}{2},\frac{3}{2}\right)$ and $\left(-\frac{1}{2},1\right)$, see
Figure 1, and in this regard observe that the criticality is 2 for
${{\nu}}_{0}=(\mathcal{G}(\frac{4}{3}),\frac{4}{3}).$ Furthermore, it is
proved in [32, Theorem B] that if ${{\nu}}_{0}=(D_{0},2)$ with
$D_{0}\in(-2,0)\setminus\\{-\frac{1}{2}\\}$ then
$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}=1$.
The same conclusion is true for any ${{\nu}}_{0}=(-F_{0},F_{0})$ with
$F_{0}\in[\frac{3}{2},2)$ thanks to [24, Theorem C]. In that paper it is also
partially proved the claim about the parameter ${{\nu}}_{0}=(-2,2)$ in
assertion $(e)$ of Theorem A. Apart from these references to previous results
we also want to point out the following issues with regard to the statement
and proof of Theorem A:

  * •

As expected, the study of the bifurcation of critical periodic orbits, either
from the inner or the outer boundary, is much more delicate when we perturb an
isochronous center. By the result of W.S. Loud, see [15], we know that there
are four nonlinear quadratic isochrones,

${{\nu}}_{1}=(0,1),\;{{\nu}}_{2}=(-1/2,2),\;{{\nu}}_{3}=(-1/2,1/2)\text{ and
}{{\nu}}_{4}=(0,1/4),$ (3)

which are located in $\Gamma_{B}.$ Chicone and Jacobs prove, see [6, Theorem
3.1], that the criticality of each isochrone ${{\nu}}_{i}$ at the inner
boundary of its period annulus (i.e., the center itself) is one. The proof of
this follows by finding a finite set of generators for the ideal of _all_ the
coefficients of the Taylor series of $P(s;{{\nu}})$ at $s=0$. In the present
paper we are able to show that ${{\nu}}_{2}$ and ${{\nu}}_{3}$ have
criticality one also at the outer boundary (i.e., the polycycle), see
Propositions 4.2 and 4.3 respectively. A crucial point to see this is that, as
we prove in [30], the flatness of the remainder in the asymptotic expansion at
$s=1$ is preserved after the derivation with respect to parameters. This
constitutes the cornerstone to obtain Lemma 4.1, which enables us to perform a
convenient division in the space of coefficients and proceed then as in the
proof of Bautin [4, §3] for the analogous result about the bifurcation of
limit cycles from the center. The isochrones ${{\nu}}_{1}$ and ${{\nu}}_{4}$
cannot be analyzed following this approach because the polycycle at the outer
boundary is not hyperbolic.

  * •

It is well-known, see [6, Theorem 3.2], that the criticality at the inner
boundary of any quadratic center is at most two and that this maximum
criticality is achieved at three parameter values, the so-called Loud points,
which we give in $(\ref{Li})$. For consistency with Chicone’s conjecture, each
one of these three parameters should have a “twin” where the maximum
criticality at the outer boundary is attained. In this paper we identify two
of these twin parameters, see assertion $(d)$ in Theorem A. We conjecture that
each pair of twins is connected by a curve that consists of local bifurcation
values at the interior, see Remarks 4 and 5.

  * •

The local bifurcation values of the period function can only occur at the
inner boundary (i.e., the center), at the outer boundary (i.e., the polycycle)
or at the interior of the period annulus, see Lemma 2.12. (With regard to the
latter, its counterpart in the context of Hilbert’s 16th Problem is the
bifurcation from a semi-stable limit cycle, which is characterized by the
sudden emergence of a double limit cycle that gives rises to two hyperbolic
limit cycles with different stability, see [12, §13.3] for instance). As
occurs with limit cycles, the identification of this third type of local
bifurcation value is out of reach for the moment and only partial results have
been obtained. Thus, in a joint paper with P. Mardešić we prove (see [23,
Theorem 4.3]) that at each Loud point there exists a germ of analytic curve
that consists of local bifurcation values at the interior. Since
$P(s;{{\nu}})$ extends analytically to $s=0$, this follows readily by applying
the Weierstrass Preparation Theorem. In the present paper, see assertion $(e)$
in Theorem A, we show the existence of two germs of curve which also consists
of local bifurcation values at the interior and that are the mirror image at
the outer boundary (i.e., at $s=1$) of those previously obtained in [23], see
Figures 5 and 6.

In another vein it is well-known (see [36, §2.2] for details) that the problem
of proving the _existence_ of a uniform bound for the number of limit cycles
in a given family, for instance Hilbert’s 16th Problem, can be replaced by a
local problem that consists in showing that the cyclicity of each limit
periodic set within the family is finite. The proof of this is by a
compactness argument and it does not provide an algorithm to compute an
explicit upper bound even if we had an explicit bound for the cyclicity of
every limit periodic set. In any case this gives a program for solving the
existential Hilbert’s 16th Problem that has been posed and implemented for the
quadratic vector fields by R. Roussarie and his collaborators (see [9, 34]).
One can of course transfer this problem to the period function and ask for the
existence of a uniform bound for the number of critical periodic orbits in the
family of quadratic centers. Similarly as it occurs in the context of limit
cycles, an affirmative answer would follow if one can prove that the
criticality of the period function at the outer boundary of any quadratic
center is finite, cf. Lemma 2.17. On account of Theorem A we are not very far
from proving the existence of this uniform bound for the family of reversible
quadratic centers. It will follow in particular if one can prove the validity
of the following more specific conjecture:

  * Conjecture.

Let $\\{X_{{{\nu}}},{{\nu}}\in\mathbb{R}^{2}\\}$ be the family of quadratic
vector fields given in $(\ref{sist_loud})$ and consider the period function of
the center at the origin. Then the following assertions are true:

    1. $(a)$

$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}=0$
for ${{\nu}}_{0}\in\\{D=-1,F\in[0,1]\\}$.

    2. $(b)$

$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}=1$
for ${{\nu}}_{0}\in\\{D=0,F\in(0,2]\\}$.

    3. $(c)$

$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}=2$
for ${{\nu}}_{0}=(0,0)$.

    4. $(d)$

There is a curve of local bifurcation values of the period function at the
interior arriving to ${{\nu}}=(0,0)$ tangent to $D=0$.

As a matter of fact to show the existence of a uniform bound for the number of
critical periodic orbits of the reversible quadratic centers it suffices to
verify that
$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}$
is finite for all ${{\nu}}_{0}=(D_{0},F_{0})$ inside the segments
$\\{-1\\}\times[0,1]$ and $\\{0\\}\times[0,2]$. To put this into context let
us recall that the differential system $(\ref{BDF})$ has no critical periodic
orbits if $B=0$ by [10, Theorem 1]. On the other hand, apart from the
reversible one, there are essentially three other families of quadratic
centers: the Hamiltonian, the codimension four $Q_{4}$ and the generalized
Lotka-Volterra systems $Q_{3}^{LV}.$ According to Chicone’s conjecture the
number of critical periodic orbits should be zero for the centers in these
three families. This is known to be true for the Hamiltonian and $Q_{4}$
families thanks to the results of Coppel and Gavrilov [7] and Zhao [41],
respectively. With regard to the family $Q_{3}^{LV}$ it is proved in [40]
that, except for a subset of codimension one in the parameter plane, the
criticality at the outer boundary is zero. It is clear then that any
contribution to the proof of the above conjecture will constitute a very
significant step forward to the existence of a uniform bound for the number of
critical periodic orbits in the _whole_ family of quadratic centers. Let us
mention in this respect that along $D=-1$ and $D=0$ the singularity at the
outer boundary of the period annulus is nilpotent. In this situation the
results of [30, 31] do not apply and new techniques must be developed.

The paper is organized in the following way. In Section 2 we recall the
definition of local bifurcation value at the outer boundary, that we introduce
in our early paper [23] to study the bifurcation diagram of the period
function of the family $\\{X_{{\nu}},{{\nu}}\in\mathbb{R}^{2}\\}$, and we
prove several results that relate it with the criticality. We also show how to
study the criticality by means of a suitable parametrization of the set of
periodic orbits near the outer boundary. Section 3 is devoted to the
asymptotic expansion of the period function near the outer boundary, which is
the cornerstone in the proof of Theorem A. To this end we prove three results
that are addressed to three different parameter subsets according to the
contour of the period annulus. As one might expect the proofs of these results
are rather long and technical. Furthermore they are based on previous tools
developed in [29, 30, 31] that need to be introduced appropriately. For these
reasons, to ease the paper’s readability we defer some proofs to Appendix A.
In Section 4 we study three distinguished parameters. On one hand the two
isochrones for which we succeed in proving that the criticality is one (see
Propositions 4.2 and 4.3) and, on the other hand, the parameter
${{\nu}}=(\mathcal{G}(\frac{4}{3}),\frac{4}{3})$, which is also rather special
because it has criticality two (see Proposition 4.4). Due to the novel
approach of its proof we think that each one of these results is of particular
interest in the context of Hilbert’s 16th Problem. Section 5 is entirely
devoted to the proof of Theorem A. Next, in Appendix A we prove the results
stated in Section 3 that we mentioned before and in Appendix B we are
concerned with the integral representation of the Beta and hypergeometric
functions, which usually appear as coefficients in the asymptotic expansions
that we obtain. Finally Appendix C is addressed to prove a technical result
that is used to study the vanishing set of two coefficients.

##  2 Criticality vs bifurcation

In this section we recap the notion of local bifurcation value of the period
function at the outer boundary as we introduced in our early paper [23]. We
relate it with the criticality, which is a more quantitative and geometric
definition, and prove a general result connecting both notions, see Lemma
2.16. More specifically our aim is to take advantage in the present paper of
the results that we obtained in [23] with regard to the period function of
Loud’s centers $(\ref{sist_loud})$ and that are not stated using the notion of
criticality. Related with this issue, our goal in this section is also to
clarify the usage of a parametrization of the period function near the outer
boundary to compute its criticality, see Lemma 2.4. Finally we give a
sufficient condition in order that a parameter is a local bifurcation value of
the period function at the interior, see Lemma 2.15.

Several results in this section are equally valid in the finitely smooth class
$\mathscr{C}^{k}$, $k\in\mathbb{N},$ the infinitely smooth class
$\mathscr{C}^{\infty}$ and the analytic class $\mathscr{C}^{\omega}$. For
simplicity in the exposition we write $\mathscr{C}^{\varpi}$ with the wild
card $\varpi\in\mathbb{N}\cup\\{\infty,\omega\\}.$ Our first result is
addressed to the regularity properties of the map
$(p,\mu)\mapsto\hat{P}(p;\mu)$ that assigns to each $\mu\in U$ and
$p\in\mathscr{P}_{\mu}$ the period $\hat{P}$ of the periodic orbit of
$X_{\mu}$ passing through the point $p.$ The result is given under a technical
assumption concerning the existence of a continuous parametrization
$\sigma(s;\mu)$ of the period annulus $\mathscr{P}_{\mu}$ near its outer
boundary $\Pi_{\mu}$. We point out that from now on, in contrast with the
notation used in the introduction, for the sake of convenience $s=0$
corresponds to $\Pi_{\mu}$ and $s=1$ to the center.

######  Lemma 2.1.

Let us fix $\varpi\in\mathbb{N}\cup\\{\infty,\omega\\}$ and consider a
$\mathscr{C}^{\varpi}$ family of planar vector fields $\\{X_{\mu}\\}_{\mu\in
U}$ such that, for each $\mu\in U$, $X_{\mu}$ has a center
$p_{\mu}\in\mathbb{R}^{2}$ with period annulus $\mathscr{P}_{\mu}$. Suppose
that there exists a continuous map $\sigma:(0,\delta)\times
U\to\mathbb{R}^{2}$ verifying, for each fixed $\mu\in U$, that

  1. $(a)$

the map $\sigma(\,\cdot\,;\mu):(0,\delta)\to\mathbb{R}^{2}$ is
$\mathscr{C}^{1}$,

  2. $(b)$

the vectors $\partial_{s}\sigma(s;\mu)$ and $X_{\mu}(\sigma(s;\mu))$ are
linearly independent for all $s\in(0,\delta),$ and

  3. $(c)$

for each compact set $K\subset\mathscr{P}_{\mu}\cup\\{p_{\mu}\\}$ there exists
$s_{K}>0$ such that $\sigma(s;\mu)\in\mathscr{P}_{\mu}\setminus K$ for all
$s\in(0,s_{K}).$

Then the following assertions hold:

  1. $1.$

$\mathscr{U}=\bigcup\limits_{\mu\in U}\mathscr{P}_{\mu}\times\\{\mu\\}$ is an
open subset of $\mathbb{R}^{2}\times U$, and

  2. $2.$

the map $(p,\mu)\mapsto\hat{P}(p;\mu)=\\{\text{period of the periodic orbit of
$X_{\mu}$ passing through $p$}\\}$ is $\mathscr{C}^{\varpi}$ on $\mathscr{U}.$

  * Proof.

We consider the family $\\{X_{\mu}\\}_{\mu\in U}$ as a single
$\mathscr{C}^{\varpi}$ vector field $Y$ on $\mathbb{R}^{2}\times U$ whose
trajectories are contained in the submanifolds $\mu=\text{constant}$. Denote
the flow of $Y$ by $\phi(t;p,\mu)=(\varphi(t;p,\mu),\mu)$. In order to prove
the first assertion, for a given $(p_{0},\mu_{0})\in\mathscr{U}$ we must show
that there is an open subset $V$ of $\mathbb{R}^{2}\times U$ such that
$(p_{0},\mu_{0})\in V\subset\mathscr{U}.$ We claim that this is true in the
particular case that there exists $s_{0}\in(0,\delta)$ such that
$\sigma(s_{0};\mu_{0})=p_{0}.$ Indeed, due to the assumption in $(b)$, note
that $Y$ is transverse to

$\Sigma_{\varepsilon}\\!:=\\{(\sigma(s;\mu),\mu);\text{
$|s-s_{0}|<\varepsilon$ and $\|\mu-\mu_{0}\|<\varepsilon$}\\}$

for all $\varepsilon>0$ small enough and that
$(p_{0},\mu_{0})\in\Sigma_{\varepsilon}.$ Then, since $\sigma:(0,\delta)\times
U\to\mathbb{R}^{2}$ is continuous, by the flow box theorem (and shrinking
$\varepsilon>0$ if necessary) it follows that

$V\\!:=\bigcup_{t\in(-\varepsilon,\varepsilon)}\phi(t;\Sigma_{\varepsilon})$

is an open subset of $\mathbb{R}^{2}\times U$. Furthermore, since
$\mathscr{U}$ is invariant by $\phi$ and
$\Sigma_{\varepsilon}\subset\mathscr{U}$ by construction, we have that
$(p_{0},\mu_{0})\in V\subset\mathscr{U}$ and this proves the claim. Let us
consider now an arbitrary $p_{0}\in\mathscr{P}_{\mu_{0}}$. Denote the periodic
orbits of $X_{\mu_{0}}$ passing through $q\\!:=\sigma(\delta/2;\mu_{0})$ and
$p_{0}$ by $\gamma_{q}$ and $\gamma_{p_{0}}$, respectively. For each $\mu\in
U$ we take the orthogonal vector field to $X_{\mu}$, say $X_{\mu}^{\bot}$,
pointing inward the periodic orbits in $\mathscr{P}_{\mu}$. We consider the
family $\\{X_{\mu}^{\bot}\\}_{\mu\in U}$ as a single $\mathscr{C}^{\varpi}$
vector field $\hat{Y}$ on $\mathbb{R}^{2}\times U$ and denote its flow by
$\hat{\phi}(t;p,\mu)=(\hat{\varphi}(t;p,\mu),\mu)$. Note that $p_{\mu}$ is
also a singular point for $X_{\mu}^{\bot}$ that, by applying the Poincaré-
Bendixson Theorem (see for instance [3]), it is easy to show to be
asymptotically stable. Observe moreover that
$\hat{\phi}(t;\mathscr{U})\subset\mathscr{U}$ for all $t\geqslant 0.$ We
define $\Gamma\\!:=\\{\hat{\varphi}(t;q,\mu_{0});\,t\geqslant
0\\}\subset\mathscr{P}_{\mu_{0}},$ which is clearly a transverse section for
$X_{\mu_{0}}$, and distinguish two cases:

    * –

Case 1: $\Gamma\cap\gamma_{p_{0}}\neq\emptyset$. In this case there exist
$t_{1},t_{2}\geqslant 0$ such that
$\hat{\varphi}(t_{2};q,\mu_{0})=\varphi(t_{1};p_{0},\mu_{0})\in\Gamma.$ Since
$q=\sigma(\delta/2;\mu_{0}),$ on account of the claim we can take an open
neighbourhood $V_{1}$ of $(q,\mu_{0})$ in $\mathbb{R}^{2}\times U$ with
$V_{1}\subset\mathscr{U}.$ Then, by the continuity of solutions with respect
to initial conditions, there exists an open neighbourhood $V_{2}$ of
$(p_{0},\mu_{0})$ such that
$\hat{\phi}\big{(}-t_{2};\phi(t_{1};V_{2})\big{)}\subset V_{1}.$ Thus
$V_{2}\subset\phi\big{(}-t_{1};\hat{\phi}(t_{2};V_{1})\big{)}\subset\mathscr{U},$
where the second inclusion follows due to the
$\hat{\phi}(t;\mathscr{U})\subset\mathscr{U}$ for all $t\geqslant 0$ and
$\phi(t;\mathscr{U})=\mathscr{U}$ for all $t\in\mathbb{R},$ together with the
fact that $V_{1}\subset\mathscr{U}.$

    * –

Case 2: $\Gamma\cap\gamma_{p_{0}}=\emptyset$. Note that in this case
$\text{Int}(\gamma_{q})\subset\text{Int}(\gamma_{p_{0}})$. (Here, given a
Jordan curve $\gamma\subset\mathbb{R}^{2}$, $\text{Int}(\gamma)$ denotes the
bounded connected component of $\mathbb{R}^{2}\setminus\\{\gamma\\}$.)
Moreover, by the assumption in $(c)$ and taking
$K=\overline{\text{Int}(\gamma_{p_{0}})}$, there exists $s_{1}\in(0,\delta/2)$
satisfying that $\sigma(s_{1};\mu_{0})\in\mathscr{P}_{\mu_{0}}\setminus K.$
Therefore, since $q=\sigma(\delta/2;\mu_{0})\in\text{Int}(\gamma_{p_{0}})$, by
continuity there exists $s_{2}\in(s_{1},\delta/2)$ such that
$\sigma(s_{2};\mu_{0})\in\gamma_{p_{0}}.$ Consequently
$\sigma(s_{2};\mu_{0})=\varphi(t_{3};p_{0},\mu_{0})$ for some
$t_{3}\in\mathbb{R}$ and on the other hand, again on account of the claim,
there exists an open neighbourhood $V_{3}$ of
$(\sigma(s_{2};\mu_{0}),\mu_{0})$ in $\mathbb{R}^{2}\times U$ with
$V_{3}\subset\mathscr{U}$. Thus, exactly as before, by continuity of solutions
with respect to initial conditions, there is an open neighbourhood $V_{4}$ of
$(p_{0},\mu_{0})$ such that
$V_{4}\subset\phi(-t_{3};V_{3})\subset\mathscr{U}.$

This proves the validity of the first assertion.

Let us prove now that the function
${\hat{P}}\\!:{\mathscr{U}}\longrightarrow{(0,+\infty)}$ defined by

$(p,\mu)\mapsto\hat{P}(p;\mu)=\\{\text{period of the periodic orbit of
$X_{\mu}$ passing through $p$}\\}$

is $\mathscr{C}^{\varpi}$. In what follows we shall use the notation $p=(x,y)$
for the components of a point of $\mathbb{R}^{2}.$ We fix
$(\hat{p},\hat{\mu})\in\mathscr{U}$ and suppose that the period of the
periodic orbit of $X_{\hat{\mu}}$ passing through
$\hat{p}=(\hat{x},\hat{y})\in\mathscr{P}_{\hat{\mu}}$ is $\hat{\tau}>0.$ Then,
due to $X(\hat{p};\hat{\mu})\\!:=X_{\hat{\mu}}(\hat{p})\neq(0,0)$, there is
$i\in\\{1,2\\}$ such that

$\partial_{t}\varphi_{i}(\hat{\tau};\hat{p},\hat{\mu})=\partial_{t}\varphi_{i}(0;\hat{p},\hat{\mu})=X_{i}(\hat{p};\hat{\mu})\neq
0.$

For simplicity in the exposition let us suppose that
$X_{1}(\hat{p};\hat{\mu})>0.$ In this case we can apply the Implicit Function
Theorem to the equation $\varphi_{1}(t;p,\mu)=x$ at
$(t,p,\mu)=(\hat{\tau},\hat{p},\hat{\mu})$ in order to obtain a
$\mathscr{C}^{\varpi}$ positive function ${S}(p;\mu)$ in a open neighbourhood
$W\subset\mathscr{U}$ of $(\hat{p},\hat{\mu})$ verifying
${S}(\hat{p};\hat{\mu})=\hat{\tau}$ and

$\left.\varphi_{1}(t;p,\mu)\right|_{t={S}(p;\mu)}=x\text{ for all $(p,\mu)\in
W.$}$ (4)

Clearly we can assume that $W$ is a cube $Q({\varepsilon_{1}})$ with center
$(\hat{p},\hat{\mu})$ and edge length $\varepsilon_{1}>0$. We diminish
$\varepsilon_{1}$ if necessary so that $X_{1}(p;\mu)>0$ for all $(p,\mu)\in
Q({\varepsilon_{1}})$. Furthermore, thanks to
${S}(\hat{p};\hat{\mu})=\hat{\tau}$ together with the continuity of ${S}$ and
$\phi$, we can take $\varepsilon_{2}\in(0,\varepsilon_{1})$ such that

$\left.\phi(t;p,\mu)\right|_{t={S}(p;\mu)}\in Q({\varepsilon_{1}})\text{ for
all $(p,\mu)\in Q({\varepsilon_{2}}).$}$

We claim that $\hat{P}=S$ on $Q({\varepsilon_{2}})$. Clearly the claim will
follow once we show that

$\left.\varphi_{2}(t;p,\mu)\right|_{t={S}(p;\mu)}=y\text{ for all $(p,\mu)\in
Q(\varepsilon_{2})$.}$

By contradiction, suppose that there exists $(\bar{p},\bar{\mu})\in
Q({\varepsilon_{2}})$ such that
$\varphi_{2}\big{(}t;\bar{p},\bar{\mu}\big{)}|_{t={S}(\bar{p};\bar{\mu})}\neq\bar{y}$.
Due to $Q(\varepsilon_{2})\subset\mathscr{U},$ the trajectory of
$X_{\bar{\mu}}$ passing through $\bar{p}$ is a periodic orbit which, for
simplicity in the exposition, we assume to travel clockwise around the center
$p_{\bar{\mu}}$ (the other case follows verbatim). That being said we consider
the piece of trajectory

$\displaystyle\ell\\!:=\left\\{\varphi(t;\bar{p},\bar{\mu});\,t\in[0,{S}(\bar{p},\bar{\mu})]\right\\}$
and the vertical segment, recall $(\ref{A4eq1})$,
$\displaystyle\Gamma\\!:=\left\\{(1-s)\bar{p}+s\varphi\big{(}{S}(\bar{p},\bar{\mu});\bar{p},\bar{\mu}\big{)};\,s\in(0,1)\right\\}\subset\\{x=\bar{x}\\}.$

Arguing on the phase portrait of $X_{\bar{\mu}}$, due to
$X_{1}(p;\bar{\mu})>0$ for all $p\in\Gamma$, if
$\varphi_{2}\big{(}{S}(\bar{p},\bar{\mu});\bar{p},\bar{\mu}\big{)}<\bar{y}$
then interior of the Jordan curve $\ell\cup\Gamma$ is a positively but not
negatively invariant subset of $\mathscr{P}_{\bar{\mu}}.$ Similarly, if
$\varphi_{2}\big{(}{S}(\bar{p},\bar{\mu});\bar{p},\bar{\mu}\big{)}>\bar{y}$
then we obtain a negatively invariant subset of $\mathscr{P}_{\bar{\mu}}$
which is not positively invariant. In both cases we get a contradiction with
the fact that $\mathscr{P}_{\bar{\mu}}$ is foliated by periodic orbits of
$X_{\bar{\mu}}$ and $Q(\varepsilon_{2})\subset\mathscr{U}.$ Consequently
$\varphi_{2}\big{(}{S}(p,\mu);p,\mu\big{)}=y$ for all $(p,\mu)\in
Q(\varepsilon_{2})$ and so the validity of the claim follows. Since
$Q(\varepsilon_{2})$ is an open neighbourhood of an arbitrary point of
$\mathscr{U}$ and $S$ is $\mathscr{C}^{\varpi}$ in $Q(\varepsilon_{2})$, the
claim implies the second assertion in the statement.

The previous result is addressed to a family $\\{X_{\mu}\\}_{\mu\in U}$ of
vector fields and this is the reason why we require the existence of a local
transverse section near the outer boundary of the period annulus $\Pi_{\mu}$
that behaves well with respect to parameters. That being said, Lemma 2.1 can
be applied to a single vector field $X$ without this requirement because a
trajectory of the orthogonal vector field $X^{\bot}$ already provides a
transverse section in the whole period annulus. Thus in order to assert that
$p\mapsto\hat{P}(p;\mu)$ is $\mathscr{C}^{\varpi}$ on $\mathscr{P}_{\mu}$ for
each fixed $\mu\in U$, it is not necessary to verify the existence of a
continuous map $\sigma:(0,\delta)\times U\to\mathbb{R}^{2}$ satisfying $(a)$,
$(b)$ and $(c).$

  * Remark 2.2.

If $X$ is a $\mathscr{C}^{\varpi}$ vector field,
$\varpi\in\mathbb{N}\cup\\{\infty,\omega\\},$ with a center then the period
function $\hat{P}$ is a first integral for the flow of $X$ on the period
annulus $\mathscr{P}$ that, by Lemma 2.1, is $\mathscr{C}^{\varpi}$.
Consequently the scalar product $\nabla\hat{P}(p)\cdot X(p)$ is zero for all
$p\in\mathscr{P}$. This implies that if $\gamma$ is a critical periodic orbit
of $X$ then the gradient $\nabla\hat{P}$ vanishes on $\gamma.$ Indeed, if
${\sigma}\\!:{(0,1)}\longrightarrow{\mathscr{P}}$ is a $\mathscr{C}^{\varpi}$
transverse section to $X$ on $\mathscr{P}$ and $P(s)\\!:=\hat{P}(\sigma(s))$
then $P^{\prime}(s)=\nabla\hat{P}(\sigma(s))\cdot\sigma^{\prime}(s).$ Thus,
since $\nabla\hat{P}(\sigma(s))\cdot X(\sigma(s))=0$, the transversality of
$\sigma$ implies that $P^{\prime}(s)=0$ if, and only if,
$\nabla\hat{P}(\sigma(s))=(0,0).$ This shows in particular that the condition
for $\gamma$ to be a critical periodic orbit is local and independent of the
particular transverse section used to parametrize the set of critical periodic
orbits near $\gamma.$ $\square$

We define next the notion that enable us to effectively study the criticality
at the outer boundary.

  * Definition 2.3.

Let $U$ be an open set of $\mathbb{R}^{N}$ and consider a family of functions
$\\{h(\,\cdot\,;\mu)\\}_{\mu\in U}$ on $(0,\varepsilon).$ Given any
$\mu_{\star}\in U$ we define $\mathcal{Z}_{0}(h(\,\cdot\,;\mu),\mu_{\star})$
to be the smallest integer $n$ having the property that there exist $\delta>0$
and a neighbourhood $V$ of $\mu_{\star}$ such that for every $\mu\in V$ the
function $h(s;\mu)$ has no more than $n$ isolated zeros on $(0,\delta)$
counted with multiplicities. $\square$

The hypothesis with regard to the local transverse section in our next result
are slightly stronger than in the previous one because we require the
continuity at $s=0$ and that $\sigma(0;\mu)$ belongs to the outer boundary
$\Pi_{\mu}$ for all $\mu\in U,$ cf. assumption $(c)$ in Lemma 2.1. We also
remark that in the statement $\hat{P}(p;\mu)$ stands for the period of the
periodic orbit of $X_{\mu}$ passing through $p\in\mathscr{P}_{\mu}.$

######  Lemma 2.4.

Let us consider a $\mathscr{C}^{\omega}$ family $\\{X_{\mu}\\}_{\mu\in U}$ of
planar polynomial vector fields such that, for each $\mu\in U$, $X_{\mu}$ has
a center $p_{\mu}\in\mathbb{R}^{2}$ with period annulus $\mathscr{P}_{\mu}$.
Let $\Pi_{\mu}\subset\mathbb{RP}^{2}$ be the outer boundary of
$\mathscr{P}_{\mu}.$ Suppose there exists a continuous map
$\sigma:[0,\delta)\times U\to\mathbb{RP}^{2}$ verifying that, for each $\mu\in
U,$

  1. $(a)$

the map $\sigma(\,\cdot\,;\mu):(0,\delta)\to\mathbb{R}^{2}$ is
$\mathscr{C}^{1}$,

  2. $(b)$

the vectors $\partial_{s}\sigma(s;\mu)$ and $X_{\mu}(\sigma(s;\mu))$ are
linearly independent for all $s\in(0,\delta)$,

  3. $(c)$

$\sigma(s;\mu)\in\mathscr{P}_{\mu}$ for all $s\in(0,\delta)$ and
$\sigma(0;\mu)\in\Pi_{\mu}$.

Then, for each fixed $\mu_{\star}\in U$, the following assertions hold:

  1. 1.

The Hausdorff distance between the outer boundaries $\Pi_{\mu}$ and
$\Pi_{\mu_{\star}}$ tends to zero as $\mu\to\mu_{\star}.$

  2. 2.

If $P(s;\mu)\\!:=\hat{P}(\sigma(s;\mu);\mu)$ for all
$(s,\mu)\in(0,\delta)\times U$, then

    1. $(2a)$

$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}\leqslant\mathcal{Z}_{0}(P^{\prime}(\,\cdot\,;\mu),\mu_{\star}).$

    2. $(2b)$

$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}\geqslant
n$ if for each open neighbourhood $V$ of $\mu_{\star}$ and $\delta>0$ there
exist $n$ different numbers $s_{1},s_{2}\ldots,s_{n}\in(0,\delta)$ and
$\hat{\mu}\in V$ such that $P^{\prime}(s_{i};\hat{\mu})=0$ for
$i=1,2,\ldots,n.$

    3. $(2c)$

$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}=0$
if, and only if, $\mathcal{Z}_{0}(P^{\prime}(\,\cdot\,;\mu),\mu_{\star})=0.$

  * Proof.

To show the first assertion note that, since $X_{\mu}$ is polynomial, we can
consider its Poincaré compactification $p(X_{\mu})$, see [3, §5] for details,
which is an analytic vector field on the sphere $\mathbb{S}^{2}$ topologically
equivalent to $X_{\mu}.$ The outer boundary $\Pi_{\mu}$ becomes then a
polycycle of $p(X_{\mu})$ that can be studied using local charts of
$\mathbb{S}^{2}.$ On account of this, the fact that
$d_{H}(\Pi_{\mu},\Pi_{\mu_{\star}})\to 0$ as $\mu\to\mu_{\star}$ follows by
the continuity of $\mu\mapsto\sigma(0;\mu)\in\Pi_{\mu}$ together with the
continuity with respect to initial conditions and parameters of the
trajectories of $p(X_{\mu}).$ The interested reader is referred to [36, Lemma
22, p. 110] for a related result for limit periodic sets.

With regard to the upper bound in $(2a)$ it is clear that if
$\mathcal{Z}_{0}(P^{\prime}(\,\cdot\,;\mu),\mu_{\star})=+\infty$ then there is
nothing to be proved. So let us assume that
$\mathcal{Z}_{0}(P^{\prime}(\,\cdot\,;\mu),\mu_{\star})=\ell\in\mathbb{Z}_{\geq
0}$ and argue by contradiction. If
$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}\geqslant\ell+1$
then there exist $\ell+1$ sequences
$\\{\gamma^{k}_{\mu_{i}}\\}_{i\in\mathbb{N}}$, $k=1,2,\ldots,\ell+1$, where
$\gamma^{1}_{\mu_{i}},\gamma^{2}_{\mu_{i}},\ldots,\gamma^{\ell+1}_{\mu_{i}}$
are different critical periodic orbits of $X_{\mu_{i}}$ for each
$i\in\mathbb{N}$, such that ${\mu_{i}}\to\mu_{\star}$ and
$d_{H}(\gamma^{k}_{\mu_{i}},\Pi_{\mu_{\star}})\to 0$ as $i\to+\infty$. Then,
due to $d_{H}(\Pi_{\mu},\Pi_{\mu_{\star}})\to 0$ as $\mu\to\mu_{\star}$ and

$d_{H}(\gamma^{k}_{\mu_{i}},\Pi_{\mu_{i}})\leqslant
d_{H}(\gamma^{k}_{\mu_{i}},\Pi_{\mu_{\star}})+d_{H}(\Pi_{\mu_{i}},\Pi_{\mu_{\star}}),$

we have $d_{H}(\gamma^{k}_{\mu_{i}},\Pi_{\mu_{i}})\to 0$ as $i\to+\infty$ for
each $k=1,2,\ldots,\ell+1.$ Since $\sigma(0;{\mu_{i}})\in\Pi_{\mu_{i}}$ and
there is a one-to-one correspondence between zeros of $P^{\prime}(s;\mu_{i})$
arbitrarily near $s=0$ and critical periodic orbits of $X_{\mu_{i}}$
arbitrarily close to $\Pi_{\mu_{i}}$ (cf. [36, Lemma 22]), this implies that
there exist $\ell+1$ sequences of positive numbers
$\\{s_{i}^{k}\\}_{i\in\mathbb{N}}$, $k=1,2,\ldots,\ell+1,$ such that
$P^{\prime}(s_{i}^{k};\mu_{i})=0$ and
$\\#\\{s_{i}^{1},s_{i}^{2},\ldots,s_{i}^{\ell+1}\\}=\ell+1$ for each
$i\in\mathbb{N}$, and $\lim_{i\to+\infty}s_{i}^{k}=0$ for each
$k=1,2,\ldots,\ell+1.$ This clearly contradicts that
$\mathcal{Z}_{0}(P^{\prime}(\,\cdot\,;\mu),\mu_{\star})=\ell$, see Definition
2. The assertion in $(2b)$ follows similarly. Indeed, on account of the
assumption and the above mentioned one-to-one correspondence between zeros of
$P^{\prime}(s;\mu)$ near $s=0$ and critical periodic orbits of $X_{\mu}$ close
to $\Pi_{\mu_{i}},$ we can construct ${n}$ sequences
$\\{\gamma^{k}_{\mu_{i}}\\}_{i\in\mathbb{N}}$, $k=1,2,\ldots,{n}$, where
$\gamma^{1}_{\mu_{i}},\gamma^{2}_{\mu_{i}},\ldots,\gamma^{{n}}_{\mu_{i}}$ are
different critical periodic orbits of $X_{\mu_{i}}$ for each $i\in\mathbb{N}$,
such that ${\mu_{i}}\to\mu_{\star}$ and
$d_{H}(\gamma^{k}_{\mu_{i}},\Pi_{\mu_{i}})\to 0$ as $i\to+\infty$. Then, using
that $d_{H}(\Pi_{\mu},\Pi_{\mu_{\star}})\to 0$ as $\mu\to\mu_{\star}$, we can
assert that $\lim_{i\to+\infty}d_{H}(\gamma^{k}_{\mu_{i}},\Pi_{\mu_{0}})=0$
for each $k=1,2,\ldots,n$, which implies
$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}\geqslant
n$, as desired. Finally the assertion in $(2c)$ follows easily from the ones
in $(2a)$ and $(2b).$ This completes the proof of the result.

Next we introduce the notion of global transverse section for a family of
period annuli. Roughly speaking it is a transverse section, joining the center
with some point at the outer boundary of the period annulus, that behaves well
with the parameters.

  * Definition 2.5.

Let us fix $\varpi\in\mathbb{N}\cup\\{\infty,\omega\\}$ and consider a
$\mathscr{C}^{\varpi}$ family $\\{X_{\mu}\\}_{\mu\in U}$ of planar vector
fields such that, for each $\mu\in U,$ $X_{\mu}$ has a center
$p_{\mu}\in\mathbb{R}^{2}$ with period annulus $\mathscr{P}_{\mu}$. Let
$\Pi_{\mu}\subset\mathbb{RP}^{2}$ be the outer boundary of
$\mathscr{P}_{\mu}.$ A _global transverse section_ for the family of period
annuli $\\{\mathscr{P}_{\mu}\\}_{\mu\in U}$ is a continuous map
$\sigma:[0,1]\times U\to\mathbb{RP}^{2}$ verifying that

    1. $(a)$

the map $\sigma(\,\cdot\,;\mu):[0,1]\to\mathbb{RP}^{2}$ is
$\mathscr{C}^{\varpi}$ for each $\mu\in U$,

    2. $(b)$

the vectors $\partial_{s}\sigma(s;\mu)$ and $X_{\mu}(\sigma(s;\mu))$ are
linearly independent for all $(s,\mu)\in(0,1)\times U$ and the map
$\partial_{s}\sigma:(0,1)\times U\to\mathbb{R}^{2}$ is continuous,

    3. $(c)$

$\sigma(s;\mu)\in\mathscr{P}_{\mu}$ for all $s\in(0,1)$,
$\sigma(0;\mu)\in\Pi_{\mu}$ and $\sigma(1;\mu)=p_{\mu}$.

When such a global transverse section exists we say that the family of period
annuli $\\{\mathscr{P}_{\mu}\\}_{\mu\in U}$ varies continuously. $\square$

  * Remark 2.6.

The period annulus of the family of Loud’s quadratic centers given in
$(\ref{sist_loud})$ varies continuously in the sense of Definition 2. Indeed,
it follows from the proof of [23, Lemma 3.2] that

$\mu=(D,F)\mapsto\xi_{\mu}\\!:=\sup\\{t>0;\,(s,0)\in\mathscr{P}_{\mu}\text{
for all $s\in(0,t)$}\\}$

is a well-defined continuous function on $\mathbb{R}^{2}.$ Moreover the point
$(\xi_{\mu},0)$ belongs to $\Pi_{\mu}$ and $0<\xi_{\mu}\leqslant 1$ for all
$\mu\in\mathbb{R}^{2}.$ Then $\sigma(s;\mu)=((1-s)\xi_{\mu},0)$ for
$(s,\mu)\in[0,1]\times\mathbb{R}^{2}$ is clearly a global transverse section.
In particular, since the Loud’s system is polynomial, the outer boundary of
the period annulus varies continuously in the Hausdorff sense by the first
assertion in Lemma 2.4. $\square$

Note, see $(b)$ in Definition 2, that we also require
$(s,\mu)\mapsto\partial_{s}\sigma(s;\mu)$ to be continuous. The reason for
this is because if we define $P(s;\mu)=\hat{P}(\sigma(s;\mu);\mu)$ then
$(s,\mu)\mapsto\partial_{s}P(s;\mu)$ is a continuous function by Lemma 2.1.
This continuity is a key point in the forthcoming results. Before that we
summarize in the next statement the properties that we get for $P(s;\mu)$ as a
consequence of Lemma 2.1 and Definition 2.

######  Corollary 2.7.

Let us fix $\varpi\in\mathbb{N}\cup\\{\infty,\omega\\}$ and consider a
$\mathscr{C}^{\varpi}$ family of planar vector fields $\\{X_{\mu}\\}_{\mu\in
U}$ such that, for each $\mu\in U$, $X_{\mu}$ has a center
$p_{\mu}\in\mathbb{R}^{2}$ with period annulus $\mathscr{P}_{\mu}$. Assume
that the family of period annuli varies continuously and let
$\sigma:[0,1]\times U\to\mathbb{RP}^{2}$ be a global transverse section for
$\\{\mathscr{P}_{\mu}\\}_{\mu\in U}$. If
$P(s;\mu)\\!:=\hat{P}(\sigma(s;\mu);\mu)$ for all $(s,\mu)\in(0,1)\times U$
then the following holds:

  1. $(a)$

$P(\,\cdot\,;\mu)\in\mathscr{C}^{\varpi}((0,1))$ for each $\mu\in U,$ and

  2. $(b)$

$P$ and $\partial_{s}P$ are continuous functions on $(0,1)\times U$.

  * Definition 2.8.

Under the assumptions of Corollary 2.7, we say that
$P(s;\mu)=\hat{P}(\sigma(s;\mu);\mu)$, which is defined for
$(s,\mu)\in(0,1)\times U,$ is a _global parametrization of the period
function_. In contrast,

$(p,\mu)\mapsto\hat{P}(p;\mu)=\\{\text{period of the periodic orbit of
$X_{\mu}$ passing through $p$}\\}$

is defined on $\bigcup_{\mu\in U}\mathscr{P}_{\mu}\times\\{\mu\\},$ which is
not so easy to handle. $\square$

One of the main goals in the present section is to relate the concept of local
bifurcation value of the period function, as introduced in [23], with the
notion of criticality, see Definition 1. As we will see the first one concerns
with the qualitative properties of the period function, whereas the second is
more geometric and quantitative. In doing so we will be able to take advantage
of the results about the bifurcation diagram of the period function of the
Loud’s centers that we obtained in [23]. For reader’s convenience we next
recall the definition of local bifurcation value of the period function.

  * Definition 2.9.

Let $\\{I_{\mu}\\}_{\mu\in U}$ be a continuous family of intervals in
$\mathbb{R}$, i.e., such that $I_{\mu}=(\ell(\mu),r(\mu))$ with
$\ell,r\in\mathscr{C}^{0}(U)$, and consider a continuous family of functions
$\\{\mbox{${F_{\mu}}\\!:{I_{\mu}}\longrightarrow{\mathbb{R}}$}\\}_{\mu\in U}.$
We say that $\mu_{0}\in U$ is a _regular value of the family_
$\\{\mbox{${F_{\mu}}\\!:{I_{\mu}}\longrightarrow{\mathbb{R}}$}\\}_{\mu\in U}$
if there exist a neighbourhood $V$ of $\mu_{0}$ and an isotopy
$\\{\mbox{${h_{\mu}}\\!:{I_{\mu}}\longrightarrow{I_{\mu_{0}}}$}\\}_{\mu\in
V},$ with $h_{\mu_{0}}=id,$ such that

$\text{sgn}\biggl{(}{F}_{\mu}(s)\biggr{)}=\text{sgn}\biggl{(}{F}_{\mu_{0}}\bigl{(}h_{\mu}(s)\bigr{)}\biggr{)}\text{
for all $s\in I_{\mu}$ and $\mu\in V,$}$ (5)

where $\mathrm{sgn}:\mathbb{R}\to\\{-1,0,1\\}$ is the extended sign function.
A parameter $\mu_{0}$ which is not regular is called a _bifurcation value._
$\square$

The endpoints of $I_{\mu}$, the domain of definition of $F_{\mu}$, depend
continuously on $\mu,$ so that $\cup_{\mu\in{U}}I_{\mu}\\!\times\\!\\{\mu\\}$
is an open subset of $\mathbb{R}\times{U}.$ Thus, by a continuous family of
functions
$\\{\mbox{${F_{\mu}}\\!:{I_{\mu}}\longrightarrow{\mathbb{R}}$}\\}_{\mu\in U}$,
we mean that the map $(s,\mu)\mapsto F_{\mu}(s)$ is continuous on
$\cup_{\mu\in{U}}I_{\mu}\\!\times\\!\\{\mu\\}$. Next we particularize the
previous definition to study the period function. To this aim note that, by
Corollary 2.7, if $\\{X_{\mu}\\}_{\mu\in U}$ is a $\mathscr{C}^{1}$ family of
vector fields with a center such that the corresponding family of period
annuli varies continuously, and we set $P(s;\mu)=\hat{P}(\sigma(s;\mu);\mu)$,
then $\\{\partial_{s}P(\,\cdot\,;\mu)\\}_{\mu\in U}$ is a continuous family of
functions on $(0,1).$

  * Definition 2.10.

Consider a $\mathscr{C}^{1}$ family of planar vector fields
$\\{X_{\mu}\\}_{\mu\in U}$ such that, for each $\mu\in U$, $X_{\mu}$ has a
center $p_{\mu}\in\mathbb{R}^{2}$ with period annulus $\mathscr{P}_{\mu}$,
that we suppose to vary continuously.

    1. $(a)$

We say that $\mu_{0}\in{U}$ is a _regular_ (respectively, _bifurcation_)
_value of the period function_ if for some global parametrization of the
period function ${P}\\!:{(0,1)\times U}\longrightarrow{(0,+\infty)}$ we have
that $\mu_{0}$ is a regular (respectively, bifurcation) value of the family
$\\{P^{\prime}(\,\cdot\,;\mu)\\!:(0,1)\to\mathbb{R}\\}_{\mu\in U}.$

    2. $(b)$

We say that $\mu_{0}\in{U}$ is a _local regular value of the period function
at the interior_ if there is some global parametrization of the period
function ${P}\\!:{(0,1)\times U}\longrightarrow{(0,+\infty)}$ satisfying that
for each $c\in(0,1)$ there exists a continuously varying neighbourhood
$I_{\mu}(c)$ of $c$ in $(0,1)$ such that $\mu_{0}$ is a regular value of the
family $\\{P^{\prime}(\,\cdot\,;\mu)\\!:I_{\mu}(c)\to\mathbb{R}\\}_{\mu\in
U}.$ A parameter which is not a local regular value at the interior is called
a _local bifurcation value at the interior._

    3. $(c)$

We say that $\mu_{0}\in{U}$ is a _local regular value of the period function
at the outer_ (respectively, _inner_) _boundary_ if for some global
parametrization of the period function ${P}\\!:{(0,1)\times
U}\longrightarrow{(0,+\infty)}$ there exists a continuously varying
neighbourhood $I_{\mu}(c)$ of $c=0$ (respectively, $c=1$) such that $\mu_{0}$
is a regular value of the family
$\\{\mbox{${P^{\prime}(\,\cdot\,;\mu)}\\!:{I_{\mu}(c)\cap(0,1)}\longrightarrow{\mathbb{R}}$}\\}_{\mu\in{U}}.$
A parameter which is not a local regular value at the outer (respectively,
inner) boundary is called a _local bifurcation value at the outer_
(respectively, _inner_) _boundary._

$\square$

  * Remark 2.11.

Let us make the following easy observations with regard to the previous
definitions:

    1. $(a)$

One can replace “some global parametrization” by “any global parametrization”.
Indeed, suppose that $\mu_{0}\in U$ is a regular value for
$\\{P^{\prime}(\,\cdot\,;\mu)\\}_{\mu\in U}$ where
$P(s;\mu)=\hat{P}(\sigma(s;\mu);\mu)$ and consider another global
parametrization $\bar{P}(s;\mu)=\hat{P}(\bar{\sigma}(s;\mu);\mu)$ of the
period function, see Definition 2. If we denote by $\tau_{\mu}(s)$ the
Poincaré map from the transverse section $\Sigma$ given by
$s\mapsto\sigma(s;\mu)$ to the transverse section $\bar{\Sigma}$ given by
$s\mapsto\bar{\sigma}(s;\mu)$ then $\tau_{\mu}$ is an increasing
diffeomorphism and $P(s;\mu)=\bar{P}(\tau_{\mu}(s);\mu),$ so that
$P^{\prime}(s;\mu)=\bar{P}^{\prime}(\tau_{\mu}(s);\mu)\tau^{\prime}_{\mu}(s).$
On account of this and following the notation in Definition 2,
$\bar{h}_{\mu}\\!:=\tau_{\mu_{0}}\circ h_{\mu}\circ\tau^{-1}_{\mu}$ is a
suitable isotopy in order to show that $\mu_{0}$ is a regular value for the
family $\\{\bar{P}^{\prime}(\,\cdot\,;\mu\\}_{\mu\in U}$ because

$\text{sgn}\big{(}\bar{P}^{\prime}(s;\mu)\big{)}=\text{sgn}\big{(}P^{\prime}(\tau_{\mu}^{-1}(s);\mu)\big{)}=\text{sgn}\big{(}P^{\prime}\big{(}(h_{\mu}\circ\tau_{\mu}^{-1})(s);\mu_{0}\big{)}\big{)}=\text{sgn}\big{(}\bar{P}^{\prime}\big{(}\bar{h}_{\mu}(s);\mu_{0}\big{)}\big{)},$

where we use that $\tau^{\prime}(s)>0.$

    2. $(b)$

In order to study if a parameter is a local regular value at the outer
boundary it is not necessary to consider a global transverse section
${\sigma}\\!:{[0,1]\times U}\longrightarrow{\mathbb{RP}^{2}}$ for the family
of period annuli. Indeed, see point $(c)$ in Definition 2, it suffices to take
a local parametrization ${\sigma}\\!:{[0,\delta)\times
U}\longrightarrow{\mathbb{RP}^{2}}$. Similarly, to study the local regular
values at the inner boundary it suffices to take a local parametrization
${\sigma}\\!:{(1-\delta,1]\times U}\longrightarrow{\mathbb{R}^{2}}$.

$\square$

As expected, $\mu_{0}$ is a bifurcation value of the period function if, and
only if, $\mu_{0}$ is either a local bifurcation value at the inner boundary,
at the outer boundary or at the interior. This is stated in the following
result and the interested reader is referred to [23, Lemma 2.7] for the proof.

######  Lemma 2.12.

Let us consider a $\mathscr{C}^{1}$ family of analytic planar vector fields
$\\{X_{\mu}\\}_{\mu\in U}$ such that, for each $\mu\in U$, $X_{\mu}$ has a
center $p_{\mu}\in\mathbb{R}^{2}$ with period annulus $\mathscr{P}_{\mu}$,
that we suppose to vary continuously. Then the bifurcation diagram of the
period function is the union of the local bifurcation diagrams at the inner
and outer boundary and in the interior.

Under the assumptions and notation in Corollary 2.7, a sufficient condition
for $\mu_{\star}\in U$ to be a local regular value of the period function at
the interior is that $P^{\prime}(s;\mu_{\star})\neq 0$ for all $s\in(0,1).$
This follows easily by the continuity of $(s,\mu)\mapsto P^{\prime}(s;\mu)$ on
$(0,1)\times U$ and a compactness argument. In case that this function is
$\mathscr{C}^{1}$ then another sufficient condition is that
$P^{\prime}(\,\cdot\,;\mu_{\star})$ has only simple zeros because the
application of the Implicit Function Theorem provides the appropriate
isotopies. Hence, in this context, the set of local bifurcation values of the
period function at the interior is contained in

$\Delta\\!:=\\{\mu\in U;\text{ there exists $s\in(0,1)$ such that
}P^{\prime}(s;\mu)=P^{\prime\prime}(s;\mu)=0\\}.$

If $P^{\prime}(s;\mu)$ was polynomial in $s$ (which is certainly not true)
then $\Delta$ would consist of those parameters $\mu_{\star}$ such that the
_discriminant_ of $P^{\prime}(s;\mu_{\star})$ is equal to zero. (Recall that
the discriminant of $q\in\mathbb{R}[x]$ is the resultant between $q(x)$ and
$q^{\prime}(x)$, see for instance [8].) One may expect on the other hand that
the parameters in $\Delta$ are always local bifurcation values of the period
function at the interior. However this is not always the case and the
following toy models show that some additional assumptions are needed to this
end.

  * Example 2.13.

Setting $N=1,$ we take $P^{\prime}$ to be $F(s;\mu)=(s-\mu)^{2}$ and
$U=(0,1).$ Then it is clear that any $\mu\in U$ is a local regular value of
$F$ at the interior (i.e., there are no local bifurcation values) but we have
that $\Delta=U.$ Note that in this case the interior of $\Delta$ is non-empty.
$\square$

  * Example 2.14.

Setting $N=2,$ we take $P^{\prime}$ to be $F(s;\mu)=(s-\mu_{1})^{3}-\mu_{2}$
and $U=(0,1)^{2}.$ Then again it turns out that any $\mu=(\mu_{1},\mu_{2})\in
U$ is a local regular value of $F$ at the interior, whereas $\Delta=\\{\mu\in
U:\mu_{2}=0\\}.$ Observe that in this case the interior of $\Delta$ is empty
but $F(\,\cdot\,;\mu)$ has zeroes of multiplicity 3. $\square$

The following result provides us with an analytical tool to study the local
bifurcation values of the period function at the interior. We emphasize that
it has the natural hypothesis in view of the previous discussion.

######  Lemma 2.15.

Let $\\{X_{\mu}\\}_{\mu\in U}$ be an analytic family of planar vector fields
such that, for each $\mu\in U$, $X_{\mu}$ has a center
$p_{\mu}\in\mathbb{R}^{2}$ with period annulus $\mathscr{P}_{\mu}$. Assume
that the family of period annuli varies continuously and let
$\sigma:[0,1]\times U\to\mathbb{RP}^{2}$ be a global transverse section for
$\\{\mathscr{P}_{\mu}\\}_{\mu\in U}$. Setting
$P(s;\mu)=\hat{P}(\sigma(s;\mu);\mu)$ for all $(s,\mu)\in(0,1)\times U$,
suppose additionally that

  1. $(a)$

the interior of $\Delta$ $($as a subset of $U\subset\mathbb{R}^{N})$ is empty,
and

  2. $(b)$

for each $\mu\in U$, the zeros of $P^{\prime}(\,\cdot\,;\mu)$ have at most
multiplicity 2.

Then each $\mu\in\Delta$ is a local bifurcation value of the period function
at the interior.

  * Proof.

Note first that, by Corollary 2.7, the function $P(\,\cdot\,;\mu)$ is analytic
on $(0,1)$ for each $\mu\in U.$ Let us take any $\mu_{0}\in\Delta.$ Then there
exists $s_{0}\in(0,1)$ such that
$P^{\prime}(s_{0};\mu_{0})=P^{\prime\prime}(s_{0};\mu_{0})=0$ and, by the
hypothesis in $(b)$, $P^{\prime\prime\prime}(s_{0};\mu_{0})\neq 0$.
Consequently $P^{\prime}(\,\cdot\,;\mu_{0})$ has a local extremum at $s=s_{0}$
and so there exists $\varepsilon>0$ small enough such that
$P^{\prime}(\,\cdot\,;\mu_{0})$ has the same sign $+1$ or $-1$ on
$(s_{0}-\varepsilon,s_{0}+\varepsilon)\setminus\\{s_{0}\\}.$ Assume by
contradiction that $\mu_{0}$ is a local regular value of the period function
at the interior. Then, taking $c=s_{0}$ in $(b)$ of Definition 2, we can
consider a neighbourhood $V$ of $\mu_{0}$, a continuously varying
neighbourhood $I_{\mu}$ of $s_{0}$ in $(0,1)$ and an isotopy
$h_{\mu}:I_{\mu}\to I_{\mu_{0}}$ for $\mu\in V$, with
$h_{\mu_{0}}=\mathrm{id}$, verifying the equality in $(\ref{sgn})$. Since
$\Delta$ has empty interior we can take $\hat{\mu}\in V\setminus\Delta$ and
define $\hat{s}\\!:=h_{\hat{\mu}}^{-1}(s_{0})\in I_{\hat{\mu}}$. On account of
this, particularizing $(\ref{sgn})$ with $\mu=\hat{\mu}$ and $s=\hat{s}$ we
deduce that $P^{\prime}(\hat{s};\hat{\mu})=0$. Accordingly, due to
$\hat{\mu}\notin\Delta,$ it follows that
$P^{\prime\prime}(\hat{s};\hat{\mu})\neq 0$. Therefore the function $s\mapsto
P^{\prime}(s;\hat{\mu})$ changes sign at $s=\hat{s}.$ This contradicts
$(\ref{sgn})$ taking $\mu=\hat{\mu}$ and $s\approx\hat{s}$ because
$P^{\prime}(\,\cdot\,;\mu_{0})$ has the same sign on
$(s_{0}-\varepsilon,s_{0}+\varepsilon)\setminus\\{s_{0}\\}.$

In the statement of our next result $p(X)$ stands for the Poincaré
compactification in $\mathbb{S}^{2}$ of a planar polynomial vector field $X$,
see [3, §5] for details. Recall also that any polycycle of an analytic vector
field can be desingularized giving a polycycle with only hyperbolic or semi-
hyperbolic vertices. By a hyperbolic polycycle we mean that its
desingularization does not have semi-hyperbolic vertices (i.e., saddle-nodes).

######  Lemma 2.16.

Consider a $\mathscr{C}^{\omega}$ family of planar polynomial vector fields
$\\{X_{\mu}\\}_{\mu\in U}$ such that, for each $\mu\in U$, $X_{\mu}$ has a
center $p_{\mu}\in\mathbb{R}^{2}$ with period annulus $\mathscr{P}_{\mu}$,
that we suppose to vary continuously. Then the following assertions hold for
any given $\mu_{\star}\in U$:

  1. $(a)$

If
$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}=0$
then $\mu_{\star}$ is a local regular value of the period function at the
outer boundary.

  2. $(b)$

Assuming additionally that the outer boundary $\Pi_{\mu_{\star}}$ is a
hyperbolic polycycle of $p(X_{\mu_{\star}})$, if $\mu_{\star}$ is a local
regular value of the period function at the outer boundary then
$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}=0$.

  * Proof.

Since the family of period annuli varies continuously, see Definition 2, we
can take a global transverse section $\sigma:[0,1]\times U\to\mathbb{RP}^{2}$
and consider the global parametrization of the period function given by
$P(s;\mu)\\!:=\hat{P}(\sigma(s;\mu);\mu)$ for $(s,\mu)\in(0,1)\times U,$ see
Corollary 2.7.

In order to show $(a)$ note that if
$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}=0$
then $\mathcal{Z}_{0}(P^{\prime}(\,\cdot\,,\mu),\mu_{\star})=0$ by assertion
$(2c)$ in Lemma 2.4. This implies, see Definition 2, the existence of
$\delta>0$ and a neighbourhood $V$ of $\mu_{\star}$ such that
$P^{\prime}(s;\mu)$ does not vanish on $(0,\delta)\times V$. Hence, since
$(s,\mu)\mapsto P^{\prime}(s;\mu)$ is continuous thanks to $(b)$ in Corollary
2.7, the function $P^{\prime}(s;\mu)$ has constant sign on $(0,\delta)\times
V$. Thus, see Definitions 2 and 2, taking $I_{\mu}=(0,\delta)$ and
$h_{\mu}=id$ it follows that $\mu_{\star}$ is a regular value of the family
$\\{P^{\prime}(\,\cdot\,;\mu):I_{\mu}\to\mathbb{R}\\}_{\mu\in U}$ as desired.
This shows the validity of the assertion in $(a)$.

Let us turn next to the assertion in $(b)$. If $\mu_{\star}$ is a local
regular value of the period function at the outer boundary then there exist a
neighbourhood $V$ of $\mu_{\star}$, a continuous strictly positive function
$\mu\mapsto\varepsilon_{\mu}$ on $V$ and an isotopy
$\\{\mbox{${h_{\mu}}\\!:{(0,\varepsilon_{\mu})}\longrightarrow{(0,\varepsilon_{\mu_{\star}})}$}\\}_{\mu\in
V}$ such that
$\mathrm{sgn}(P^{\prime}(s;\mu))=\mathrm{sgn}(P^{\prime}(h_{\mu}(s));\mu_{\star})$
for all $s\in(0,\varepsilon_{\mu})$ and $\mu\in V$. From this point we
distinguish two cases:

    1. 1.

If the center of $X_{\mu_{\star}}$ is not isochronous then, by applying [26,
Theorem 1.1], the zeros of $P^{\prime}(s;\mu_{\star})$ do not accumulate to
$s=0.$ Let us remark that to apply this result we take into account that the
transverse section $\sigma(\,\cdot\,;\mu_{\star})$ is analytic at $s=0$, see
Definition 2, and the hypothesis that $\Pi_{\mu_{\star}}$ is a hyperbolic
polycycle of $p(X_{\mu_{\star}})$. Hence there exists $\rho>0$ such that
$P^{\prime}(s;\mu_{\star})\neq 0$ for all $s\in(0,\rho).$ Thus, since we can
suppose without loss of generality that $\varepsilon_{\mu_{\star}}\in(0,\rho)$
and $\delta\\!:=\inf\\{\varepsilon_{\mu}:\mu\in V\\}>0$, it follows that
$P^{\prime}(s;\mu)\neq 0$ on $(0,\delta)\times V,$ which implies (see
Definition 2) that $\mathcal{Z}_{0}(P^{\prime}(\,\cdot\,,\mu),\mu_{\star})=0$.
Therefore, by assertion $(2c)$ in Lemma 2.4,
$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}=0$.

    2. 2.

If the center of $X_{\mu_{\star}}$ is isochronous then
$P^{\prime}(\,\cdot\,;\mu_{\star})\equiv 0.$ Hence
$\mathrm{sgn}(P^{\prime}(s;\mu))=\mathrm{sgn}(P^{\prime}(h_{\mu}(s));\mu_{\star})=0$
for all $s\in(0,\varepsilon_{\mu})$ and $\mu\in V$. Thus
$P^{\prime}(\,\cdot\,;\mu)$ has not isolated zeros for all $\mu\in V$ and
consequently, see Definition 2,
$\mathcal{Z}_{0}(P^{\prime}(\,\cdot\,,\mu),\mu_{\star})=0$. Then
$\mathrm{Crit}\bigl{(}(\Pi_{\mu_{\star}},X_{\mu_{\star}}),X_{\mu}\bigr{)}=0$
by $(2c)$ in Lemma 2.4.

This shows $(b)$ and completes the proof of the result.

We conclude this section by showing that, as we explain in the introduction,
Theorem A leaves us very close to the proof of the existence of an upper bound
for the number of critical periodic orbits in the family
$\\{X_{{\nu}},{{\nu}}\in\mathbb{R}^{2}\\}$. In this respect we note that there
are parameter values ${{\nu}}\in\mathbb{R}^{2}$ for which $X_{{\nu}}$ has
another center $p_{\nu}$ apart from the one at the origin (see for instance
[23, Figure 4]). The bound also holds for the critical periodic orbits of this
second center because one can always find an invertible affine transformation
${g}\\!:{\mathbb{R}^{2}}\longrightarrow{\mathbb{R}^{2}}$ with
$g(p_{{\nu}})=(0,0)$ such that the push-forward of $X_{{\nu}}$ by $g$ verifies
$g_{*}(X_{{\nu}})=\beta X_{\hat{{\nu}}}$ for some
$\hat{{\nu}}\in\mathbb{R}^{2}$ and $\beta\neq 0.$

######  Lemma 2.17.

Consider the family of vector fields
$\\{X_{{\nu}},{{\nu}}\in\mathbb{R}^{2}\\}$ given in $(\ref{sist_loud})$. If
$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}$
is finite for every ${{\nu}}_{0}\in\mathbb{R}^{2}$ then there exists
$N\in\mathbb{N}$ such that the center at the origin of $X_{{\nu}}$ has at most
$N$ critical periodic orbits for all ${{\nu}}\in\mathbb{R}^{2}.$

  * Proof.

By Lemma 2.1,
$\mathscr{U}\\!:=\bigcup_{{{\nu}}\in\mathbb{R}^{2}}\mathscr{P}_{{\nu}}\times\\{{{\nu}}\\}$
is an open subset of $\mathbb{R}^{2}\times\mathbb{R}^{2}$ and the map

$(p,{{\nu}})\mapsto\hat{P}(p;{{\nu}})=\\{\text{period of the periodic orbit of
$X_{{\nu}}$ passing through $p$}\\}$

is analytic on $\mathscr{U}.$ We define
$P(s;{{\nu}})\\!:=\hat{P}\big{(}\big{(}(1-s)\xi_{{\nu}},0\big{)};{{\nu}}\big{)}$
for each $(s,{{\nu}})\in(0,1)\times\mathbb{R}^{2}$, see Remark 2, which
provides us with a suitable global parametrization of the period function. Let
us note in particular that $\partial_{s}^{k}P(s;{{\nu}})$ is a continuous
function on $(0,1)\times\mathbb{R}^{2}$ for each $k\in\mathbb{N}$. Moreover,
by [19, Theorem A], we know that if ${{\nu}}=(D,F)\notin
K\\!:=[-7,2]\times[0,4]$ then then the center at the origin of $X_{{\nu}}$ has
no critical periodic orbits. Consequently, if for each fixed
${{\nu}}\in\mathbb{R}^{2}$ we define $N_{{\nu}}$ to be the number of isolated
zeros of $s\mapsto P^{\prime}(s;{{\nu}})$ on the interval $(0,1)$ counted
without multiplicities, the result will follow once we prove that

$\sup_{{{\nu}}\in K}(N_{{\nu}})<+\infty.$

Let us advance that this will be a consequence of the compactness of
$[0,1]\times K.$ With this end in view we fix any
$(s_{\star},{{\nu}}_{\star})\in[0,1]\times K$ and observe that three different
situations may occur:

    1. $(a)$

Case $\mathbf{s_{\star}=1}$. As a consequence of the result of Chicone and
Jacobs, see [6, Theorem 3.1], there exist $\varepsilon,\delta>0,$ depending on
${{\nu}}_{\star},$ such if ${{\nu}}\in
B_{\varepsilon}({{\nu}}_{\star})\\!:=\\{{{\nu}}\in\mathbb{R}^{2}:\|{{\nu}}-{{\nu}}_{\star}\|<\varepsilon\\}$
then the number of isolated roots of $P^{\prime}(s;{{\nu}})=0$ with
$s\in(1-\delta,1)$ is at most 2 (counted with multiplicities).

    2. $(b)$

Case $\mathbf{s_{\star}=0}$. Since
$\ell\\!:=\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{\star}},X_{{{\nu}}_{\star}}),X_{{{\nu}}}\bigr{)}<+\infty$
by assumption, $(2b)$ in Lemma 2.4 implies that there exist
$\varepsilon,\delta>0$ (depending on ${{\nu}}_{\star}$ again) such if
${{\nu}}\in B_{\varepsilon}({{\nu}}_{\star})$ then the number of isolated
roots of $P^{\prime}(s;{{\nu}})=0$ with $s\in(0,\delta)$ is at most $\ell$
(counted without multiplicities).

    3. $(c)$

Case $\mathbf{s_{\star}\in(0,1)}$.

      1. $(c1)$

If the center of $X_{{{\nu}}_{\star}}$ is not isochronous then there exists
$k\in\mathbb{N}$, depending on $(s_{\star},{{\nu}}_{\star})$, such that
$\partial_{s}^{k}P(s_{\star};{{\nu}}_{\star})\neq 0.$ By continuity there is a
neighbourhood $V$ of $(s_{\star},{{\nu}}_{\star})$ such that
$\partial_{s}^{k}P(s;{{\nu}})\neq 0$ for all $(s,{{\nu}})\in V.$ Hence the
application of Rolle’s Theorem shows that there exist $\varepsilon,\delta>0$
such if ${{\nu}}\in B_{\varepsilon}({{\nu}}_{\star})$ then the number of roots
of $P^{\prime}(s;{{\nu}})=0$ with $s\in(s_{\star}-\delta,s_{\star}+\delta)$ is
at most $k$ (counted with multiplicities).

      2. $(c2)$

Let us suppose finally that the center of $X_{{{\nu}}_{\star}}$ is
isochronous. Since
$\big{(}((1-s)\xi_{{{\nu}}_{\star}},0),{{{\nu}}_{\star}}\big{)}\in\mathscr{U}$,
and by taking for instance the flow of the orthogonal vector field
$X_{{\nu}}^{\bot}$, there exists a transverse section
$\bar{s}\mapsto\sigma(\bar{s};{{\nu}})$ given by an analytic map

${\sigma}\\!:{(-\delta_{1},\delta_{1})\times
B_{\varepsilon_{1}}({{\nu}}_{\star})}\longrightarrow{\mathscr{U}}$

and such that
$\sigma(0;{{\nu}}_{\star})=\big{(}((1-s_{\star})\xi_{{{\nu}}_{\star}},0),{{{\nu}}_{\star}}\big{)}.$
We then define
$\bar{P}(\bar{s};{{\nu}})\\!:=\hat{P}(\sigma(\bar{s};{{\nu}}))$, which is
clearly analytic on $(-\delta_{1},\delta_{1})\times
B_{\varepsilon_{1}}({{\nu}}_{\star})$. We can thus compute its Taylor’s series
at $\bar{s}=0$,

$\bar{P}(\bar{s};{{\nu}})=\sum_{i=0}^{\infty}a_{i}({{\nu}})\bar{s}^{i},$

where each $a_{i}$ is an analytic function on
$B_{\varepsilon_{1}}({{\nu}}_{\star})$ with $a_{i}({{\nu}}_{\star})=0$.
Working in the local ring $\mathbb{R}\\{{{\nu}}\\}_{{{\nu}}_{\star}}$ of
convergent power series at ${{\nu}}_{\star}$, we consider the ideal
$\mathfrak{B}\\!:=\big{(}a_{i},i\in\mathbb{N}\big{)}$. The ring is Noetherian
and so there exists $\ell\in\mathbb{N}$ such that
$\mathfrak{B}=(a_{1},a_{2},\ldots,a_{\ell}).$ Verbatim the proof of Chicone
and Jacobs for [6, Theorem 2.2] (see also the result of Roussarie in [36,
§4.3.1] for a similar result for the displacement map), there exist analytic
functions $h_{i}(\bar{s};{{\nu}})$ in a neighbourhood of $(0,{{\nu}}_{\star})$
with $h_{i}(0;{{\nu}})\equiv 1$ for $i=1,2,\ldots,\ell$ such that we can write

$\bar{P}^{\prime}(\bar{s};{{\nu}})=\sum_{i=1}^{\ell}a_{i}({{\nu}})\bar{s}^{i-1}h_{i}(\bar{s};{{\nu}}).$

Now, setting
$\psi_{i}(\bar{s};{{\nu}})\\!:=\bar{s}^{i-1}h_{i}(\bar{s};{{\nu}})$ and
proceeding just like the proof of [6, Theorem 2.2], one can apply the well-
known derivation-division algorithm and use recursively Rolle’s Theorem to
show that there exist $\delta_{2},\varepsilon_{2}>0$ small enough such that if
${{\nu}}\in B_{\varepsilon_{2}}({{\nu}}_{\star})$ then the ordered set
$(\psi_{1},\psi_{2},\ldots,\psi_{\ell})$ is an extended complete Cheybshev
system for $\bar{s}\in(-\delta_{2},\delta_{2})$, see [13] for a definition.
Accordingly if ${{\nu}}\in B_{\varepsilon_{2}}({{\nu}}_{\star})$ then either
$\bar{P}^{\prime}(\,\cdot\,;{{\nu}})\equiv 0$ or
$P^{\prime}(\bar{s};{{\nu}})=0$ has at most $\ell-1$ roots with
$\bar{s}\in(-\delta_{2},\delta_{2})$ counted with multiplicities. Using the
original parametrization of the period function, this shows the existence of
$\delta_{3},\varepsilon_{3}>0$ small enough such that if ${{\nu}}\in
B_{\varepsilon_{3}}({{\nu}}_{\star})$ then the number of isolated roots of
$P^{\prime}(s;{{\nu}})=0$ with
$s\in(s_{\star}-\delta_{3},s_{\star}+\delta_{3})$ is at most $\ell-1$ taking
multiplicities into account.

Since in each one of the possible cases there is a neighbourhood of
$(s_{\star},{{\nu}}_{\star})$ where the number of critical periods is finite,
the result follows by taking a finite subcover of $[0,1]\times K.$

##  3 Asymptotic expansion of the period function

From now on we focus on the quadratic family
$\\{X_{{\nu}},{{\nu}}\in\mathbb{R}^{2}\\}$ given in $(\ref{sist_loud})$ and
study the period function of the center at the origin. In this section we give
its asymptotic expansion near the outer boundary $\Pi_{{\nu}}$ for parameters
$\nu$ inside three specific sets (see Figure 1):

$\displaystyle\Gamma_{1}=$
$\displaystyle\textstyle\big{\\{}D=-\frac{1}{2},F\in(\frac{1}{2},1)\big{\\}}\cup\big{\\{}F=\frac{1}{2},D\in(-1,0)\big{\\}},$
$\displaystyle\Gamma_{2}=$
$\displaystyle\textstyle\big{\\{}F=2,D\in(-2,0)\big{\\}}\cup\big{\\{}D=\mathcal{G}(F):F\in(1,\frac{3}{2})\big{\\}}$
and $\displaystyle\Gamma_{3}=$
$\displaystyle\big{\\{}F=1,D\in(-1,0)\big{\\}}.$

In all the cases the period annulus $\mathscr{P}_{{\nu}}$ is unbounded. Since
the vector field $X_{\nu}$ is polynomial, in order to study the behaviour of
the trajectories near infinity one can use its Poincaré compactification
$p(X_{\nu}),$ which is an analytic vector field on the sphere $\mathbb{S}^{2}$
topologically equivalent to $X_{\nu}$, see [3, §5] for details. The outer
boundary $\Pi_{\nu}$ is a polycycle of $p(X_{\nu})$ that can be studied using
local charts of $\mathbb{S}^{2}.$ In doing so one obtains (see [23, Figure 4])
the different phase portraits in the dehomogeneized Loud’s family
$\\{X_{\nu},\nu\in\mathbb{R}^{2}\\}$. For the parameter values studied in this
section it occurs that the polycycle $\Pi_{\nu}$ of $p(X_{\nu})$ is hyperbolic
if $\nu\in\Gamma_{1}\cup\Gamma_{2}$ and has a saddle-node singularity if
$\nu\in\Gamma_{3}.$ With regard to the phase portrait, it happens that the
affine part of $\Pi_{\nu}$ is a straight line for $\nu\in\Gamma_{1},$ whereas
it is a branch of a hyperbola for $\nu\in\Gamma_{2}.$ These are the reasons
why we split the parameters under consideration in these three subsets, which
are studied in the forthcoming subsections. Concerning the behaviour of the
period function near $\Pi_{\nu}$, the dichotomy between local regular value
and local bifurcation value (see Definition 2) is solved for any
${{\nu}}\in\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}$ thanks to the results in
[22, 23, 25, 38]. In these papers it is computed the asymptotic expansion of
the period function to second order, which usually suffices to tackle the
regular/bifurcation dichotomy. However in order to study the criticality we
need here to go further and compute the third, and even the fourth, order
expansion. Let us advance that the asymptotic expansions for
$\nu\in\Gamma_{1}$ are given in Proposition 3.2 and the ones for
$\nu\in\Gamma_{2}$ in Proposition 3.3. Being the proof of both results rather
long and technical, for the sake of paper’s readability we postpone them to
Appendix A, where we also summarize the fundamental results and definitions
from [29, 30, 31] that we shall use here. Among them we point out the notion
of $L$-flatness $\mathcal{F}_{L}^{\infty}({{\nu}}_{0})$, see Definition A.1,
used in the remainder, and the Écalle-Roussarie compensator
$\omega(s;\alpha),$ that is a deformation of the logarithm used in the
monomial scale in which the asymptotic expansion is given.

  * Definition 3.1.

The function defined for $s>0$ and $\alpha\in\mathbb{R}$ by means of

$\omega(s;\alpha)\\!:=\left\\{\begin{array}[]{ll}\frac{s^{-\alpha}-1}{\alpha}&\text{if
$\alpha\neq 0,$}\\\\[2.0pt] -\log s&\text{if $\alpha=0,$}\end{array}\right.$

is called the _Écalle-Roussarie compensator_. In the sequel we shall also use
the notation $\omega_{\alpha}(s)=\omega(s;\alpha)$. $\square$

The asymptotic expansion for $\nu\in\Gamma_{3}$ is given in Proposition 3.6
and its proof is of a different nature due to the occurrence of a saddle-node
bifurcation at the polycycle.

###  3.1 Study of $\\{D=-1/2,F\in(1/2,1)\\}$ and $\\{F=1/2,D\in(-1,0)\\}$

Figure 2 shows the phase portrait in the Poincaré disc of the vector field
$X_{{\nu}}$ in $(\ref{sist_loud})$ for ${{\nu}}$ varying inside

$V\\!:=\left\\{(D,F)\in\mathbb{R}^{2}:D\in(-1,0),\,F\in(0,1)\right\\}.$

We take transverse sections $\Sigma_{1}$ and $\Sigma_{2}$ parametrized by
$s\mapsto(1-s,0)$ and $s\mapsto(-1/s,0)$ with $s>0$, respectively, and

[l(0mm),r(0mm),t(0mm),b(0mm),]fig1(0.8)
[l]-5,28;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{2}}$
[l]32,28;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{1}}$

Figure 2: Phase portrait in the Poincaré disc of $X_{{\nu}}$ for ${{\nu}}\in
V$, where for convenience the center at $(0,0)$ is shifted to the left and the
vertical invariant line is $\\{x=1\\}$.

define $T(s;{{\nu}})$ to be the time that spends the solution of $X_{{\nu}}$
with initial condition at $(1-s,0)\in\Sigma_{1}$ to arrive at $\Sigma_{2}.$
Thanks to the symmetry of $X_{{\nu}}$ with respect to $\\{y=0\\}$, it turns
out that the period of the periodic orbit passing through
$(1-s,0)\in\Sigma_{1}$ is precisely $2T(s;{{\nu}})$. Consequently the
emergence/disappearance of critical periodic orbits from $\Pi_{{\nu}}$
corresponds to zeros of $T^{\prime}(s;{{\nu}})$ bifurcating from $s=0,$ more
concretely to the number
$\mathcal{Z}_{0}(T^{\prime}(\,\cdot\,;{{\nu}}),{{\nu}}_{\star})$ as introduced
in Definition 2. A key point to study these bifurcations is that
$T(s;{{\nu}})$ is the Dulac time associated to the passage through a
hyperbolic saddle, which is at infinity (see Figure 2 again). Therefore we can
apply [30, Theorem A] to obtain the asymptotic expansion of $T(s;{{\nu}})$ at
$s=0$ and use then [31, Theorem A] to compute its first coefficients
$T_{ij}({{\nu}})$. Next result gathers all this information, where
$\Gamma(\,\cdot\,)$ denotes the gamma function.

######  Proposition 3.2.

Let $T(s;{{\nu}})$ be the Dulac time of the passage from $\Sigma_{1}$ to
$\Sigma_{2}$ of the saddle at infinity of the vector field $X_{{\nu}}$ in
$(\ref{sist_loud})$ for ${{\nu}}\in V$. Then the coefficients $T_{00},$
$T_{01},$ $T_{10}$ and $T_{20}$ in its asymptotic expansion at $s=0$ are
meromorphic functions on $V$ that can be written as

$\begin{array}[]{lcl}T_{00}({{\nu}})=\frac{\pi}{2\sqrt{F(D+1)}},&&T_{01}({{\nu}})=\rho_{1}({{\nu}})\frac{\Gamma\big{(}-\frac{\lambda}{2}\big{)}}{\Gamma\big{(}\frac{1-\lambda}{2}\big{)}},\\\\[10.0pt]
T_{10}({{\nu}})\textstyle=\rho_{2}({{\nu}})(2D+1)\frac{\Gamma\big{(}1-\frac{1}{2\lambda}\big{)}}{\Gamma\big{(}\frac{3}{2}-\frac{1}{2\lambda}\big{)}},&&T_{20}({{\nu}})=\frac{\sqrt{\pi}}{\sqrt{2F}}\frac{\Gamma\big{(}\frac{1}{2}-\frac{1}{\lambda}\big{)}}{\Gamma\big{(}1-\frac{1}{\lambda}\big{)}}+\rho_{3}({{\nu}})(2D+1),\end{array}$

where $\lambda({{\nu}})=\frac{F}{1-F}$ is the hyperbolicity ratio of the
saddle,

$\textstyle\rho_{1}({{\nu}})=\frac{\sqrt{\pi}}{2(1-F)}\left(\frac{F}{D+1}\right)^{\frac{\lambda+1}{2}}\left(\frac{D}{F-1}\right)^{\frac{\lambda}{2}}\text{
and }\rho_{2}({{\nu}})=\frac{\sqrt{\pi}}{2\sqrt{F(D+1)^{3}}},$

and $\rho_{3}$ is an analytic function on
$V\cap\left\\{\frac{2}{3}<F<1\right\\}.$ In addition the following holds:

  1. $(a)$

If ${{\nu}}_{0}\in V\cap\left\\{\frac{2}{3}<F<1\right\\}$ then, for all
$\upsilon>0$ small enough,

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{10}({{\nu}})s+T_{20}({{\nu}})s^{2}+\mathcal{F}_{L_{0}-\upsilon}^{\infty}({{\nu}}_{0})$

with $L_{0}=\min\big{(}3,\lambda({{\nu}}_{0})\big{)}$. Moreover
$T_{10}(-\frac{1}{2},F)=0$ and $T_{20}(-\frac{1}{2},F)>0$ for all
$F\in(\frac{2}{3},1)$.

  2. $(b)$

If ${{\nu}}_{0}\in V\cap\left\\{\frac{1}{2}<F<\frac{2}{3}\right\\}$ then, for
all $\upsilon>0$ small enough,

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{10}({{\nu}})s+T_{01}({{\nu}})s^{\lambda}+\mathcal{F}^{\infty}_{2-\upsilon}({{\nu}}_{0}).$

Furthermore $T_{10}(-\frac{1}{2},F)=0$ and $T_{01}(-\frac{1}{2},F)>0$ for all
$F\in(\frac{1}{2},\frac{2}{3})$.

  3. $(c)$

If ${{\nu}}_{0}\in V\cap\left\\{F=\frac{2}{3}\right\\}$ then, for all
$\upsilon>0$ small enough,

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{10}({{\nu}})s+T_{201}^{2}({{\nu}})s^{2}\omega_{2-\lambda}(s)+T_{200}^{2}({{\nu}})s^{2}+\mathcal{F}_{3-\upsilon}^{\infty}({{\nu}}_{0}),$

where $T_{200}^{2}$ and $T_{201}^{2}$ are analytic functions in a
neighbourhood of $V\cap\\{F=\frac{2}{3}\\}$. Moreover
$T_{10}(-\frac{1}{2},\frac{2}{3})=0$ and
$T_{201}^{2}(-\frac{1}{2},\frac{2}{3})\neq 0$.

  4. $(d)$

If ${{\nu}}_{0}\in V\cap\left\\{F=\frac{1}{2}\right\\}$ then, for all
$\upsilon>0$ small enough,

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{101}^{1}({{\nu}})s\omega_{1-\lambda}(s)+T_{100}^{1}({{\nu}})s+\mathcal{F}_{2-\upsilon}^{\infty}({{\nu}}_{0}),$

where

$T_{101}^{1}({{\nu}})=-\rho_{4}({{\nu}})(F-1/2)^{2}\text{ and
}T_{100}^{1}({{\nu}})=\rho_{5}({{\nu}})(D+1/2)+\rho_{6}({{\nu}})(F-1/2)$

for some analytic positive functions $\rho_{i}$ in a neighbourhood of
$V\cap\\{F=\frac{1}{2}\\}$ with
$\rho_{5}(-\frac{1}{2},\frac{1}{2})=\rho_{6}(-\frac{1}{2},\frac{1}{2}).$

As we already explained, the proof of this result is postponed to Appendix A.
The monomial order in each one of these asymptotic expansions is with respect
to the strict partial order $\prec_{{{\nu}}_{0}}$ given in [29]. Let us recall
in its regard that we write $f\prec_{{{\nu}}_{0}}g$ in case that

$\lim_{(s,{{\nu}})\to(0,{{\nu}}_{0})}\frac{g(s;\lambda)}{f(s;\lambda)}=0.$

For the monomials under consideration this order is preserved after derivation
with respect to $s$, and so it is the good flatness properties of the
remainder. Thus, as it occurs with the Taylor’s series of an analytic
function, an upper bound for the number of zeros of $T^{\prime}(s;{{\nu}})$
that can bifurcate from $s=0$ follows by identifying the first non-vanishing
coefficient in the asymptotic expansion. For the proof and a precise statement
of this result, which essentially follows by using the well-known derivation-
division algorithm, the reader is referred to [29, Theorem C].

###  3.2 Study of $\\{F=2,D\in(-2,0)\\}$ and
$\\{D=\mathcal{G}(F):F\in(1,3/2)\\}$

Figure 3 shows the phase portrait in the Poincaré disc of the vector field
$X_{{\nu}}$ in $(\ref{sist_loud})$ for ${{\nu}}$ varying inside

[l(0mm),r(0mm),t(0mm),b(0mm)]dib3(1)
[l]-3,22;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{2}}$
[l]10,22;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{1}}$
[l]65,22;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{2}}$
[l]82,22;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{1}}$

Figure 3: Phase portrait in the Poincaré disc of $X_{{\nu}}$ for ${{\nu}}\in
W$ with $D<-1$ (left) and $D>-1$ (right), where the center at $(0,0)$ is
shifted to the left, the vertical invariant line is $\\{x=1\\}$ and the
hyperbola $\\{\frac{1}{2}y^{2}-q(x)=0\\}$ appears in boldface type.

$W\\!:=\\{(D,F)\in\mathbb{R}^{2}:F+D>0,\,D<0\text{ and }F>1\\}.$

In this case the outer boundary of the period annulus of the center at $(0,0)$
is contained in the union of the line at infinity and an invariant hyperbola
$\mathcal{C}\\!:=\\{\frac{1}{2}y^{2}-q(x)=0\\}$, where $q(x)=ax^{2}+bx+c$ with

$a\\!:=\frac{D}{2(1-F)},\mbox{ }b\\!:=\frac{D-F+1}{(1-F)(1-2F)}\mbox{ and
}c\\!:=\frac{F-D-1}{2F(1-F)(1-2F)}.$

One can verify that if ${{\nu}}\in W$ then $q$ has two distinct real zeros,
that we shall denote by $p_{1}$ and $p_{2}$ taking $p_{1}<p_{2}.$ That being
said, we place two transverse sections $\Sigma_{1}$ and $\Sigma_{2}$
parametrized by $s\mapsto(p_{1}-s,0)$ and $s\mapsto(-1/s,0)$ with $s>0$,
respectively, and define $T(s;{{\nu}})$ to be the time that takes to the
solution of $X_{{\nu}}$ with initial condition at $(p_{1}-s,0)\in\Sigma_{1}$
to arrive at $\Sigma_{2}.$ Then $T(s;{{\nu}})$ is the Dulac time associated to
the passage through a hyperbolic saddle at infinity, so that we can apply the
results in [30, 31] to obtain its asymptotic expansion at $s=0.$ This is
important for the proof of Theorem A because, exactly as in the previous case,
the symmetry of $X_{{\nu}}$ with respect to $\\{y=0\\}$ implies that the
period of the periodic orbit passing through $(1-s,0)\in\Sigma_{1}$ is
$2T(s;{{\nu}})$. With regard to our next result we remark that
$\frac{1-p_{2}}{1-p_{1}}<1$ for all ${{\nu}}\in W$, which is relevant since
the hypergeometric function ${}_{2}F_{1}(a,b;c;\,\cdot\,)$ is holomorphic on
$\mathbb{C}\setminus[1,+\infty)$, see Appendix C. Let us also mention that
$\mathrm{B}(\cdot,\cdot)$ is the beta function.

######  Proposition 3.3.

Let $T(s;{{\nu}})$ be the Dulac time of the passage from $\Sigma_{1}$ to
$\Sigma_{2}$ of the saddle at infinity of the vector field $X_{{\nu}}$ in
$(\ref{sist_loud})$ for ${{\nu}}\in W$. Then the coefficients $T_{00},$
$T_{01},$ $T_{10}$ and $T_{20}$ in its asymptotic expansion at $s=0$ are
meromorphic functions on $W$ that can be written as

$\displaystyle T_{00}({{\nu}})$
$\displaystyle\textstyle=\frac{\sqrt{2}}{\sqrt{a}(1-p_{1})}\,{}_{2}F_{1}\big{(}1,\frac{1}{2};\frac{3}{2};\frac{1-p_{2}}{1-p_{1}}\big{)},$
$\displaystyle T_{01}({{\nu}})$
$\displaystyle\textstyle=\rho_{1}({{\nu}})\mathrm{B}\big{(}-\lambda,\frac{1}{2}\big{)},$
$\displaystyle T_{10}({{\nu}})$
$\displaystyle\textstyle=\rho_{2}({{\nu}})\mathrm{B}\big{(}1-\frac{1}{\lambda},-\frac{1}{2}\big{)}{}_{2}F_{1}\big{(}-1-\frac{1}{\lambda},-\frac{1}{2};\frac{1}{2}-\frac{1}{\lambda};\frac{1-p_{2}}{1-p_{1}}\big{)}$
and $\displaystyle T_{20}({{\nu}})$
$\displaystyle\textstyle=\rho_{3}({{\nu}})\mathrm{B}\big{(}1-\frac{2}{\lambda},-\frac{3}{2}\big{)}{}_{2}F_{1}\big{(}-\frac{2}{\lambda}-3,-\frac{3}{2};-\frac{1}{2}-\frac{2}{\lambda};\frac{1-p_{2}}{1-p_{1}}\big{)}+\rho_{4}({{\nu}})T_{10}({{\nu}}),$

where $\lambda({{\nu}})=\frac{1}{2(F-1)}$ is the hyperbolicity ratio of the
saddle and, for $i=1,2,3,4$, $\rho_{i}$ is an analytic positive function on
$W$. In addition the following holds:

  1. $(a)$

If ${{\nu}}_{0}\in W\cap\left\\{1<F<\frac{5}{4}\right\\}$ then, for all
$\upsilon>0$ small enough,

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{10}({{\nu}})s+T_{20}({{\nu}})s^{2}+\mathcal{F}_{L_{0}-\upsilon}^{\infty}({{\nu}}_{0})$

with $L_{0}=\min(3,\lambda({{\nu}}_{0}))$. Moreover $T_{20}({{\nu}})\neq 0$
for all ${{\nu}}\in W\cap\left\\{1<F<\frac{5}{4}\right\\}$ such that
$T_{10}({{\nu}})=0.$

  2. $(b)$

If ${{\nu}}_{0}\in W\cap\left\\{\frac{5}{4}<F<\frac{3}{2}\right\\}$ then, for
all $\upsilon>0$ small enough,

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{10}({{\nu}})s+T_{01}({{\nu}})s^{\lambda}+T_{20}({{\nu}})s^{2}+\mathcal{F}_{L_{0}-\upsilon}^{\infty}({{\nu}}_{0})$

with $L_{0}=\lambda({{\nu}}_{0})+1,$ and there exists a unique
${{\nu}}_{\star}\in W\cap\left\\{\frac{5}{4}<F<\frac{3}{2}\right\\}$ such that
$T_{10}({{\nu}}_{\star})=0$ and $T_{01}({{\nu}}_{\star})=0$. Furthermore
$T_{20}({{\nu}}_{\star})<0$, the gradients of $T_{01}$ and $T_{10}$ at
${{\nu}}_{\star}$ are linearly independent, and
${{\nu}}_{\star}=(D_{\star},\frac{4}{3})$ with
$D_{\star}=\mathcal{G}(\frac{4}{3})\approx-1.128$.

  3. $(c)$

If ${{\nu}}_{0}\in W\cap\left\\{F=\frac{5}{4}\right\\}$ then, for all
$\upsilon>0$ small enough,

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{10}({{\nu}})s+T_{201}^{2}({{\nu}})s^{2}\omega_{2-\lambda}(s)+T^{2}_{200}({{\nu}})s^{2}+\mathcal{F}_{3-\upsilon}^{\infty}({{\nu}}_{0}),$

where $T_{200}^{2}$ and $T_{201}^{2}$ are analytic functions in a
neighbourhood of $W\cap\\{F=\frac{5}{4}\\}$. Moreover
$T_{10}(D,\frac{5}{4})=0$ if and only if $D=-1$, and
$T_{201}^{2}(-1,\frac{5}{4})\neq 0$.

  4. $(d)$

If ${{\nu}}_{0}\in W\cap\left\\{F=2\right\\}$ then, for all $\upsilon>0$ small
enough,

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{01}({{\nu}})s^{\lambda}+T^{\frac{1}{2}}_{101}({{\nu}})s\omega_{1-2\lambda}(s)+T_{100}^{\frac{1}{2}}({{\nu}})s+\mathcal{F}_{3/2-\upsilon}^{\infty}({{\nu}}_{0}),$

where $T_{100}^{\frac{1}{2}}$ and $T_{101}^{\frac{1}{2}}$ are analytic
functions in a neighbourhood of $W\cap\\{F=2\\}$. Moreover $T_{01}(D,2)=0$ for
all $D\in(-2,0)$, $T_{101}^{\frac{1}{2}}(D,2)=0$ if and only if
$D=-\frac{1}{2}$, and the gradients of $T_{01}$ and $T_{101}^{\frac{1}{2}}$
are linearly independent at $(-\frac{1}{2},2)$.

The proof of this result is postponed to Appendix A.

  * Remark 3.4.

The asymptotic expansions in Proposition 3.3 were already given in [23,
Theorem 3.6] but only to second order. In that result it is given, among
others, the expression of the coefficient $T_{10}({{\nu}})$ in terms of a
definite improper integral. Furthermore, see [23, Proposition 3.11], it is
proved by applying the Implicit Function Theorem that the set of those
${{\nu}}\in W_{1}\\!:=W\cap\\{F<3/2\\}$ such that $T_{10}({{\nu}})=0$ is the
graphic of an analytic function $D=\mathcal{G}(F).$ This is the function that
appears in assertion $(b)$ of Proposition 3.3. Thanks to the results in
Appendix B we can now identify the improper integral as a hypergeometric
function, so that we can write

$\textstyle\big{\\{}{{\nu}}\in
W_{1}:D=\mathcal{G}(F)\big{\\}}=\big{\\{}{{\nu}}\in
W_{1}:{}_{2}F_{1}\big{(}-1-\frac{1}{\lambda},-\frac{1}{2};\frac{1}{2}-\frac{1}{\lambda};\frac{1-p_{2}}{1-p_{1}}\big{)}=0\big{\\}},$

where $p_{1}$ and $p_{2}$ with $p_{1}<p_{2}$ are the real roots of $q(x)=0$
and $\lambda({{\nu}})=\frac{1}{2(F-1)}$. $\square$

  * Remark 3.5.

In the statement of Proposition 3.3 we refer to some positive functions
$\rho_{i}\in\mathscr{C}^{\omega}(W).$ Let us mention that in the proof we show
that

$\begin{array}[]{lcl}\rho_{1}({{\nu}})=\frac{1}{2\sqrt{2a}}\frac{(p_{2}-p_{1})^{\frac{1}{2(F-1)}}}{(F-1)(1-p_{1})^{\frac{F}{F-1}}}&&\rho_{2}({{\nu}})=\frac{1}{2\sqrt{2a}}\frac{1}{(p_{2}-p_{1})(1-p_{1})}\\\\[15.0pt]
\rho_{3}({{\nu}})=\frac{3}{8\sqrt{2a}}\frac{1}{(p_{2}-p_{1})^{2}(1-p_{1})}&&\rho_{4}({{\nu}})=\frac{p_{1}-1+2F(p_{2}-p_{1})}{(p_{2}-p_{1})(p_{1}-1)}\end{array}$

We do not use the explicit expressions in this paper but they may be relevant
for future applications. $\square$

###  3.3 Study of $\\{F=1,D\in(-1,0)\\}$

Our aim in this section is to study the period function of the center at the
origin of $X_{{\nu}}$ for ${{\nu}}=(D,F)$ with $F\approx 1$ and $D\in(-1,0)$.

[l(0mm),r(0mm),t(0mm),b(0mm)]dib4(1)
[l]-4,22;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{2}}$
[l]4,44;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{2}^{n}}$
[l]23,37;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{1}^{n}}$
[l]68,41;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{2}^{n}}$
[l]86,35;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{1}^{n}}$
[l]17,22;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{1}}$
[l]63,22;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{2}}$
[l]82,22;${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\Sigma_{1}}$

Figure 4: Phase portrait of $X_{{\nu}}$ for
${{\nu}}=(D,F)\in(-1,0)\times(0,+\infty)$ in the Poincaré disc with
$F\leqslant 1$ (left) and $F>1$ (right). In this case, contrary to the
previous ones, the singularity at infinity for $F=1$ is not a hyperbolic
saddle but a saddle-node.

To this end we introduce transverse sections $\Sigma_{1}$ and $\Sigma_{2}$
parametrized, respectively, by means of

$\sigma_{1}(s;{{\nu}})\\!:=\left\\{\begin{array}[]{cl}(1-s,0)&\text{ if
$F\leqslant 1,$}\\\\[2.0pt] (p_{1}-s,0)&\text{ if
$F>1,$}\end{array}\right.\text{ and }\sigma_{2}(s;{{\nu}})\\!:=(-1/s,0),$

where recall that $q(x)=a(x-p_{1})(x-p_{2})$ with $p_{1}<p_{2}$ for $F>1$. One
can also check that $\lim_{F\to 1^{+}}p_{1}=1.$ For each
${{\nu}}=(D,F)\in(-1,0)\times(0,+\infty)$ we define $T(s;{{\nu}})$ as the time
that spends the solution of $X_{{\nu}}$ starting at
$\sigma_{1}(s;{{\nu}})\in\Sigma_{1}$ to arrive at $\Sigma_{2}.$ A key feature
of this Dulac time is that the singularity for $F=1$ is not a hyperbolic
saddle but a saddle-node. Our next result gives the asymptotic expansion of
$T(s;{{\nu}})$ at $s=0$ for $F\approx 1$. We point out that this is relevant
for the proof of Theorem A because the period of the periodic orbit of
$X_{{\nu}}$ passing through $\sigma_{1}(s;{{\nu}})\in\Sigma_{1}$ is precisely
$2T(s;{{\nu}})$ due to the symmetry of the vector field.

######  Proposition 3.6.

Let $T(s;{{\nu}})$ be the Dulac time of the passage from $\Sigma_{1}$ to
$\Sigma_{2}$ of the saddle-node unfolding at infinity $\\{X_{{\nu}}\\}.$ Then
there is an open neighbourhood $\mathscr{U}$ of $(-1,0)\times\\{1\\}$ such
that

$T(s;{{\nu}})=T_{0}({{\nu}})+T_{1}({{\nu}})s+T_{2}({{\nu}})s^{2}+s^{2}h(s;{{\nu}}),$

where $T_{0},T_{1},T_{2}\in\mathscr{C}^{0}(\mathscr{U})$ and, setting
$\Theta=s\partial_{s}$, $\lim_{s\to 0^{+}}\Theta^{k}h(s;{{\nu}})=0$ uniformly
on compact sets of $\mathscr{U}$ for $k=0,1,2$. Moreover $T_{1}(D,1)=0$ if,
and only if, $D=-\frac{1}{2}$. Finally $T_{2}(-1/2,1)\neq 0.$

  * Proof.

To study the saddle-node bifurcation that occurs at infinity we work in the
projective plane $\mathbb{RP}^{2}$ and perform the change of coordinates

$(u,v)=p(x,y)\\!:=\left(\frac{1-x}{y},\frac{1}{y}\right).$

The meromorphic extension of $X_{{\nu}}$ in these coordinates is given by

$\tilde{X}_{{\nu}}\\!:=p_{*}X_{{\nu}}=\frac{1}{v}\Big{(}uP(u,v;{{\nu}})\partial_{u}+vQ(u,v;{{\nu}})\partial_{v}\Big{)}$

with

$\displaystyle P(u,v;{{\nu}})$ $\displaystyle=1-F-Du^{2}+(2D+1)uv-(D+1)v^{2}$
and $\displaystyle Q(u,v;{{\nu}})$
$\displaystyle=-F-Du^{2}+(2D+1)uv-(D+1)v^{2}.$

Our first goal is to show that we can bring locally the saddle-node unfolding
to a convenient normal form in order that we can apply the tools developed in
[25] to study the asymptotic expansion of its Dulac map and Dulac time. With
this aim, some long but easy computations show that the local analytic change
of coordinates given by

$(z,w)=\Psi(u,v)\\!:=\left(\frac{u}{\sqrt{g(u,v)}},\frac{v}{\sqrt{g(u,v)}}\right),$

where
$g(u,v)\\!:={\frac{\left(2D+1\right)}{\left(2F-1\right)D}}uv-{\frac{\left(D+1\right)}{2FD}}{v}^{2}-\frac{1}{2D},$
brings the vector field $\tilde{X}_{{\nu}}$ to

$\bar{X}_{{\nu}}\\!:=\frac{1}{w\bar{U}(z,w;{{\nu}})}\bigl{(}z(z^{2}-2(F-1))\partial_{z}-w(2F-z^{2})\partial_{w}\bigr{)},$

with
$\bar{U}(z,w;{{\nu}})\\!:=\left({\frac{\left(2D+1\right)}{2(2F-1)}}zw-{\frac{\left(D+1\right)}{4F}}{w}^{2}-\frac{D}{2}\right)^{-\frac{1}{2}}$.
A technical assumption in order to apply the results from [25] is that for
each ${{\nu}}$ the Taylor’s series of $(z,w)\mapsto\bar{U}(z,w;{{\nu}})$ at
$(0,0)$ is absolutely convergent for all $(z,w)\in[-1,1]^{2}$. This is not
fulfilled unless we perform a rescaling which is only well defined provided
that ${{\nu}}$ varies inside a compact subset of $(-1,0)\times(0,+\infty)$ and
this forces us to work locally. For this reason, as a first step in the proof,
we will show a local version of the statement. More concretely, that for each
${{\nu}}_{\star}=(D_{\star},1)$ with $D_{\star}\in(-1,0)$ there exists an open
ball
$B_{{{\nu}}_{\star}}=\\{{{\nu}}\in\mathbb{R}^{2}:\|{{\nu}}-{{\nu}}_{\star}\|<\delta\\}$
such that

$T(s;{{\nu}})=T_{0}^{{{\nu}}_{\star}}({{\nu}})+T_{1}^{{{\nu}}_{\star}}({{\nu}})s+T_{2}^{{{\nu}}_{\star}}({{\nu}})s^{2}+s^{2}h^{{{\nu}}_{\star}}(s;{{\nu}}),$

with $T_{i}^{{{\nu}}_{\star}}$ continuous functions on $B_{{{\nu}}_{\star}}$
and $\lim_{s\to 0^{+}}\Theta^{k}h^{{{\nu}}_{\star}}(s;\nu)=0$ uniformly on
$B_{{{\nu}}_{\star}}$ for $k=0,1,2$. To begin with we take $\delta>0$ small
enough so that the closure of $B_{{{\nu}}_{\star}}$ is inside
$(-1,0)\times(\frac{1}{2},+\infty)$ and define

$r\\!:=\inf\left\\{\left|\frac{(D+1)}{DF}w^{2}-\frac{2(2D+1)}{D(2F-1)}zw\right|^{-1/2}:|z|\leqslant
1,|w|\leqslant 1\text{ and
}\|{{\nu}}-{{\nu}}_{\star}\|\leqslant\delta\right\\},$ (6)

which is strictly positive. The pull-back of $\bar{X}_{{\nu}}$ by the
rescaling $\rho(z,w)\\!:=(rz,rw)$ can now be written as in [25, Eq. 13]
because one can easily verify that

$\rho^{*}\bar{X}_{{\nu}}=\frac{1}{wU(z,w;{{\nu}})}\left(z(z^{2}-\varepsilon)\partial_{z}-w(2F/r^{2}-z^{2})\partial_{w}\right)\text{
with $\varepsilon\\!:=2(F-1)/r^{2}$}$

and where the Taylor’s series of

$U(z,w;{{\nu}})\\!:=\frac{\bar{U}(rz,rw;{{\nu}})}{r}=\frac{-D}{2r}\left(1+r^{2}\left(\frac{(D+1)}{2FD}w^{2}-\frac{(2D+1)}{D(2F-1)}zw\right)\right)^{-\frac{1}{2}}$

at $(z,w)=(0,0)$ is absolutely convergent for all $(z,w)\in[-1,1]^{2}$ and
${{\nu}}\in B_{{{\nu}}_{\star}}$ since, on account of $(\ref{p3eq7})$,

$r^{2}\left|\frac{(D+1)}{2FD}w^{2}-\frac{(2D+1)}{D(2F-1)}zw\right|\leqslant\frac{1}{2}\text{
for all $(z,w,{{\nu}})\in[-1,1]^{2}\times B_{{{\nu}}_{\star}}.$}$

In these new rescaled coordinates, that we still denote by $(z,w)$ for
simplicity, the period annulus is inside the quadrant $\\{w\geqslant
0,z\geqslant\vartheta_{\varepsilon}\\}$ where

$\vartheta_{\varepsilon}\\!:=\left\\{\begin{array}[]{cl}\sqrt{\varepsilon}&\text{
if $\varepsilon\geqslant 0$},\\\ 0&\text{ if
$\varepsilon<0$}.\end{array}\right.$

Setting $\Psi_{\star}\\!:=\rho^{-1}\circ\Psi,$ we take two auxiliary
transverse sections $\Sigma_{1}^{n}\\!:=\Psi_{\star}^{-1}(\\{w=1\\})$ and
$\Sigma_{2}^{n}\\!:=\Psi_{\star}^{-1}(\\{z=1\\})$ parameterized by
$\sigma_{1}^{n}(s;{{\nu}})\\!:=\Psi_{\star}^{-1}(s+\vartheta_{\varepsilon},1)$
and $\sigma_{2}^{n}(s;{{\nu}})\\!:=\Psi_{\star}^{-1}(1,s)$, respectively (see
Figure 4). We define $\mathcal{T}(s;{{\nu}})$ and $\mathscr{D}(s;{{\nu}})$ to
be the Dulac time and Dulac map of $\tilde{X}_{{\nu}}$ from $\Sigma_{1}^{n}$
to $\Sigma_{2}^{n},$ respectively. We remark that, by construction,
$\mathcal{T}(s;{{\nu}})$ is the time that the solution of
$\rho^{*}\bar{X}_{{\nu}}$ starting at the point
$(s+\vartheta_{\varepsilon},1)$ spends to arrive at $\\{z=1\\}$ and that the
intersection point is precisely $(z,w)=(1,\mathscr{D}(s;{{\nu}}))$. In this
regard, since $w=\mathscr{D}(z;{{\nu}})$ is a trajectory of the vector field
$z(z^{2}-\varepsilon)\partial_{z}+w(2F/r^{2}-z^{2})\partial_{w}$, see [25, p.
6417] for details, the application of $(b)$ in Corollary A of [25] with
$\\{\mu=2,\ell=k=2,\lambda=2F/r^{2}\\}$ shows that

$\mathscr{D}(s;{{\nu}})=s^{2}\mathcal{I}(B_{{{\nu}}_{\star}}),$ (7)

by shrinking $\delta>0$ if necessary. Here, and in what follows,
$\mathcal{I}(B_{{{\nu}}_{\star}})$ stands for some function $h(s;{{\nu}})$
verifying that $\lim_{s\to 0^{+}}\Theta^{k}h(s;{{\nu}})=0$ uniformly on
${{\nu}}\in B_{{{\nu}}_{\star}}$ for $k=0,1,2$. Furthermore, by applying
Corollary B in the same paper with $\\{\mu=2,\ell=k=2\\}$ and shrinking
$\delta>0$ again we can assert that

$\mathcal{T}(s;{{\nu}})=b_{0}({{\nu}})+b_{1}({{\nu}})s+b_{2}({{\nu}})s^{2}+s^{2}\mathcal{I}(B_{{{\nu}}_{\star}})$
(8)

with $b_{i}\in\mathscr{C}^{0}(B_{{{\nu}}_{\star}})$ for $i=0,1,2.$ Working in
the original $(x,y)$ coordinates, we consider next the transition times
$T_{1}(\,\cdot\,;{{\nu}})$ and $T_{2}(\,\cdot\,;{{\nu}})$ of $X_{{\nu}}$ from
$\Sigma_{1}$ to $p^{-1}(\Sigma_{1}^{n})$ and from $p^{-1}(\Sigma_{2}^{n})$ to
$\Sigma_{2}$, respectively. We define moreover $R(\,\cdot\,;{{\nu}})$ to be
the transition map from $\Sigma_{1}$ to $p^{-1}(\Sigma_{1}^{n})$. Accordingly

$T(s)=T_{1}(s)+\big{(}\mathcal{T}\circ
R\big{)}(s)+\big{(}T_{2}\circ\mathscr{D}\circ R\big{)}(s),$ (9)

where we omit the dependence on ${{\nu}}$ for the sake of shortness. By [22,
Lemma 3.2], we have that $T_{2}(s;{{\nu}})$ an analytic function at
$\\{0\\}\times B_{{{\nu}}_{\star}}$ with $T_{2}(0;{{\nu}})=0.$ Observe at this
point that, setting

$\xi_{{\nu}}\\!:=\left\\{\begin{array}[]{cl}0&\text{ if $F\leqslant 1,$}\\\
1-p_{1}&\text{ if $F>1,$}\end{array}\right.$

we can write the parametrization of $\Sigma_{1}$ as
$\sigma_{1}(s;{{\nu}})=(1-\xi_{{\nu}}-s,0).$ We claim that there exist two
functions $f(\hat{s};{{\nu}})$ and $g(\hat{s};{{\nu}})$, analytic at
$\\{0\\}\times B_{{{\nu}}_{\star}},$ such that

$T_{1}(s;{{\nu}})=f(s+\xi_{{\nu}};{{\nu}})\text{ and
}R(s;{{\nu}})=g(s+\xi_{{\nu}};{{\nu}})-\vartheta_{\varepsilon}.$ (10)

To show this let us consider two additional transverse sections
$\hat{\Sigma}_{1}$ and $\hat{\Sigma}_{1}^{n}$ parameterized respectively by
$\hat{\sigma}_{1}(\hat{s};{{\nu}})\\!:=(1-\hat{s},0)$ and
$\hat{\sigma}_{1}^{n}(\hat{s};{{\nu}})\\!:=(\Psi_{\star}\circ
p)^{-1}(\hat{s},1),$ which clearly are analytic at $\\{0\\}\times
B_{{{\nu}}_{\star}}$. Moreover it is clear that they are related with
$\Sigma_{1}$ and $\Sigma_{1}^{n}$ through
$\sigma_{1}(s;{{\nu}})=\hat{\sigma}_{1}(s+\xi_{{\nu}};{{\nu}})$ and
$\sigma_{1}^{n}(s;{{\nu}})=\hat{\sigma}_{1}^{n}(s+\vartheta_{\varepsilon};{{\nu}}).$
That being said, the claim follows by noting that if we choose
$f(\hat{s};{{\nu}})$ and $g(\hat{s};{{\nu}})$ to be, respectively, the
transition time and transition map of $X_{{\nu}}$ from $\hat{\Sigma}_{1}$ to
$\hat{\Sigma}_{1}^{n}$, which are clearly analytic at $\\{0\\}\times
B_{{{\nu}}_{\star}},$ then the equalities in $(\ref{p3eq1})$ hold. Note
moreover that $g(\xi_{{\nu}};{{\nu}})=\vartheta_{\varepsilon}$ since
$R(0;{{\nu}})=0.$ On account of the claim, by considering the second order
Taylor’s development of $f(\hat{s};{{\nu}})$ and $g(\hat{s};{{\nu}})$ at
$\hat{s}=\xi_{{\nu}}$, respectively, we get

$T_{1}(s;{{\nu}})=a_{0}({{\nu}})+a_{1}({{\nu}})s+a_{2}({{\nu}})s^{2}+s^{2}\mathcal{I}(B_{{{\nu}}_{\star}})\text{
and
}R(s;{{\nu}})=c_{1}({{\nu}})s+c_{2}({{\nu}})s^{2}+s^{2}\mathcal{I}(B_{{{\nu}}_{\star}})$

with $a_{i},c_{i}\in\mathscr{C}^{0}(B_{{{\nu}}_{\star}})$ and where we also
use that ${{\nu}}\mapsto\xi_{{\nu}}$ is a continuous function. The combination
of the second expression above with $(\ref{p3eq2})$ and $(\ref{p3eq3})$ yields

$\big{(}\mathscr{D}\circ
R\big{)}(s)=s^{2}\mathcal{I}(B_{{{\nu}}_{\star}})\text{ and
}\big{(}\mathcal{T}\circ
R\big{)}(s)=\hat{b}_{0}({{\nu}})+\hat{b}_{1}({{\nu}})s+\hat{b}_{2}({{\nu}})s^{2}+s^{2}\mathcal{I}(B_{{{\nu}}_{\star}}),$

respectively, with $\hat{b}_{i}\in\mathscr{C}^{0}(B_{{{\nu}}_{\star}}).$
Summing up, since $\big{(}T_{2}\circ\mathscr{D}\circ
R\big{)}(s)=s^{2}\mathcal{I}(B_{{{\nu}}_{\star}})$ due to
$T_{2}(0;{{\nu}})=0,$ from $(\ref{p3eq4})$ we can assert that

$T(s;{{\nu}})=T_{0}^{{{\nu}}_{\star}}({{\nu}})+T_{1}^{{{\nu}}_{\star}}({{\nu}})s+T_{2}^{{{\nu}}_{\star}}({{\nu}})s^{2}+s^{2}h^{{{\nu}}_{\star}}(s;{{\nu}})$
(11)

for some functions $T^{{{\nu}}_{\star}}_{i}$ that are continuous on
$B_{{{\nu}}_{\star}}$ and some
$h^{{{\nu}}_{\star}}\in\mathcal{I}(B_{{{\nu}}_{\star}})$. This concludes the
proof of the local version of the statement, in which we remark that the
coefficients $T_{i}^{{{\nu}}_{\star}}({{\nu}})$ and the remainder
$s^{2}h^{{{\nu}}_{\star}}(s;{{\nu}})$ depend by construction on
${{\nu}}_{\star}.$ Our next step will be to globalize them and to this end we
define

$\mathscr{U}\\!:=\bigcup_{{{\nu}}_{\star}\in(-1,0)\times\\{1\\}}B_{{{\nu}}_{\star}}$

which is clearly an open neighbourhood of $(-1,0)\times\\{1\\}$. Let us
consider now any ${{\nu}}_{1},{{\nu}}_{2}\in(-1,0)\times\\{1\\}$ such that
$B_{{{\nu}}_{1}}\cap B_{{{\nu}}_{2}}\neq 0.$ Then, from $(\ref{p3eq5})$, we
get that

$T_{0}^{{{\nu}}_{1}}({{\nu}})-T_{0}^{{{\nu}}_{2}}({{\nu}})+\big{(}T_{1}^{{{\nu}}_{1}}({{\nu}})-T_{1}^{{{\nu}}_{2}}({{\nu}})\big{)}s+\big{(}T_{2}^{{{\nu}}_{1}}({{\nu}})-T_{2}^{{{\nu}}_{2}}({{\nu}})\big{)}s^{2}+s^{2}\big{(}h^{{{\nu}}_{1}}(s;{{\nu}})-h^{{{\nu}}_{2}}(s;{{\nu}})\big{)}=0$

for all $s>0$ small enough and ${{\nu}}\in B_{{{\nu}}_{1}}\cap
B_{{{\nu}}_{2}}$. Since
$h^{{{\nu}}_{1}}-h^{{{\nu}}_{2}}\in\mathcal{I}(B_{{{\nu}}_{1}}\cap
B_{{{\nu}}_{2}}),$ taking the limit $s\to 0^{+}$ on both sides of the above
equality we deduce that $T_{0}^{{{\nu}}_{1}}=T_{0}^{{{\nu}}_{2}}$ on
$B_{{{\nu}}_{1}}\cap B_{{{\nu}}_{2}}$. Similarly, but taking the first and
second derivatives with respect to $s$, respectively, we get that
$T_{1}^{{{\nu}}_{1}}=T_{1}^{{{\nu}}_{2}}$ and
$T_{2}^{{{\nu}}_{1}}=T_{2}^{{{\nu}}_{2}}$ on $B_{{{\nu}}_{1}}\cap
B_{{{\nu}}_{2}}$. Hence, for $i=0,1,2,$ the local functions
$T_{i}^{{{\nu}}_{\star}}\in\mathscr{C}^{0}(B_{{{\nu}}_{\star}})$ for
${{\nu}}_{\star}\in(-1,0)\times\\{1\\}$ glue together into a well defined
continuous function $T_{i}$ on $\mathscr{U}$. Exactly the same argument shows
that the local functions
$h^{{{\nu}}_{\star}}\in\mathcal{I}(B_{{{\nu}}_{\star}})$ for
${{\nu}}_{\star}\in(-1,0)\times\\{1\\}$ glue together into a well defined
function $h(s;{{\nu}})$ satisfying that $\lim_{s\to
0^{+}}\Theta^{k}h(s;\nu)=0$ uniformly on compact sets of $\mathscr{U}$ for
$k=0,1,2$. To show this last assertion it suffices to take a finite subcover
$B_{{{\nu}}_{1}}\cup\ldots\cup B_{{{\nu}}_{n}}$ of the given compact subset of
$\mathscr{U}$ and use that
$h|_{B_{{{\nu}}_{i}}}\in\mathcal{I}(B_{{{\nu}}_{i}})$ for $i=1,2,\ldots,n.$

So far we have proved the first assertion in the statement. Let us turn to the
proof of the second one. To this end the key point is that for those
${{\nu}}_{0}\in\mathscr{U}\cap\\{F<1\\}$ we can also apply $(a)$ in
Proposition 3.2 to obtain that

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{10}({{\nu}})s+T_{20}({{\nu}})s^{2}+\mathcal{F}_{3-\upsilon}^{\infty}({{\nu}}_{0}),$
(12)

where, setting $\lambda({{\nu}})=\frac{F}{1-F}$,

$\textstyle
T_{10}({{\nu}})=\frac{\sqrt{\pi}(2D+1)}{2\sqrt{F(1+D)^{3}}}\frac{\Gamma\left(1-\frac{1}{2\lambda}\right)}{\Gamma\left(\frac{3}{2}-\frac{1}{2\lambda}\right)}\text{
and
}T_{20}(-\frac{1}{2},F)=\frac{\sqrt{\pi}}{\sqrt{2F}}\frac{\Gamma\big{(}\frac{1}{2}-\frac{1}{\lambda}\big{)}}{\Gamma\big{(}1-\frac{1}{\lambda}\big{)}}.$

Hence, since $T_{i}\in\mathscr{C}^{0}(\mathscr{U})$, from $(\ref{p3eq5})$ and
$(\ref{p3eq6})$ we can assert that

$T_{1}(D,1)=\lim_{F\to 1^{-}}T_{10}(D,F)=\frac{2D+1}{(1+D)^{3/2}},$

where we also use that $\lim_{F\to
1^{-}}\frac{\Gamma\left(1-\frac{1}{2\lambda}\right)}{\Gamma\left(\frac{3}{2}-\frac{1}{2\lambda}\right)}=\frac{\Gamma(1)}{\Gamma(\frac{3}{2})}=\frac{2}{\sqrt{\pi}}$.
Consequently, as desired, $T_{1}(D,1)=0$ if and only if $D=-\frac{1}{2}$. The
same argument shows that

$T_{2}(-1/2,1)=\lim_{F\to
1^{-}}T_{20}(-1/2,F)=\frac{\sqrt{\pi}}{\sqrt{2}}\frac{\Gamma\big{(}\frac{1}{2}\big{)}}{\Gamma\big{(}1\big{)}}\neq
0,$

and this completes the proof of the result.

##  4 Distinguished cases

This section is devoted to study three specific parameters. Recall that among
the quadratic centers there are four nonlinear isochrones, see
$(\ref{isochronous})$. Chicone and Jacobs show in [6, Theorem 3.1] that the
criticality of the period function at the inner boundary (i.e., the center) of
$\mathscr{P}$ is exactly 1 for each one of the nonlinear isochrones. In this
section we prove that for two of them, namely $\nu=(-\frac{1}{2},2)$ and
$\nu=(-\frac{1}{2},\frac{1}{2}),$ the criticality at the outer boundary (i.e.,
the polycycle) is also 1, see Propositions 4.2 and 4.3, respectively. In the
same vein it is also well-known that the criticality at the inner boundary of
any quadratic center is at most two, see [6, Theorem 3.2]. This maximum
criticality is achieved in three parameter values, the so-called Loud points,
which following the notation in [6] are given by $\nu=L_{i}$ with

$\textstyle
L_{1}\\!:=\left(-\frac{3}{2},\frac{5}{2}\right),\;L_{2}\\!:=\left(\frac{-11+\sqrt{105}}{20},\frac{15-\sqrt{105}}{20}\right)\text{
and
}L_{3}\\!:=\left(\frac{-11-\sqrt{105}}{20},\frac{15+\sqrt{105}}{20}\right).$
(13)

As we already explained in the introduction, we conjecture that the
criticality at the outer boundary of any quadratic center is at most two, and
that there are only three parameter values where this maximum criticality is
attained. In this paper we identify and prove the validity of the conjecture
for two of these parameters. We investigate one of them in this section, see
Proposition 4.4. The other one was already studied in [24] and we postpone its
treatment until the proof of Theorem A.

The following is a sort of division theorem within the class of flat functions
that will be used to study the criticality at the outer boundary for the
above-mentioned isochrones. In its statement, and in what follows, we use the
notation $0_{n}=(0,0,\ldots,0)\in\mathbb{R}^{n}$ for the sake of shortness.

######  Lemma 4.1.

Let us fix $K\in\mathbb{N}\cup\\{\infty\\}$, $L\geqslant 0$ and
$n\in\mathbb{N}.$ If
$f(s;\mu_{1},\ldots,\mu_{n})\in\mathcal{F}^{K}_{L}(0_{n})$ verifies that

$f(s;\mu_{1},\ldots,\mu_{k-1},0,\ldots,0)\equiv 0\text{, for some
$k\in\\{1,2,\ldots,n\\}$,}$

then there exist $f_{k},\ldots,f_{n}\in\mathcal{F}_{L}^{K-1}(0_{n})$ such that
$f=\sum_{i=k}^{n}\mu_{i}f_{i}$.

  * Proof.

We proceed by induction on $n\in\mathbb{N}.$ For the base case $n=1$ we take
$f(s;\mu_{1})\in\mathcal{F}_{L}^{K}(0_{1})$ with $f(s;0)\equiv 0$ and define
$f_{1}(s;\mu_{1})\\!:=\int_{0}^{1}\partial_{2}f(s;\mu_{1}t)dt,$ so that
$f=\mu_{1}f_{1}.$ To show that $f_{1}\in\mathcal{F}_{L}^{K-1}(0_{1})$ we use
that, by hypothesis (see Definition A.1), for every
$\nu=(\nu_{0},\nu_{1})\in\mathbb{Z}^{2}_{\geq 0}$ with
$|\nu|=\nu_{0}+\nu_{1}\leqslant K-1$ there exist a neighbourhood
$V\subset\mathbb{R}$ of $0$ and $C,s_{0}>0$ such that
$|\partial^{\nu_{0}}_{s}\partial^{\nu_{1}+1}_{\mu_{1}}f(s;\mu_{1})|\leqslant
Cs^{L-\nu_{0}}$ for every $\mu_{1}\in V$ and $s\in(0,s_{0})$. On account of
this and applying the Dominated Convergence Theorem [37, Theorem 11.30],

$|\partial^{\nu}f_{1}(s;\mu_{1})|\leqslant\int_{0}^{1}\left|\partial^{\nu}(\partial_{2}f(s;\mu_{1}t))\right|dt\leqslant\int_{0}^{1}|\partial_{s}^{\nu_{0}}\partial_{2}^{\nu_{1}+1}f(s;\mu_{1}t)|t^{\nu_{1}}dt\leqslant\frac{C}{\nu_{1}+1}s^{L-\nu_{0}}$

for every $\mu_{1}\in V$ and $s\in(0,s_{0}).$ Hence
$f_{1}\in\mathcal{F}_{L}^{K-1}(0_{1})$. To prove the inductive step we suppose
that $n>1$ and consider
$f(s;\mu_{1},\ldots,\mu_{n})\in\mathcal{F}^{K}_{L}(0_{n})$ verifying that
$f(s;\mu_{1},\ldots,\mu_{k-1},0,\ldots,0)\equiv 0$ for some
$k\in\\{1,2,\ldots,n\\}$. It is clear that we can write

$\displaystyle
f(s;\mu_{1},\ldots,\mu_{n-1},\mu_{n})-f(s;\mu_{1},\ldots,\mu_{n-1},0)=\mu_{n}f_{n}(s;\mu_{1},\ldots,\mu_{n})$
(14) with $\displaystyle
f_{n}(s;\mu_{1},\ldots,\mu_{n})\\!:=\int_{0}^{1}\partial_{n+1}f(s;\mu_{1},\ldots,\mu_{n-1},\mu_{n}t)dt.$

Similarly as for the base case, taking $f\in\mathcal{F}_{L}^{K}(0_{n})$ into
account, one can easily show that $f_{n}\in\mathcal{F}_{L}^{K-1}(0_{n})$.
Since $f(s;\mu_{1},\ldots,\mu_{n-1},0)|_{\mu_{k}=\ldots=\mu_{n-1}=0}\equiv 0,$
by the inductive hypothesis there exist

$\displaystyle
f_{k}(s;\mu_{1},\ldots,\mu_{n-1}),\ldots,f_{n-1}(s;\mu_{1},\ldots,\mu_{n-1})\in\mathcal{F}_{L}^{K-1}(0_{n-1})$
such that $\displaystyle
f(s;\mu_{1},\ldots,\mu_{n-1},0)=\sum_{i=k}^{n-1}\mu_{i}f_{i}(s;\mu_{1},\ldots,\mu_{n-1}).$

Due to $\mathcal{F}_{L}^{K-1}(0_{n-1})\subset\mathcal{F}_{L}^{K-1}(0_{n}),$
see Definition A.1, the combination of this identity with $(\ref{l1eq1})$
shows that $f=\sum_{i=k}^{n}\mu_{i}f_{i}$ with
$f_{k},\ldots,f_{n}\in\mathcal{F}_{L}^{K-1}(0_{n})$ as desired. This shows the
inductive step and concludes the proof of the result.

We state next our first result about the bifurcation of critical periodic
orbits from the outer boundary of an isochronous center. With regard to its
proof let us advance that, after a convenient division in the space of
coefficients, we proceed as in the proofs of Bautin [4, §3] and Chicone and
Jacobs [6, Theorem 2.2] for the analogous results about limit cycles and
critical periods, respectively, bifurcating from the center. Here we tackle
the bifurcation from the polycycle, which is more challenging because,
contrary to the center, the period function cannot be analytically extended
there. To overcome this difficulty it is crucial the fact that the flatness of
the remainder in the asymptotic expansion is preserved after the derivation
with respect to the parameters.

######  Proposition 4.2.

If ${{\nu}}_{0}=(-\frac{1}{2},2)$ then
$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}=1$.

  * Proof.

We show first the upper bound
$\mathrm{Crit}\bigl{(}(\Pi_{{{\nu}}_{0}},X_{{{\nu}}_{0}}),X_{{{\nu}}}\bigr{)}\leqslant
1$, which constitutes the difficult part of the proof. To this end, following
the notation introduced in Section 3.2, we define $P(s;{{\nu}})$ to be the
period of the periodic orbit of $X_{{\nu}}$ passing through the point
$(p_{1}-s,0)$. Thanks to the reversibility of $X_{{\nu}}$ with respect to
$\\{y=0\\}$ it turns out that $P(s;{{\nu}})=2T(s;{{\nu}})$ where
$T(\,\cdot\,;{{\nu}})$ is the Dulac time that we consider in Proposition 3.3.
Thus, by applying $(d)$ in that result and setting
$\lambda({{\nu}})=\frac{1}{2(F-1)}$, we can assert that

$T(s;{{\nu}})=T_{00}({{\nu}})+T_{01}({{\nu}})s^{\lambda}+T^{\frac{1}{2}}_{101}({{\nu}})s\omega_{1-2\lambda}(s)+T_{100}^{\frac{1}{2}}({{\nu}})s+r_{1}(s;{{\nu}}),$

where $r_{1}\in\mathcal{F}_{3/2-\upsilon}^{\infty}({{\nu}}_{0})$ for all
$\upsilon>0$ small enough, the coefficients are analytic in a neighbourhood of
${{\nu}}_{0}=(-\frac{1}{2},2)$ and, moreover, the gradients of $T_{01}$ and
$T_{101}^{\frac{1}{2}}$ are linearly independent at ${{\nu}}_{0}$. Since one
can verify that
$\partial_{s}s\omega_{\alpha}(s)=(1-\alpha)\omega_{\alpha}(s)-1$, the
derivation of the above equality yields

$s^{1-\lambda}T^{\prime}(s;{{\nu}})=\lambda T_{01}({{\nu}})+2\lambda
T_{101}^{\frac{1}{2}}({{\nu}})s^{1-\lambda}\omega_{1-2\lambda}(s)+\big{(}T_{100}^{\frac{1}{2}}({{\nu}})-T_{101}^{\frac{1}{2}}({{\nu}})\big{)}s^{1-\lambda}+r_{2}(s;{{\nu}})$

where, by using Lemmas A.3 and A.4 in [29], the remainder
$r_{2}\\!:=s^{1-\lambda}\partial_{s}r_{1}$ belongs to
$\mathcal{F}_{1-\upsilon}^{\infty}({{\nu}}_{0})$ because
$\partial_{s}r_{1}\in\mathcal{F}_{1/2-\upsilon}^{\infty}({{\nu}}_{0})$ and, on
the other hand,
$s^{1-\lambda}\in\mathcal{F}_{1/2-\upsilon}^{\infty}({{\nu}}_{0})$ due to
$\lambda({{\nu}}_{0})=1/2.$ Note furthermore that
$\hat{{\nu}}=\Psi({{\nu}})\\!:=\big{(}\lambda({{\nu}})T_{01}({{\nu}}),2\lambda({{\nu}})T_{101}^{\frac{1}{2}}({{\nu}})\big{)}$
is local analytic change of coordinates at ${{\nu}}_{0}=(-\frac{1}{2},2)$ such
that $\Psi({{\nu}}_{0})=(0,0)$. We can thus write

$\mathscr{R}_{1}(s;\hat{{\nu}})\\!:=\left.s^{1-\lambda({{\nu}})}T^{\prime}(s;{{\nu}})\right|_{{{\nu}}=\Psi^{-1}(\hat{{\nu}})}=\hat{{\nu}}_{1}+\hat{{\nu}}_{2}s^{1-\hat{\lambda}}\omega_{1-2\hat{\lambda}}(s)+a(\hat{{\nu}})s^{1-\hat{\lambda}}+h(s;\hat{{\nu}}),$
(15)

where we set $\hat{\lambda}(\hat{{\nu}})\\!:=\lambda(\Psi^{-1}(\hat{{\nu}}))$
for shortness and define

$a(\hat{{\nu}})\\!:=\big{(}T_{100}^{\frac{1}{2}}-T_{101}^{\frac{1}{2}}\big{)}(\Psi^{-1}(\hat{{\nu}}))\text{
and
}h(s;\hat{{\nu}})\\!:=r_{2}(s;\Psi^{-1}(\hat{{\nu}}))\in\mathcal{F}^{\infty}_{1-\upsilon}(0_{2}).$

Recall at this point that the center at the origin of $X_{{{\nu}}_{0}}$ is
isochronous, so that $T^{\prime}(s;{{\nu}}_{0})\equiv 0.$ Consequently, due to
$\Psi({{\nu}}_{0})=(0,0)$,

$a(0,0)=0\text{ and }h(s;0,0)\equiv 0.$

By the Weierstrass Division Theorem (see for instance [11, Theorem 1.8]), the
first equality implies that
$a(\hat{{\nu}})=\hat{{\nu}}_{1}a_{1}(\hat{{\nu}})+\hat{{\nu}}_{2}a_{2}(\hat{{\nu}})$
with $a_{1}$ and $a_{2}$ analytic functions at $(0,0).$ On the other hand, by
Lemma 4.1,
$h(s;\hat{{\nu}})=\hat{{\nu}}_{1}h_{1}(s;\hat{{\nu}})+\hat{{\nu}}_{2}h_{2}(s;\hat{{\nu}})$
with $h_{i}\in\mathcal{F}^{\infty}_{1-\nu}(0_{2}).$ Therefore, from
$(\ref{p32eq1})$,

$\mathscr{R}_{1}(s;\hat{{\nu}})=\hat{{\nu}}_{1}\Big{(}1+a_{1}(\hat{{\nu}})s^{1-\hat{\lambda}}+h_{1}(s;\hat{{\nu}})\Big{)}+\hat{{\nu}}_{2}\Big{(}s^{1-\hat{\lambda}}\omega_{1-2\hat{\lambda}}(s)+a_{2}(\hat{{\nu}})s^{1-\hat{\lambda}}+h_{2}(s;\hat{{\nu}})\Big{)}$

Since $h_{i}\in\mathcal{F}^{\infty}_{1-\upsilon}(0_{2}),$
$h_{i}(s;\hat{{\nu}})$ tends to zero uniformly for $\hat{{\nu}}\approx(0,0)$
as $s\to 0^{+}$ (see Definition A.1). Due to $\lambda({{\nu}}_{0})=1/2,$ this
is also the case of $s^{1-\hat{\lambda}}$ and
$s^{1-\hat{\lambda}}\omega_{1-2\hat{\lambda}}(s)$ by $(c)$ of Lemma A.4 in
[29]. Hence there exists a neighbourhood $U$ of $(0,0)$ such that $\lim_{s\to
0^{+}}(1+a_{1}(\hat{{\nu}})s^{1-\hat{\lambda}}+h_{1}(s;\hat{{\nu}}))=1$
uniformly on $U$. Accordingly, the function

$\displaystyle\mathscr{R}_{2}(s;\hat{{\nu}})\\!:=$
$\displaystyle\,\frac{\mathscr{R}_{1}(s;\hat{{\nu}})}{1+a_{1}(\hat{{\nu}})s^{1-\hat{\lambda}}+h_{1}(s;\hat{{\nu}})}=\hat{{\nu}}_{1}+\hat{{\nu}}_{2}\ell(s;\hat{{\nu}})$
(16) with $\displaystyle\ell(s;{{\nu}})\\!:=$
$\displaystyle\,\frac{s^{1-\hat{\lambda}}\omega_{1-2\hat{\lambda}}(s)+a_{2}(\hat{{\nu}})s^{1-\hat{\lambda}}+h_{2}(s;\hat{{\nu}})}{1+a_{1}(\hat{{\nu}})s^{1-\hat{\lambda}}+h_{1}(s;\hat{{\nu}})}$

belongs to the class $\mathscr{C}^{\infty}_{s>0}(U)$, see Definition A.1.

We claim that, by shrinking $U$, there exists $s_{0}>0$ such that
$\mathscr{R}_{2}(s;\hat{{\nu}})$ has at most one zero on $(0,s_{0})$, counted
with multiplicities, for all $\hat{{\nu}}=(\hat{{\nu}}_{1},\hat{{\nu}}_{2})\in
U\setminus\\{(0,0)\\}.$ Indeed, to show this note first that if
$\hat{{\nu}}_{2}=0$ then $\mathscr{R}_{2}(s;\hat{{\nu}})=\hat{{\nu}}_{1}\neq
0,$ so that there is nothing to be proved in this case. Let us study
consequently the case $\hat{{\nu}}_{2}\neq 0$. To this end we observe that
$\mathscr{R}_{2}^{\prime}(s;\hat{{\nu}})=\hat{{\nu}}_{2}\ell^{\prime}(s;\hat{{\nu}})$
where, using a more compact notation,

$\displaystyle\ell^{\prime}(s;\hat{{\nu}})$
$\displaystyle=\partial_{s}\left(\frac{s^{1-\hat{\lambda}}\omega_{1-2\hat{\lambda}}+a_{2}s^{1-\hat{\lambda}}+\mathcal{F}^{\infty}_{1-\upsilon}}{1+a_{1}s^{1-\hat{\lambda}}+\mathcal{F}^{\infty}_{1-\upsilon}}\right)$
$\displaystyle=\frac{\omega_{1-2\hat{\lambda}}}{s^{\hat{\lambda}}}\left(\frac{\hat{\lambda}+\frac{a_{2}-a_{2}\hat{\lambda}-1}{\omega_{1-2\hat{\lambda}}}+\mathcal{F}^{\infty}_{\frac{1}{2}-\upsilon^{\prime}}}{1+a_{1}s^{1-\hat{\lambda}}+\mathcal{F}^{\infty}_{1-\upsilon}}-\frac{\big{(}s^{1-\hat{\lambda}}+a_{2}\frac{s^{1-\hat{\lambda}}}{\omega_{1-2\hat{\lambda}}}+\mathcal{F}^{\infty}_{1-\upsilon^{\prime}}\big{)}\big{(}(1-\hat{\lambda})a_{1}+\mathcal{F}^{\infty}_{\frac{1}{2}-\upsilon^{\prime}}\big{)}}{(1+a_{1}s^{1-\hat{\lambda}}+\mathcal{F}^{\infty}_{1-\upsilon})^{2}}\right).$

Here we use the identity
$\partial_{s}s^{b}\omega_{\alpha}(s)=s^{b-1}((b-\alpha)\omega_{\alpha}(s)-1)$
and that, by Lemmas A.3 and A.4 in [29], we have
$1/\omega_{1-2\hat{\lambda}}\in\mathcal{F}^{\infty}_{-\upsilon}(0_{2})$ and
$s^{-\hat{\lambda}}\in\mathcal{F}_{-1/2-\upsilon}^{\infty}(0_{2})$ for all
$\upsilon>0$ small enough due to $\hat{\lambda}(0,0)=\frac{1}{2}$ and,
moreover, that the inclusion
$\mathcal{F}_{L}^{\infty}\mathcal{F}_{L^{\prime}}^{\infty}\subset\mathcal{F}_{L+L^{\prime}}^{\infty}$
holds. We also remark that, by $(a)$ of Lemma A.4 in [29],

$\lim_{s\to
0^{+}}\frac{1}{\omega_{1-2\hat{\lambda}}(s)}=\frac{|1-2\hat{\lambda}|-(1-2\hat{\lambda})}{2}\text{
uniformly for $\hat{{\nu}}\approx(0,0)$.}$

On account of this, from the above expression of $\ell^{\prime}$ we obtain
that

$\lim_{s\to
0^{+}}\frac{s^{\hat{\lambda}}\ell^{\prime}(s;\hat{{\nu}})}{\omega_{1-2\hat{\lambda}}(s)}=b(\hat{{\nu}})\text{
uniformly for $\hat{{\nu}}\approx(0,0)$,}$

where