# Number of limit cycles for planar systems with invariant algebraic curves

Armengol Gasull  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<EMAIL_ADDRESS>and  Hector Giacomini  Institut
Denis Poisson. Université de Tours, C.N.R.S. UMR 7013. 37200 Tours, France.
<EMAIL_ADDRESS>
###### Abstract.

For planar polynomials systems the existence of an invariant algebraic curve
limits the number of limit cycles not contained in this curve. We present a
general approach to prove non existence of periodic orbits not contained in
this given algebraic curve. When the method is applied to parametric families
of polynomial systems that have limit cycles for some values of the
parameters, our result leads to effective algebraic conditions on the
parameters that force non existence of the periodic orbits. As applications we
consider several families of quadratic systems: the ones having some quadratic
invariant algebraic curve, the known ones having an algebraic limit cycle, a
family having a cubic invariant algebraic curve and other ones. For any
quadratic system with two invariant algebraic curves we prove a finiteness
result for its number of limit cycles that only depends on the degrees of
these curves. We also consider some families of cubic systems having either a
quadratic or a cubic invariant algebraic curve and a family of Liénard
systems. We also give a new and simple proof of the known fact that quadratic
systems with an invariant parabola have at most one limit cycle. In fact, what
we show is that this result is a consequence of the similar result for
quadratic systems with an invariant straight line.

###### Key words and phrases:

Invariant algebraic curve; Limit cycle; Periodic orbit; Quadratic system;
Cubic system; Liénard system

###### 2010 Mathematics Subject Classification:

Primary: 34C07. Secondary: 37C27

##  1\. Introduction

The study of the number of limit cycles of polynomial differential systems has
attracted the interest of many researchers due to the unexpected difficulties
that it presents. In fact, even in the case of quadratic systems (QS) the
problem remains open. Perhaps the most influential works for this very
particular case have been the Coppel survey ([18]) and the Ye Yanqian et alt.
book ([42]).

To advance in particular subfamilies of QS people has started studying QS with
additional properties. One of these properties has been the existence of a
particular invariant algebraic curve for the QS. One of the reasons is the
general belief that for any polynomial system the existence of an invariant
algebraic curve for it limits the number of limit cycles not contained on this
curve.

So a first celebrated family has been QS with an invariant straight line. With
some effort people has been able to prove the existence of at most one limit
cycle for this particular case, and also its hyperbolicity, see [12, 13, 17,
19, 39].

Afterwards the attention has moved to QS with irreducible invariant curves of
degree two. In this case it was not difficult prove that no limit cycles
coexist with hyperbolas or ellipses, although, in the later case, the ellipse
itself can be a limit cycle, see [11, 42] and also the proofs of Theorem 1.2
and Proposition 6.4. In case of QS having an invariant parabola it took more
effort to prove the existence and uniqueness of the limit cycle, see [14, 43].

It is remarkable that it is possible to prove that result as a direct
consequence of the same result for QS with an invariant straight line. This
gives a simple and alternative proof to that of [43].

######  Theorem 1.1.

A quadratic system with an invariant parabola and a limit cycle can be
transformed into another quadratic system with an invariant straight line. As
a consequence, quadratic systems with an invariant parabola have at most one
limit cycle and when it exists it is hyperbolic.

Continuing in the world of QS with invariant algebraic curves several
directions have been appeared: studying the degrees of the possible invariant
curves, the existence of algebraic limit cycles (limit cycles included in an
invariant algebraic curve), the number of limit cycles of families with higher
degree invariant curves, …. Our second result considers the question of the
non existence of other limit cycles when the QS has an algebraic limit cycle.
In [32, 33] the authors study a related problem. They give conditions on a QS
to ensure that it has at most one algebraic limit cycle.

We face the following natural problem, see for instance [29, 30]: Can an
algebraic and a non-algebraic limit cycle coexist in a QS? Although it is yet
an open question, it is known that all known QS exhibiting an algebraic limit
cycle have no other limit cycle. A proof for the known cases at that time was
given in [8].

Our second main result proves with an unified method this fact for all the
eight known families with one algebraic limit cycle. The eight families that
we will study, together with the expression of their corresponding algebraic
limit cycles are listed in the proof of the following theorem. Here, we only
make some few comments about them, see the nice paper [1] for more details.
The researchers who have found these families are: Qin; Yablonskii; Filiptsov;
Chavarriga; Christopher, Llibre and Świrszcz (two cases), Chavarriga, Llibre
and Sorolla; and Alberich-Carramiñana, Ferragut and Llibre.

In fact, for completeness, our proof studies all the eight cases, although
from the results of [1] it is clear these eight cases can be reduced to four
of them because the other ones can be obtained from these four via suitable
Cremona transformations (birational automorphisms) and changes of time. See
more details in [1, Fig. 10].

######  Theorem 1.2.

The eight families of quadratic systems detailed in the proof, and that
nowadays include all the cases of known quadratic systems with an algebraic
limit cycle, have at most one limit cycle and when it exists it is this
algebraic limit cycle.

We do not study here the hyperbolicity of the limit cycles considered in the
above theorem. This question is studied in [27].

The general approach that we use to prove the eight cases of the above theorem
is detailed in Section 2, see Theorem 2.1, and essentially consists on a
method for effectively finding a kind of Lyapunov functions for systems with
an invariant algebraic curve. As we will see, these functions are actually not
Lyapunov functions because only the condition on not changing sign of its
temporal derivative is required. The ideas of this approach go back to
Poincaré and have also been used in some cases in [5, 8]. In fact, in this
work we will show that the method provides a systematic approach to study many
families of planar differential systems, not necessarily QS, with invariant
algebraic curves.

Our third main result deals with QS with two generalized invariant curves and
gives an upper bound for their number of limit cycles. The same question but
with a different point of view was treated in [6, 7] but our results are
different of the ones given in these works. See for instance [21] for several
examples of polinomial systems with generalized invariant curves. The
definition of these type of curves and what we mean when we say that two of
them are different is deferred to the following section. Here we only comment
that they include usual invariant algebraic curves.

######  Theorem 1.3.

Consider a quadratic system with two different generalized invariant curves.
Let $M$ be the number of limit cycles contained in them. Then its maximum
number of limit cycles is $2M+2.$

As we will see, our proof starts with the result of [36] that shows that if a
QS has two different nests of limit cycles then one of the critical points is
surrounded by a single limit cycle. A similar result, but with a bigger upper
bound for the number of limit cycles, could be obtained under our hypotheses
without using that result.

Another key point in our proof is to prove that all the limit cycles not
contained in the generalized invariant curves are hyperbolic and have the same
stability. As we will see, to prove this fact we find some algebraic relations
between the characteristic exponents of the periodic orbits and some integrals
involving the cofactors of the curves. It is also possible to prove this
result by using the generalized Bendixson-Dulac Theorem, see for instance [24,
25].

We remark that although most probably this upper bound is not sharp (in fact,
people believe that this type of QS does not have limit cycles) it is never
easy to found upper bounds for the number of limit cycles of a given QS. We
also remark that although after the work of Bamon ([2]) the finiteness of the
number of limit cycles for a given QS is known, the existence of a uniform
bound for all QS is yet an open problem.

We also prove the following corollaries:

######  Corollary 1.4.

If we add to the hypotheses of Theorem 1.3 that the two generalized invariant
curves do not have isolated real points, then the upper bound of the theorem
can be reduce to be $2M$ and moreover all these limit cycles are nested.

Notice that this corollary extends the classical result that Lotka-Volterra
QS, that is QS with two non parallel invariant straight lines, do not have
limit cycles. This is so because in this case $M=0$ and of course these curves
do not contain real isolated points. See also Subsection 6.2 for a different
and direct proof.

The following result gives in the case of invariant algebraic curves an upper
bound only in terms of the degrees of these curves and extends again the
result for Lotka-Volterra QS.

######  Corollary 1.5.

Consider a quadratic system with two different invariant irreducible algebraic
curves of degrees $m_{1}$ and $m_{2}$. Then its maximum number of limit cycles
is $2([m_{1}/2]+[m_{2}/2])+2,$ where $[\,\,]$ is the integer part function.
Moreover, if the two algebraic invariant curves do not have real isolated
points, then the upper bound reduces to $2([m_{1}/2]+[m_{2}/2]).$

In Corollary 5.3 we also prove that under some more additional conditions a QS
with two generalized invariant curves have in some cases at most two limit
cycles and in other ones do not have limit cycles. We do not include the
precise statements in this introduction because we would need to introduce
some more notations and definitions.

In Section 6, we apply Theorem 2.1 to other QS to prove non existence of limit
cycles, and also to give conditions for non existence of limit cycles for
several parametric families of QS having some invariant algebraic curve, such
that for some values of their parameters they have a limit cycle not contained
in this curve.

Finally, to illustrate that Theorem 2.1 is also useful to families not
necessarily quadratic we dedicate Section 7 to other parametric families of
planar systems that have some invariant algebraic curve and limit cycles, not
contained in these curves, for some values of the parameters. Specifically, in
Subsection 7.1 we study two families of cubic systems and in Subsection 7.2 we
give some results on Liénard systems.

##  2\. Preliminary results

Consider a polynomial differential system

$\dot{x}=P(x,y),\quad\dot{y}=Q(x,y)$ (1)

and denote by $\mathcal{X}=(P,Q)$ its associated vector field. Recall that it
is said that it has an invariant algebraic curve $f(x,y)=0$ if $f$ is
irreducible and it holds that

$\dot{f}(x,y)=\frac{\partial f(x,y)}{\partial x}P(x,y)+\frac{\partial
f(x,y)}{\partial y}Q(x,y)=k(x,y)f(x,y),$ (2)

for some polynomial $k(x,y),$ called the cofactor of $f.$ Note that if the set
$\\{(x,y)\in\mathbb{R}^{2}\,:\,f(x,y)=0\\}$ is not empty, then it is invariant
by the flow of (1). Observe also that if the degree of $\mathcal{X}$ (i. e.,
the maximum of the degrees of $P$ and $Q$) is $n,$ then the degree of $k$ is
at most $n-1.$

A simple, but key result for proving our results will be the next theorem. As
we have already explained it is used in some particular cases in [5, 8] and
here it is presented as a systematic method for studying the non existence of
periodic orbits of polynomial differential systems with invariant algebraic
curves.

######  Theorem 2.1.

Consider the polynomial differential system (1) and assume that it has an
invariant algebraic curve $f(x,y)=0$ with cofactor $k(x,y).$ For each
$\alpha\in\mathbb{R}$ and each polynomial $g(x,y)$ define the new polynomial

$N_{\alpha,g}(x,y)=\alpha k(x,y)g(x,y)+\frac{\partial g(x,y)}{\partial
x}P(x,y)+\frac{\partial g(x,y)}{\partial y}Q(x,y).$

Then, if for some $\alpha$ and $g,$ $N_{\alpha,g}$ does not change sign and
vanishes only on some algebraic curve that it is not invariant by the flow,
then the only limit cycles of the system (if any) are included in the
invariant algebraic curve $f(x,y)=0.$

Moreover, if for some $\alpha$ and $g,$ $N_{\alpha,g}(x,y)\equiv 0$ then
$g(x,y)|f(x,y)|^{\alpha},$ if it is not piecewise constant, is a first
integral of (1) and the same conclusion holds.

###### Proof.

We will use the following well know fact: If for some open set
$\mathcal{U}\subset\mathbb{R}^{2},$ there exists a class $\mathcal{C}^{1}$
function such that $v:\mathcal{U}\to\mathbb{R},$ and

$\dot{v}(x,y)=\frac{\partial v(x,y)}{\partial x}P(x,y)+\frac{\partial
v(x,y)}{\partial y}Q(x,y)$ (3)

does not vanish then the system (1) does not have periodic orbits totally
contained in $\mathcal{U}.$ This is so because while the solution is in
$\mathcal{U}$ the function $t\rightarrow v(x(t),y(t)),$ where $(x(t),y(t))$ is
any solution of the differential equation, is monotonous. Clearly, this fact
prevents the existence of periodic orbits. If the right hand side of (3)
vanishes on some curve, but does not change it sign, then the same holds,
unless this curve is invariant by the flow.

Since $f(x,y)=0$ is an invariant algebraic curve, each of the connected
components of $\mathcal{U}\setminus\\{f(x,y)=0\\}$ is invariant. For proving
the theorem we apply the above result by taking
$v(x,y)=g(x,y)|f(x,y)|^{\alpha}$ and $\mathcal{U}$ any of these components.
Some computations give that

$\dot{v}(x,y)=|f(x,y)|^{\alpha}N_{\alpha,g}(x,y)$

and hence the result follows under the hypotheses on $N_{\alpha,g}.$ Clearly
when $N_{\alpha,g}(x,y)\equiv 0$ then $\dot{v}(x,y)\equiv 0$ and $v$ is a
first integral of (1). Then although periodic orbits can exist, the only
possible limit cycle are contained in $\\{f(x,y)=0\\},$ as we wanted to prove.
∎

For practical applications one idea to prove that $N_{\alpha,g}$ does not
change sign, and that works in many situations, is to introduce some
parameters in the unknown function $g,$ in such a way that a clever choice of
them allows to prove the existence of some $r\in\mathbb{R}$ and some
polynomial $w$ such that $N_{\alpha,g}(x,y)=rw^{2}(x,y).$ Another useful
approach is to try to force that $N$ only depends on one of the variables and
then impose that all its roots be double.

Now we give two remarks with two extensions of Theorem 2.1.

######  Remark 2.2.

When system (1) has two different invariant irreducible algebraic curves
$f_{i}(x,y)=0,i=1,2,$ with respective cofactors $k_{i}(x,y),$ a similar
approach can be used. For each $\alpha_{1},\alpha_{2}\in\mathbb{R}$ and each
polynomial $g(x,y)$ we can define the new polynomial

$\displaystyle N_{\alpha_{1},\alpha_{2},g}(x,y)$
$\displaystyle=\big{(}\alpha_{1}k_{1}(x,y)+\alpha_{2}k_{2}(x,y)\big{)}g(x,y)$
$\displaystyle\quad+\frac{\partial g(x,y)}{\partial x}P(x,y)+\frac{\partial
g(x,y)}{\partial y}Q(x,y).$

Then under the same hypotheses that in Theorem 2.1 for this new function $N$
the same conclusions hold.

Given any smooth function $f(x,y)$, not necessarily polynomial, we will say
that $f(x,y)=0$ is a generalized invariant curve for $\mathcal{X}$ if
$\dot{f}(x,y)=k(x,y)f(x,y),$ as in (2), with $k$ also a polynomial of degree
at most $n-1,$ that is, one less that the degree of the vector field. This $k$
is called its cofactor. Given two generalized invariant curves we will say
that they are different if it do not exist any $\alpha,\beta\in\mathbb{R}$
such that $|f_{1}(x,y)|=\alpha|f_{2}(x,y)|^{\beta}.$

######  Remark 2.3.

Theorem 2.1 and Remark 2.2 also hold if in their respective statements the
invariant algebraic curves are replaced by generalized invariant curves.

##  3\. Proof of Theorem 1.1

It is not restrictive to suppose that the invariant parabola $\mathcal{P}$ is
$f(x,y)=y-x^{2}=0.$ It is well known that QS having the invariant parabola
$\mathcal{P}$ can be written as

$\begin{cases}\dot{x}=a+bx+hy+c(y-x^{2})+exy=P(x,y),\\\
\dot{y}=2x(a+bx+hy)+d(y-x^{2})+2ey^{2}=Q(x,y),\end{cases}$ (4)

where $a,b,c,d,e,h$ are arbitrary real parameters, see [14]. In fact the
cofactor of $f$ is $k(x,y)=-2cx+2ey+d.$ Moreover, the critical points on
$\mathcal{P}$ are of the form $(x_{0},x_{0}^{2})$ where $x_{0}$ is one of the
real roots of

$P(x,x^{2})=ex^{3}+hx^{2}+bx+a=0.$

Next, we split our proof in two cases:

  1. (i)

The above equation has no real roots. Then in particular $e=0.$

  2. (ii)

The above equation has some real root, say $x=x_{0}.$

We split again case (i) in two subcases $c=0$ and $c\neq 0$ and in both of
them we will apply Theorem 2.1 for proving the non existence of limit cycles.

Consider first the case (i) with $c=0.$ Then, since $e=0,$ by taking $g=1$ and
$\alpha=1$ in that theorem we get that $N(x,y)=k(x,y)=d,$ and the result
follows.

For the situation (i) with $c\neq 0,$ recall that $e=0.$ We consider

$g(x,y)=2ac^{2}+bcd+d^{2}h-2ch(dx-cy)\quad\mbox{and}\quad\alpha=h/c$

and recall that $k(x,y)=-2cx+d.$ Then some computations give that

$N(x,y)=\frac{h(2cx-d)^{2}(bc+dh)}{c}$

and again the result follows.

In case (ii) we will prove that if there exists a limit cycle it is unique and
hyperbolic by transforming these QS to new QS with an invariant straight line.
For these QS, as we have already commented, the uniqueness and hyperbolicity
follows by the results of [12, 13, 17, 19, 39].

Following [14] we start proving that in case (ii) it is not restrictive to
consider $a=0$ in (4). Take the new coordinates $X=x-x_{0}$ and
$Y=x_{0}^{2}-2x_{0}x+y.$ In these coordinates $y-x^{2}=0$ is transformed into
$Y-X^{2}=0.$ Moreover, system (4) writes as

$\begin{cases}\dot{X}=BX+HY+C(Y-X^{2})+eXY,\\\
\dot{Y}=2X(BX+HY)+D(Y-X^{2})+2eY^{2},\end{cases}$

where

$\displaystyle B$ $\displaystyle=b+2hx_{0}+3ex_{0}^{2},\quad C=c-2ex_{0},$
$\displaystyle D$ $\displaystyle=d-2cx_{0}+2ex_{0}^{2},\quad H=h+3ex_{0}.$

Hence, from now on, we consider system (4) with $a=0.$ Let $\gamma$ be a limit
cycle of this system. It is totally contained in one of the six connected
components of
$\mathbb{R}^{2}\setminus\left(\mathcal{P}\cup\\{x=0\\}\cup\\{y=0\\}\right).$
This is so because $\mathcal{P}$ is invariant and moreover

$\dot{x}|_{x=0}=(c+h)y,\quad\mbox{and}\quad\dot{y}|_{y=0}=(2b-d)x^{2}.$

Call $\mathcal{V}$ this connected component which, of course, must contain a
critical point of index $+1.$ Hence in particular the new birrational change
of variables

$X=\frac{x}{y-x^{2}},\quad Y=\frac{y}{y-x^{2}}$

is well defined on the region $\mathcal{V}$ where $\gamma$ lies, and its
inverse is

$x=\dfrac{Y-1}{X},\quad y=\dfrac{Y(Y-1)}{X^{2}}.$

It transforms the parabola $y-x^{2}=0$ into the straight line $Y-1=0.$ By
introducing a new time $s$ such that $ds/dt=X,$ and writing
$Z^{\prime}=dZ/ds=XdZ/dt=X\dot{Z},$ after some computations we get that

$\begin{cases}X^{\prime}=X\dot{X}=-cX+eY+(b-d)X^{2}+(2c+h)XY-
eY^{2},\\\\[5.69046pt]
Y^{\prime}=X\dot{Y}=(Y-1)\big{(}(2b-d)X+2(c+h)Y\big{)},\end{cases}$

which is a quadratic system with the invariant straight line $Y-1=0,$ as we
wanted to prove.

##  4\. Proof of Theorem 1.2

For each of the cases the result will be a consequence of Theorem 2.1 by
choosing suitable $\alpha$ and $g$ in such a way that
$N_{\alpha,g}(x,y)=rw^{2}(x,y),$ for some $r\in\mathbb{R}$ and some polynomial
$w.$

We skip all the computations and we simply write $P,Q,$ the algebraic curve
$f=0$ and its cofactor $k,$ a suitable constant $\alpha$ and function $g,$ and
finally the corresponding $N=N_{\alpha,g}.$

For some of the systems some given values of the parameters have to be
omitted. For instance, the value $a=-10/3$ in Case 4. These values could be
studied separately, but for the sake of shortness and because for them the
invariant algebraic curve does not contain any limit cycle we do not study
these particular values.

Case 1: Qin system with algebraic limit cycle of degree 2, see [37]:

$\displaystyle P$ $\displaystyle=-y(ax+by+c)-(x^{2}+y^{2}-1),\quad
Q=x(ax+by+c),$ $\displaystyle f$ $\displaystyle=x^{2}+y^{2}-1,\quad
k=-2x,\quad\alpha=b/2,\quad g=c+by,\quad N=abx^{2}.$

We remark that the corresponding QS has only $x^{2}+y^{2}-1=0$ as a limit
cycle when $a\neq 0,$ $c^{2}+4(b+1)>0$ and $a^{2}+b^{2}<c^{2}$ but the proof
that there are no other limit cycles works for all values of the parameters. A
similar fact also holds for all the other cases. In fact the range of
parameters for which each invariant algebraic curve contains a limit cycle is
detailed in [1].

Case 2: Yablonskii system with algebraic limit cycle of degree 4, see [41]:

$\displaystyle P$ $\displaystyle=-4abcx-(a+b)y+3(a+b)cx^{2}+4xy,$
$\displaystyle\begin{split}Q&=(a+b)abx-4abcy+(4abc^{2}-3(a+b)^{2}/2+4ab)x^{2}\\\
&\quad+8(a+b)cxy+8y^{2},\end{split}$ $\displaystyle f$
$\displaystyle=(y+cx^{2})^{2}+x^{2}(x-a)(x-b),\,\,\,k=4(-2abc+3c(a+b)x+4y),$
$\displaystyle g$ $\displaystyle=2c(a-3b)(3a-b)(y+cx^{2})-ab(a+b-4x)^{2},$
$\displaystyle\alpha$ $\displaystyle=-1/2,\quad
N=-c\big{(}2ab(a+b)+(2ab-3a^{2}-3b^{2})x\big{)}^{2}.$

Case 3: Filipstov system with algebraic limit cycle of degree 4, see [20]:

$\displaystyle P$ $\displaystyle=6(1+a)x+2y-6(2+a)x^{2}+12xy,$ $\displaystyle
Q$ $\displaystyle=15(1+a)y+3a(1+a)x^{2}-2(9+5a)xy+16y^{2},$ $\displaystyle f$
$\displaystyle=3(1+a)(ax^{2}+y)^{2}+2y^{2}(2y-3(1+a)x),$ $\displaystyle k$
$\displaystyle=6(5(1+a)-(8+4a)x+8y),$ $\displaystyle\alpha$
$\displaystyle=-1/2,\quad g=2y+(3+5a)x^{2},\quad
N=-\big{(}3(1+a)x-4y\big{)}^{2}.$

Case 4: Chavarriga system with algebraic limit cycle of degree 4, see [10]:

$\displaystyle P$ $\displaystyle=5x+6x^{2}+4(1+a)xy+ay^{2},\quad
Q=x+2y+4xy+(2+3a)y^{2},$ $\displaystyle f$
$\displaystyle=x^{2}+x^{3}+x^{2}y+2axy^{2}+2axy^{3}+a^{2}y^{4},\quad
a\neq-10/3,$ $\displaystyle k$
$\displaystyle=2\big{(}5+9x+(5+6a)y\big{)},\quad\alpha=-\frac{7+3a}{3(10+3a)},$
$\displaystyle g$ $\displaystyle=-5+(21+9a)x-(35+15a)y,$ $\displaystyle N$
$\displaystyle=\frac{2(7+3a)}{3(10+3a)}\big{(}5+9x+(5+6a)y\big{)}^{2}.$

Case 5: Chavarriga, Llibre and Sorolla system with algebraic limit cycle of
degree 4, see [10]:

$\displaystyle P$ $\displaystyle=2(1+2x-2ax^{2}+6xy),\quad
Q=8-3a-14ax-2axy-8y^{2},$ $\displaystyle f$
$\displaystyle=1/4+x-x^{2}+ax^{3}+xy+x^{2}y^{2},\quad k=4(2-3ax+2y),$
$\displaystyle\alpha$ $\displaystyle=1/3,\quad g=-5+3ax/2+y,\quad
N=-4(2-3ax+2y)^{2}/3.$

Case 6: Christopher, Llibre and Świrszcz system with algebraic limit cycle of
degree 5, see [16]:

$\displaystyle P$
$\displaystyle=28x+2(16-a^{2})(a+12)x^{2}+6(3a-4)xy-\frac{12}{a+4}y^{2},\quad
a\neq-4,$ $\displaystyle Q$
$\displaystyle=2(16-a^{2})x+8y+(16-a^{2})(a+12)xy+2(5a-12)y^{2},$
$\displaystyle\begin{split}f&=\,x^{2}+(16-a^{2})x^{3}+(a-2)x^{2}y-\frac{2}{a+4}xy^{2}-\frac{1}{4}(4-a)(a+12)x^{2}y^{2}\\\
&\quad+\frac{8-a}{a+4}xy^{3}+\frac{1}{(a+4)^{2}}y^{4}+\frac{a+12}{a+4}xy^{4}-\frac{6}{(a+4)^{2}}y^{5},\end{split}$
$\displaystyle k$
$\displaystyle=56+6(16-a^{2})(a+12)x+4(13a-24)y,\,\alpha=-\frac{3+4a}{15(3+a)},a\neq-3,$
$\displaystyle g$ $\displaystyle=28+(-144-192a+9a^{2}+12a^{3})x+(42+56a)y,$
$\displaystyle N$
$\displaystyle=-\frac{2(3+4a)}{15(3+a)}\big{(}28+(576+48a-36a^{2}-3a^{3})x+(-48+26a)y\big{)}^{2}.$

Case 7: Christopher, Llibre and Świrszcz system with algebraic limit cycle of
degree 6, see [16]:

$\displaystyle P$ $\displaystyle=28a(a-30)x+y+168a^{2}x^{2}+3xy,$
$\displaystyle\begin{split}Q&=16a(a-30)\big{(}14a(a-30)x+5y+84a^{2}x^{2}\big{)}+24a(17a-6)xy+6y^{2},\end{split}$
$\displaystyle\begin{split}f&=48a^{3}(a-30)^{4}x^{2}+24a^{2}(a-30)^{3}xy+3a(a-30)^{2}y^{2}\\\
&\quad+64a^{3}(a-30)^{3}(9a-4)x^{3}+24a^{2}(a-30)^{2}(9a-4)x^{2}y\\\
&\quad+18a(a-30)(a-2)xy^{2}-7y^{3}+576a^{3}(a-30)^{2}(a-2)^{2}x^{4}\\\
&\quad+144a^{2}(a-30)(a-2)^{2}x^{3}y+27a(a-2)^{2}x^{2}y^{2}\\\
&\quad-3456a^{3}(a-30)(a-2)^{2}(2a+3)x^{5}-432a^{2}(a-2)^{2}(2a+3)x^{4}y\\\
&\quad+3456a^{3}(a-2)^{2}(a+12)(2a+3)x^{6},\end{split}$ $\displaystyle k$
$\displaystyle=168a(a-30)+1008a^{2}x+18y,\quad\alpha=-1/3,$
$\displaystyle\begin{split}g&=-16a(a-30)^{2}-24a(a-30)(7a-30)x\\\
&\quad-72a(360-78a+5a^{2})x^{2}+3(30+a)y,\end{split}$ $\displaystyle N$
$\displaystyle=896a^{2}(a-30)(a-30+6ax)^{2}.$

We remark that there is a misprint when this system in written in [1]. There
the coefficient of $y$ in $Q$ is typed as $516a(a-30)$ instead of the correct
value $80a(a-30).$

Case 8: Alberich-Carramiñana, Ferragut and Llibre system with algebraic limit
cycle of degree 5, see [1]:

$\displaystyle P$
$\displaystyle=-8x+\frac{a}{2}(a-16)y-(5a-64)x^{2}+\frac{a}{8}(a^{2}-256)xy,$
$\displaystyle Q$
$\displaystyle=-28y+\frac{24}{a}x^{2}-3(3a-32)xy+\frac{a}{4}(a^{2}-256)y^{2},$
$\displaystyle\begin{split}f&=\,ay^{2}-4x^{2}y+\frac{a}{2}(a-12)xy^{2}-\frac{a^{2}}{4}(a-16)y^{3}+\frac{4}{a}x^{4}\\\
&\quad+(24-a)x^{3}y+\frac{a}{16}(a^{2}-256)x^{2}y^{2}-\frac{24}{a}x^{5}+(a+16)x^{4}y,\end{split}$
$\displaystyle k$
$\displaystyle=-56-2(13a-152)x+\frac{3a}{4}(a^{2}-256)y,\quad\alpha=\frac{26-4a}{15(a-2)},\,a\neq
2,$ $\displaystyle g$ $\displaystyle=112+56(2a-13)x+3a(2a-13)(a-16)y,$
$\displaystyle N$
$\displaystyle=\frac{2a-13}{60(a-2)}\big{(}-224+(1216-104a)x+a(3a^{2}-768)y\big{)}^{2}.$

##  5\. Proof of Theorem 1.3

This section is devoted to prove Theorem 1.3, Corollary 1.5 and other related
results. When we say that the sign of a number, say $h,$ gives the stability
of a limit cycle or a critical point we mean that when the sign of $h$ is
positive (resp. negative) this object is a repellor (resp. an attractor).

###### Proof of Theorem 1.3.

From the classical results on QS ([18, 42]) and the results of [36] we know
that each limit cycle of a QS surrounds exactly one critical point, that must
be of focus type. Moreover, there are at most two nests of concentric limit
cycles and in this case one of the nests is formed by a single limit cycle. In
other words, the configurations of limit cycles for a QS are either $L$ limit
cycles surrounding a focus, or $L$ limit cycles surrounding a focus and one
more limit cycle surrounding another focus. Moreover, from [2] and for a given
QS, $L$ is finite.

Let $f_{i}(x,y)=0,i=1,2$ be the two different generalized invariant curves,
and let $k_{i}(x,y)=a_{i}+b_{i}x+c_{i}y,i=1,2$ be their respective cofactors.
Let $\operatorname{div}(\mathcal{X}(x,y))=a+bx+cy$ be the divergence of the
vector field associated to the QS. It is well known ([42]) that the
characteristic exponent of a limit cycle $\gamma$ is

$h=\int_{0}^{T}\operatorname{div}(\mathcal{X})(x(t),y(t))\,{\rm
d}t=\int_{0}^{T}\big{(}a+bx(t)+cy(t)\big{)}\,{\rm d}t,$

where $(x(t),y(t))$ is the time parameterization of $\gamma$ and $T$ is its
period. Recall that if $h\neq 0$ then $\gamma$ is hyperbolic and its sign
gives its stability.

For convenience we introduce the sets
$\mathcal{Z}_{i}=\\{f_{i}(x,y)=0\\},i=1,2,$
$\mathcal{Z}_{1,2}=\mathcal{Z}_{1}\cup\mathcal{Z}_{2}$ and also two real
numbers $X$ and $Y,$ as

$X=\int_{0}^{T}x(t)\,{\rm d}t,\quad Y=\int_{0}^{T}y(t)\,{\rm
d}t\quad\mbox{and}\quad T=\int_{0}^{T}{\rm d}t.$

Let us study $h$ when $\gamma$ is not included in $\mathcal{Z}_{1,2}.$ In this
case, by integrating the equalities $\dot{f}_{i}(x,y)/f_{i}(x,y)=k_{i}(x,y)$
over $\gamma$ it holds that

$\int_{0}^{T}k_{i}(x(t),y(t))\,{\rm
d}t=\int_{0}^{T}\big{(}a_{i}+b_{i}x(t)+c_{i}y(t)\big{)}\,{\rm d}t=0,\quad
i=1,2.$

By joining all the above equalities, for
$\gamma\cap\mathcal{Z}_{1,2}=\emptyset,$ it holds that

$A\left(\begin{matrix}T\\\ X\\\ Y\end{matrix}\right)=\left(\begin{matrix}h\\\
0\\\ 0\end{matrix}\right),\quad\mbox{where}\quad
A=\left(\begin{matrix}a&b&c\\\ a_{1}&b_{1}&c_{1}\\\
a_{2}&b_{2}&c_{2}\end{matrix}\right).$

When $(b_{1}c_{2}-b_{2}c_{1})\det(A)\neq 0$ then

$h=\frac{\det(A)}{b_{1}c_{2}-b_{2}c_{1}}T\neq 0$

and as a consequence all periodic orbits not contained in $\mathcal{Z}_{1,2}$
are hyperbolic limit cycles and with the same stability. In particular, only
one limit can exist in the region delimited by two consecutive limit cycles
contained in $\mathcal{Z}_{1,2},$ or between the critical point and the first
limit cycle, or surrounding the last limit cycle of the nest. In any case, the
maximum number of limit cycles that surround one focus is 1, while surrounding
the other one the maximum number is $M$ contained in $\mathcal{Z}_{1,2}$ and
$M+1$ not in $\mathcal{Z}_{1,2}.$ This makes a total of $2M+2$ limit cycles,
as we wanted to see.

To end the proof we need to study the cases whether
$\det(A)(b_{1}c_{2}-b_{2}c_{1})=0.$ In this situation we will prove that the
QS does not have limit cycles outside $\mathcal{Z}_{1,2}.$

Assume first that $\operatorname{det}(A)=0.$ Then, there exist real numbers
$u,v$ and $w$ such that
$uk_{1}(x,y)+vk_{2}(x,y)+w\operatorname{div}(\mathcal{X})\equiv 0.$ By using
the well known formula

$\operatorname{div}\Big{(}|f_{1}(x,y)|^{p}|f_{2}(x,y)|^{q}\mathcal{X}\Big{)}\\\
=|f_{1}(x,y)|^{p}|f_{2}(x,y)|^{q}\Big{(}pk_{1}(x,y)+qk_{2}(x,y)+\operatorname{div}(\mathcal{X})\Big{)}$

and by taking $p=u/w$ and $q=v/w,$ when $w\neq 0,$ we get that the divergence
is identically zero and the QS can not have limit cycles outside
$\mathcal{Z}_{1,2}.$ When $w=0$ we can apply Remark 2.2 with $f_{1},f_{2},$
$\alpha_{1}=u,$ $\alpha_{2}=v$ and $g(x,y)\equiv 1$ to prove the non existence
of limit cycles outside $\mathcal{Z}_{1,2}$ because
$N_{\alpha_{1},\alpha_{2},g}(x,y)\equiv 0$ and
$|f_{1}(x,y)|^{p}|f_{2}(x,y)|^{q}$ is not identically constant.

Finally, when $b_{1}c_{2}-b_{2}c_{1}=0,$ notice that
$c_{2}k_{1}(x,y)-c_{1}k_{2}(x,y)=a_{1}c_{2}-c_{1}a_{2}.$ Hence we can apply
again Remark 2.2 with $f_{1},f_{2}$ and now $\alpha_{1}=c_{2},$
$\alpha_{2}=-c_{1}$ and $g(x,y)\equiv 1,$ obtaining that
$N_{\alpha_{1},\alpha_{2},g}(x,y)\equiv a_{1}c_{2}-c_{1}a_{2}.$ Therefore, the
non existence of limit cycles outside $\mathcal{Z}_{1,2}$ follows again. ∎

###### Proof of Corollary 1.4.

In this proof we will use the same notations that in the proof of Theorem 1.3.
Let us show first that the new hypothesis force the QS to have at most one
critical point outside the generalized invariant curves. We know that
$\dot{f}_{i}(x,y)=k_{i}(x,y)f_{i}(x,y),$ $i=1,2.$ Let $(\bar{x},\bar{y})$ be a
critical point not contained in these curves. Since
$\dot{f}_{i}(x,y)|_{(x,y)=(\bar{x},\bar{y})}=0$ and
$f_{i}(\bar{x},\bar{y})\neq 0,$ we get that $k_{i}(\bar{x},\bar{y})=0,$
$i=1,2.$ Writing $k_{i}(x,y)=a_{i}+b_{i}x+c_{i}y,$ $i=1,2,$ we obtain that
$(\bar{x},\bar{y})$ must be a solution of the lineal system

$a_{1}+b_{1}\bar{x}+c_{1}\bar{y}=0,\quad a_{2}+b_{2}\bar{x}+c_{2}\bar{y}=0.$
(5)

Then, either the system does not have solution, or it has exactly one
solution, or it has infinitely many. In the first two situations we are done.
In the third one, as in the proof of Theorem 1.3, we can apply Theorem 2.1 to
prove that the QS does not have limit cycles because both cofactors are
proportional.

Since the QS does not have isolated critical points in the generalized
invariant curves we have already proved that all the limit cycles are nested
and surround $(\bar{x},\bar{y}).$ Hence, arguing as in the proof of Theorem
1.3, the maximum number of limit cycles is $2M+1.$

To decrease by one this new upper bound we will prove that between the
critical point $(\bar{x},\bar{y})$ and the first limit cycle of the nest
belonging to $\mathcal{Z}_{1}\cup\mathcal{Z}_{2}$ the QS does not have limit
cycles. Assume that such a $\gamma$ do exist. We will prove that $\gamma$ and
the critical point $(\bar{x},\bar{y})$ have the same stability giving rise to
a contradiction. By the proof of Theorem 1.3 we know that we can restrict our
attention to the case $(b_{1}c_{2}-b_{2}c_{1})\det(A)\neq 0$ and that the
stability of this $\gamma$ is given by the sign of
$\det(A)T/(b_{1}c_{2}-b_{2}c_{1}).$ To study the stability of the focus point
$(\bar{x},\bar{y})$ it suffices to know the sign of

$d=\operatorname{div}(\mathcal{X})(\bar{x},\bar{y})=a+b\bar{x}+c\bar{y}.$

By using this equation and (5) we get that

$A\left(\begin{matrix}1\\\ \bar{x}\\\
\bar{y}\end{matrix}\right)=\left(\begin{matrix}d\\\ 0\\\ 0\end{matrix}\right)$

and as a consequence $d=\det(A)/(b_{1}c_{2}-b_{2}c_{1}),$ proving the desired
result. ∎

###### Proof of Corollary 1.5.

By Theorem 1.3 it suffices to prove that $M=[m_{1}/2]+[m_{2}/2].$ To prove
this equality we will show that when $f(x,y)=0$ is an algebraic invariant
curve of degree $m$ of a QS then its maximum number of nested ovals is
$[m/2].$ This is a simple consequence of the fact that a straight line passing
by a point surrounded by all these ovals cuts the curve $f$ at most at $m$
points and that each oval gives rises at least to two of these cuts. In fact,
exactly to two cuts because it is known that periodic orbits of QS are convex.
The upper bound of $2([m_{1}/2]+[m_{2}/2])$ limit cycles when the invariant
algebraic curves do not contain real isolated points follows from Corollary
1.4. ∎

In [9] it is proved that there are QS with invariant algebraic curves of
arbitrarily high degree and not being Darboux integrable. Their example is
$\dot{x}=1,$ $\dot{y}=2m+2xy+y^{2}$ and the curve has degree $m+1$. For other
similar examples see for instance [15, 35]. Nevertheless, under generic
conditions on the QS, the higher degree of the invariant algebraic invariant
curves is 4, see [4]. In fact, in that paper the author proves that if a
polynomial system of degree $n$ does not have dicritic critical points (see
the paper for a definition) then the maximum degree of any invariant algebraic
curve is $n+2.$ By using this result and Corollary 1.5 we obtain a new result.

######  Corollary 5.1.

Consider a quadratic system with two different invariant irreducible algebraic
curves and without dicritic critical points. Then its maximum number of limit
cycles is ten. Moreover, if the two algebraic invariant curves do not have
real isolated points, then this upper bound reduces to eight.

###### Proof.

From the results of [4] we know that the maximum degrees of the two invariant
algebraic curves is four. Then, taking $m_{1}=m_{2}=4$ in Corollary 1.5 the
result follows. ∎

Similarly, if we consider the classes of QS studied in [32, 33], which have at
most one algebraic limit cycle, we have the following result because we can
apply Theorem 1.3 and Corollary 1.4 with $M=1$.

######  Corollary 5.2.

Consider a quadratic system with two different invariant irreducible algebraic
curves and satisfying the hypotheses of [32, 33]. Then its maximum number of
limit cycles is four. Moreover, if the two algebraic invariant curves do not
have real isolated points, then this upper bound reduces to two.

Recall that from the proof of Theorem 1.3 we know that under its hypotheses
all the limit cycles of the QS that are not in generalized invariant curves
are hyperbolic and have the same stability. In the following result we prove
that something similar holds for all limit cycles contained in
$\mathcal{Z}_{1}$ and also for the ones contained in $\mathcal{Z}_{2}.$
Moroever, when these three stabilities coincide the upper bound for the number
of limit cycles drastically decreases.

######  Corollary 5.3.

Assume that the hypotheses of Theorem 1.3 hold. Let
$k_{i}(x,y)=a_{i}+b_{i}x+c_{i}y$ be the cofactors of the generalized invariant
curves of the QS, $f_{i}(x,y)=0,$ $i=1,2$ and let
$\operatorname{div}(\mathcal{X}(x,y))=a+bx+cy$ be its divergence. Assume that
$\Delta_{1,2}\big{(}\Delta_{1,2}-\Delta_{2}\big{)}\big{(}\Delta_{1,2}+\Delta_{1})\operatorname{det}(A)\neq
0,$ where

$\Delta_{1,2}=b_{1}c_{2}-b_{2}c_{1}\quad\mbox{and}\quad\Delta_{i}=bc_{i}-cb_{i},\,i=1,2.$

Then it holds that

  1. (i)

The stability of the limit cycles contained in $\\{f_{1}(x,y)=0\\}$ is given
by the sign of $(\Delta_{1,2}-\Delta_{2})\operatorname{det}(A).$

  2. (ii)

The stability of the limit cycles contained in $\\{f_{2}(x,y)=0\\}$ is given
by the sign of $(\Delta_{1,2}+\Delta_{1})\operatorname{det}(A).$

  3. (iii)

The stability of the limit cycles not contained in
$\\{f_{1}(x,y)=0\\}\cup\\{f_{2}(x,y)=0\\}$ is given by the sign of
$\Delta_{1,2}\operatorname{det}(A).$

Moreover, if the three quantities $\Delta_{1,2},$ $\Delta_{1,2}-\Delta_{2}$
and $\Delta_{1,2}+\Delta_{1}$ have the same sign, then the maximum number of
limit cycles of the QS is two. Furthermore, in this last situation, if the two
generalized invariant curves do not contain isolated real points then the QS
does not have limit cycles.

###### Proof.

In this proof we will use the same notations that in the proof of Theorem 1.3.
Recall that if a limit cycle does not intersect $\mathcal{Z}_{1,2}$ then its
characteristic exponent is $h=\operatorname{det}(A)T/\Delta_{1,2},$ that is
not zero under our hypotheses.

Let $\gamma$ be a limit cycle contained in $\mathcal{Z}_{1}.$ By using the
results of [27] we know that

$h=\int_{0}^{T}\operatorname{div}(\mathcal{X})(x(t),y(t))\,{\rm
d}t=\int_{0}^{T}k_{1}(x(t),y(t))\,{\rm d}t.$

Equivalently

$h=\int_{0}^{T}\big{(}a+bx(t)+cy(t)\big{)}\,{\rm
d}t=\int_{0}^{T}\big{(}a_{1}+b_{1}x(t)+c_{1}y(t)\big{)}\,{\rm d}t.$

The above equalities write as $h=aT+bX+cY=a_{1}T+b_{1}X+c_{1}Y.$ Moreover
since $\gamma\cap\mathcal{Z}_{2}=\emptyset,$ $a_{2}T+b_{2}X+c_{2}Y=0.$ In
short, for $\gamma\subset\mathcal{Z}_{1}$ it holds that

$A\left(\begin{matrix}T\\\ X\\\ Y\end{matrix}\right)=\left(\begin{matrix}h\\\
h\\\ 0\end{matrix}\right).$

Then,

$h=\frac{\det(A)}{\Delta_{1,2}-\Delta_{2}}T\neq 0.$

Similarly, if $\gamma\subset\mathcal{Z}_{2},$

$h=\frac{\det(A)}{\Delta_{1,2}+\Delta_{1}}T\neq 0.$

Hence, under our hypotheses, all possible limit cycles share stability and as
a consequence, at most one limit cycle can surround each of the critical
points. In short, 2 is the maximum number of limit cycles, as we wanted to
prove.

The non existence of limit cycles when the generalized invariant curves do not
contain real isolated points follows from Corollary 1.4. ∎

We remark that the results of Theorem 1.3 and its corollaries could be
extended to polynomial vector fields of degree $n$ having enough different
generalized invariant curves (in fact $n(n+1)/2-1$ curves). It can be proved
that all the limit cycles outside of these curves have the same stability and
an upper bound of how many of them possesses the system only depends on the
number of limit cycles contained in the given curves and on their
distribution. Also, generically, the stabilities of the limit cycles contained
in the generalized invariant curves can be explicitly obtained. Notice also
that the possible configurations of limit cycles are much more complicated
than in the QS case.

##  6\. More results on quadratic systems

In this section we collect a miscellany of results about QS where the common
point is that their proofs are based on the method introduced in Theorem 2.1.

###  6.1. Quadratic systems with an invariant hyperbola

As we have already explained, QS with an invariant hyperbola do not have limit
cycles, see [11]. We include a simple proof based on our approach. We consider
that the hyperbola is $f(x,y)=x^{2}-y^{2}-1=0,$ and we take the normal form
considered in [23],

$\begin{cases}\dot{x}=y(a+bx+cy)+u(x^{2}-y^{2}-1)=P(x,y),\\\
\dot{y}=x(a+bx+cy)+v(x^{2}-y^{2}-1)=Q(x,y).\end{cases}$ (6)

Then its cofactor is $2(ux-vy).$

######  Proposition 6.1.

Quadratic systems with an invariant hyperbola do not have limit cycles.

###### Proof.

We use the same approach and notation that in the proof of Theorem 1.2. It is
not restrictive to consider that the QS is written as in (6). When $u^{2}\neq
v^{2}$ we take in Theorem 2.1,

$g(x,y)=a^{2}(v^{2}-u^{2})^{2}+a(v^{2}-u^{2})(uc+vb)(vx-
uy),\,\,\alpha=\frac{uc+vb}{2(v^{2}-u^{2})}.$

Then

$N(x,y)=a(uc+vb)(ub+vc)(ux-vy)^{2}$

and the result follows for this case.

When $u^{2}=v^{2}=0,$ trivially the QS does not have limit cycles. Finally,
when $v=\sigma u,$ with $\sigma\in\\{+1,-1\\},$ it holds that the straight
line $f_{2}(x,y)=y-\sigma x=0$ is also invariant. Observe that the critical
points of the QS must satisfy

$0=yQ(x,y)-xP(x,y)=\sigma u(y-\sigma x)(x^{2}-y^{2}-1).$

Hence all the critical points lay on invariant algebraic curves, and since any
periodic orbit $\gamma$ must surround one of these points, such a $\gamma$ can
not exist by the theorem of uniqueness of solutions for these differential
systems. ∎

###  6.2. Lotka-Volterra QS

From the classical Bendixson-Dulac theorem, Bautin already proved that QS with
two non parallel invariant straight lines do not have limit cycles, see for
instance [3] or [24, Sec. 1.2]. The point in that proof is to use a Dulac
function of the form $|x|^{\alpha}|y|^{\beta}.$ Here we will present a
different proof by using the extension of Theorem 2.1 given in Remark 2.2.

######  Proposition 6.2.

The Lotka-Volterra QS

$\begin{cases}\dot{x}=x(a_{0}+a_{1}x+a_{2}y),\\\
\dot{y}=y(b_{0}+b_{1}x+b_{2}y)\end{cases}$ (7)

does not have limit cycles.

###### Proof.

We define several quantities associated to (7). Set
$\Delta_{i,j}=a_{i}b_{j}-a_{j}b_{i},$
$D=a_{{0}}a_{{1}}b_{{2}}-a_{{0}}b_{{1}}b_{{2}}-a_{{1}}a_{{2}}b_{{0}}+a_{{1}}b_{{0}}b_{{2}}$
and

$C=-a_{{0}}a_{{1}}b_{2}^{2}+a_{{0}}b_{{1}}b_{2}^{2}+2\,a_{{1}}a_{{2}}b_{{0}}b_{{2}}-a_{{1}}b_{{0}}b_{2}^{2}-a_{{2}}^{2}b_{{0}}b_{{1}}.$

First, let us prove that when $a_{2}b_{2}\Delta_{0,2}\Delta_{1,2}=0$ the
system does not have periodic orbits. When $a_{2}=0$ the first differential
equation does not depend on $y,$ that is, $\dot{x}=x(a_{0}+a_{1}x),$ and hence
the behaviour of $x(t)$ can not be periodic unless it is identically constant.
Something similar happens when $b_{2}=0.$ When $\Delta_{1,2}=0$ system (7)
either does not have critical points outside the axes or it has a full line of
critical poins. In any case, periodic orbits are not possible. Similarly, when
$\Delta_{0,2}=0$ the same happens.

When $a_{2}b_{2}\Delta_{0,2}\Delta_{1,2}\neq 0$ we will apply Remark 2.2 with
$f_{1}(x,y)=x,$ $f_{2}(x,y)=y,$ $k_{1}(x,y)=a_{0}+a_{1}x+a_{2}y,$
$k_{2}(x,y)=b_{0}+b_{1}x+b_{2}y,$

$\displaystyle\alpha_{1}=\frac{b_{2}C}{a_{2}\Delta_{0,2}\Delta_{1,2}},\quad\alpha_{2}=\frac{b_{2}(a_{2}-b_{2})\Delta_{0,1}}{\Delta_{0,2}\Delta_{1,2}}$

and

$g(x,y)=b_{2}C\Delta_{1,2}x-a_{2}b_{2}(a_{2}-b_{2})\Delta_{1,2}\Delta_{0,2}y-C(a_{2}-b_{2})\Delta_{0,2}.$

Straightforward computations give that

$N_{\alpha_{1},\alpha_{2},g}(x,y)=\frac{b_{2}(a_{2}-b_{2})CD}{a_{2}\Delta_{0,2}\Delta_{1,2}}\big{(}\Delta_{1,2}x+\Delta_{0,2}\big{)}^{2}.$

Hence, when $b_{2}(a_{2}-b_{2})CD\neq 0$ the QS does not have periodic orbits.
When $b_{2}(a_{2}-b_{2})CD=0$ it can have periodic orbits but not limit cycles
because it has a smooth first integral. For instance, the classical Lotka-
Volterra system, that corresponds to $a_{1}=b_{2}=0,$ $a_{0}>0,b_{1}>0$ and
$a_{2}<0,b_{0}<0,$ has a global center in the first quadrant. Notice that in
this case $D=0.$ ∎

###  6.3. A QS with an invariant algebraic curve of degree four

In [22] the authors list QS having some classical invariant algebraic curve.
Our method applies to several of them. As an example we consider the QS

$\begin{cases}\dot{x}=-2k_{1}rx+4k_{2}ry+k_{1}x^{2}-8k_{2}xy,\\\
\dot{y}=-3k_{1}ry+3k_{2}x^{2}+2k_{1}xy-16k_{2}y^{2},\end{cases}$ (8)

that has the invariant algebraic curve $f(x,y)=-2rx^{3}+x^{4}+4r^{2}y^{2},$
called curve Antiversiera, with cofactor $k(x,y)=-6k_{1}r+4k_{1}x-32k_{2}y.$

######  Proposition 6.3.

The QS (8) does not have limit cycles.

###### Proof.

By using Theorem 2.1 with $\alpha=-1/4$ and $g(x,y)=r-2x$ we get that
$N_{\alpha,g}(x,y)=\frac{3}{2}k_{1}r^{2}.$ Hence, when $rk_{1}\neq 0$ it does
not have periodic orbits. When $rk_{1}=0$ it can have periodic orbits but not
limit cycles because it has the first integral $g^{4}(x,y)/f(x,y).$ For
instance, when $k_{1}=0$ and $r\neq 0$ the QS has a center. ∎

###  6.4. A QS with an invariant algebraic curve of degree twelve

In [10] the authors prove that the QS

$\begin{cases}\dot{x}=1+x^{2}+xy,\\\
\dot{y}=\dfrac{57}{2}-\dfrac{81}{2}x^{2}+3y^{2},\end{cases}$ (9)

has the invariant algebraic curve

$f(x,y)=-5488\,{y}^{4}-32\,x\left(35125\,{x}^{2}-1029\right){y}^{3}+f_{2}(x){y}^{2}+f_{1}(x)y+f_{0}(x),$

where

$\displaystyle f_{2}(x)$
$\displaystyle=-375000{x}^{6}+5058000{x}^{4}-711288{x}^{2}-98784,$
$\displaystyle f_{1}(x)$
$\displaystyle=8x\left(1953125{x}^{8}+140625{x}^{6}+2932875{x}^{4}-1515789{x}^{2}+40284\right),$
$\displaystyle f_{0}(x)$
$\displaystyle=48828125{x}^{12}-23437500{x}^{10}+41343750{x}^{8}-97906500{x}^{6}$
$\displaystyle\quad+71546517{x}^{4}-7246584{x}^{2}-442368,$

with cofactor $k(x,y)=12(x+y)$ and that it is not Liouvillian integrable. We
show that it does not have periodic orbits.

######  Proposition 6.4.

The quadratic system (9) does not have periodic orbits.

###### Proof.

A detailed study of the curve $f(x,y)=0$ shows that it does not contain closed
ovals, although it contains the isolated points $(1,-2)$ and $(-1,2).$ In
fact, these two points are the unique critical points of the QS, and are foci
with opposite stabilities. To prove that the QS does not have periodic orbits
in $\mathbb{R}^{2}\setminus\\{f(x,y)=0\\}$ we apply Theorem 2.1 with
$g(x,y)=x$ and $\alpha=-1/12.$ Then $N_{\alpha,g}(x,y)\equiv 1$ and the result
follows. ∎

###  6.5. Nonexistence of limit cycles for QS with an invariant parabola

We apply Theorem 2.1 to give a simple test for non existence of limit cycles
for the family of QS with an invariant parabola (4),

$\begin{cases}\dot{x}=a+bx+hy+c(y-x^{2})+exy,\\\
\dot{y}=2x(a+bx+hy)+d(y-x^{2})+2ey^{2}.\end{cases}$

Now, we prove the following result:

######  Proposition 6.5.

Associated to the QS (4), define

$\displaystyle\Delta=\Big{(}$ $\displaystyle
16{a}^{2}{e}^{3}+\big{(}(-8bc-16hb-12cd-8dh)a+(2b-d)^{3}\big{)}e^{2}$
$\displaystyle-2(c+h)\big{(}4c(c+2h)a-8{b}^{2}c+6bcd-4bdh-3c{d}^{2}\big{)}e$
$\displaystyle+8c(c+h)^{2}(bc+dh)\Big{)}\Big{(}(2b-d)e+2c^{2}+2hc\Big{)}e.$

Then if $\Delta\geq 0,$ system (4) does not have periodic orbits.

###### Proof.

Recall that for the above QS the invariant parabola is $f(x,y)=y-x^{2}=0,$ and
its cofactor $k(x,y)=-2cx+2ey+d.$ For shortness we will write $\mathbf{A}$ to
refer to the set of parameters $a,d,c,d,e,h$ and $\mathbf{G}$ to refer to the
set of parameters $g_{0},g_{1},g_{2},\alpha.$ If in Theorem 2.1 we take
$g(x,y)=g_{0}+g_{1}x+g_{2}y,$ it holds that

$N:=N_{\alpha,g}(x,y)=\sum_{0\leq i+j\leq
2}n_{i,j}(\mathbf{A},\mathbf{G})x^{i}y^{j},$

where $n_{i,j}$ are polynomials in their variables. The idea is to search a
condition among the parameters $\mathbf{A}$ in such a way that the free
parameters $\mathbf{G}$ can be written in terms of the parameters
$\mathbf{A},$ $\mathbf{G}=\mathbf{G}(\mathbf{A}),$ and moreover

$N=\sum_{0\leq i+j\leq
2}n_{i,j}(\mathbf{A},\mathbf{G}(\mathbf{A}))x^{i}y^{j}=r\left(\sum_{0\leq
i+j\leq 1}w_{i,j}(\mathbf{A})x^{i}y^{j}\right)^{2},$ (10)

where $w_{i,j}$ are polynomials on the variables $\mathbf{A}$ and
$r\in\mathbb{R}.$

We claim that this condition is $\Delta\geq 0.$ From this claim, the above
equality, and Theorem 2.1, our result follows.

Let us give some hints for the proof of the claim. We skip the details and the
cumbersome expressions. A first observation is that a necessary condition for
the existence of $r$ and the polynomials $w_{i,j}$ in (10) is that the
discriminant ($\operatorname{disc}$) of $N,$ either as a quadratic polynomial
in $x,$ or as a quadratic polynomial in $y,$ are identically zero. For
instance, the condition

$\operatorname{disc}_{y}(N)=s_{0}(\mathbf{A},\mathbf{G})+s_{1}(\mathbf{A},\mathbf{G})x+s_{2}(\mathbf{A},\mathbf{G})x^{2}\equiv
0$ (11)

has to be satisfied. The equation $s_{1}(\mathbf{A},\mathbf{G})=0$ is of the
form

$s_{1}(\mathbf{A},\mathbf{G})=u_{0}(\mathbf{A},\mathbf{G})+u_{1}(\mathbf{A},\mathbf{G})g_{0}=0,$

for some polynomials $u_{0}$ and $u_{1}$ that do not depend on $g_{0}.$ Hence,
when $u_{1}(\mathbf{A},\mathbf{G})\neq 0,$ the value $g_{0}$ can be obtained
from all the other parameters $\mathbf{A}$ and $\mathbf{G}.$ By using this
value of $g_{0}$ we get that the equality (11) writes as

$\operatorname{disc}_{y}(N)=t_{0}(\mathbf{A},\mathbf{G})+t_{2}(\mathbf{A},\mathbf{G})x^{2}\equiv
0,$

where the parameter $g_{0}$ is no more in $\mathbf{G},$ $t_{0}$ is a rational
function, with $u_{1}(\mathbf{A},\mathbf{G})$ in its denominator, and $t_{2}$
a polynomial one. Next, studying the system of equations

$t_{0}(\mathbf{A},\mathbf{G})=0,\quad t_{2}(\mathbf{A},\mathbf{G})=0,$

we conclude that it has solutions if

$(-2be-2{c}^{2}-2ch+ed)g_{1}+(4ae-2cd-2dh)g_{2}=0$

and $\alpha$ satisfies a quadratic polinomial equation of the form

$v_{0}(\mathbf{A})+v_{1}(\mathbf{A})\alpha+v_{2}(\mathbf{A})\alpha^{2}=0.$
(12)

Hence our problem, when $u_{1}(\mathbf{A},\mathbf{G})\neq 0,$ has been
essentially reduced to the existence of a real value $\alpha$ satisfying the
above quadratic equation. Finally, using that if we have a suitable set of
values $g_{0},g_{1}$ and $g_{2},$ for which (10) is satisfied, the same holds
for $\beta g_{0},\beta g_{1}$ and $\beta g_{2},$ for any $\beta\in\mathbb{R},$
we can eliminate the presence of denominators during all the process and the
condition $\operatorname{disc}_{y}(N)\equiv 0$ follows for all the values of
the parameters. The discriminant of (12) with respect to $\alpha$ is precisely
$4((2b-d)e+2c^{2}+2hc)^{2}\Delta,$ and so, a suitable $\alpha$ exists when
$\Delta\geq 0.$ ∎

Notice the non existence result of Proposition 6.5 also provides, through the
birrational transformation used in its proof, conditions for non existence of
limit cycles of some QS with an invariant straight line.

For some cases of QS with an invariant straight line a direct proof of non
existence of limit cycles for some values of the parameters also can be
addressed with our approach. Next results presents an example for the simple
1-parametric subcase,

$\begin{cases}\dot{x}=dx-y+\frac{1}{4}x^{2}+\frac{1}{5}xy-\frac{1}{3}y^{2},\\\
\dot{y}=x(1+y),\end{cases}$ (13)

that has the invariant straight line $f(x,y)=1+y=0$ with cofactor $k(x,y)=x.$
The results of [28, Thm 2’], only apply to QS without critical points on this
invariant line and only one singularity at infinity, the one corresponding to
this line. For our system these hypotheses correspond to
$d\in(1/5-\sqrt{6}/3,1/5+\sqrt{6}/3).$ By using this result and the properties
of the rotated families of vector fields, if $d^{\pm}=(3\pm\sqrt{6})/15,$ it
holds that the system has a limit cycle only when $d\in(0,d^{-}),$ where
$d^{-}\approx 0.0367,$ and it surrounds the origin or when $d\in(d^{+},3/5),$
where $d^{+}\approx 0.3633,$ and it surrounds $(0,-3)$ and always it is unique
and hyperbolic. We prove a much weaker result, but with a simple proof.

######  Proposition 6.6.

System (13), with $d\geq 1/5,$ does not have periodic orbits surrounding the
origin.

###### Proof.

We will apply Theorem 2.1 with $\alpha=0$ and
$g(x,y)=g_{0}(y)+g_{1}(y)x+g_{2}(y)x^{2},$ and a slightly different approach
that in previous cases considered in this paper, for studying when $N_{0,g}$
does not change sign. Notice that with this point of view, in this case there
is no need to introduce the parameter $\alpha$ in the approach because the
term $|f(x,y)|^{\alpha}$ introduced in the proof of Theorem 2.1 reduces to
$(y+1)^{\alpha}$ in the region $y+1>0$ and this function can be thought to be
already contained in the functions $g_{i}(y).$ If we compute $N=N_{0,g}$ we
obtain

$\displaystyle N(x,y)$
$\displaystyle=\Big{(}(1+y)g_{2}^{\prime}(y)+\frac{1}{2}g_{2}(y)\Big{)}x^{3}$
$\displaystyle\quad+\Big{(}(1+y)g_{1}^{\prime}(y)+\frac{1}{4}g_{1}(y)+\Big{(}2d+\frac{2}{5}y\Big{)}g_{2}(y)\Big{)}x^{2}$
$\displaystyle\quad+\Big{(}(1+y)g_{0}^{\prime}(y)+\Big{(}d+\frac{1}{5}y\Big{)}g_{1}(y)-2y\Big{(}1+\frac{1}{3}y\Big{)}g_{2}(y)\Big{)}x$
$\displaystyle\quad-\frac{1}{3}y(y+3)g_{1}(y).$

∎

To impose that $N$ does not depend on $x$ we have to solve three ordinary
differential equations, one for each $g_{i},i=0,1,2.$ By imposing that
$g_{2}(0)=1$ and $g_{1}(0)=0$ we obtain that

$g_{2}(y)=\frac{1}{\sqrt{1+y}},\quad
g_{1}(y)=-\frac{8}{15}\frac{(15d-4)(\sqrt[4]{1+y}-1)+y}{\sqrt{1+y}}.$

We do not need to find explicitly $g_{0},$ but only to know its existence for
all $y+1>0.$ This follows from the differential equation associated to the
coefficient of $x$ in $N.$ By taking these functions

$N(x,y)=-\frac{1}{3}y(y+3)g_{1}(y)=\frac{8}{45}\frac{y+3}{\sqrt{1+y}}yM(y),$

where $M(y)=(15d-4)(\sqrt[4]{1+y}-1)+y.$ Hence the result will follow if we
prove that $yM(y)\geq 0$ for $y\geq-1,$ when $d\geq 1/5.$ This can be seen by
studying the function $M$ and its derivative. Notice that $M(-1)=3-15d\leq 0$
and that at $y\simeq 0,$ $M(y)=\frac{15}{4}dy+O(y^{2}).$

###  6.6. Nonexistence of limit cycles for some QS with an invariant cubic

It is known that there are two families of QS with an invariant cubic and that
have a (unique) limit cycle not contained on this curve, see for instance [6,
40, 44]. We consider one of these families and give conditions on its
parameters to have non existence of periodic orbits.

We take the system

$\begin{cases}\dot{x}=\dfrac{1}{2}(1-x)x-\dfrac{L}{2}x^{2}+\dfrac{1}{2}u(-1+2x+xy),\\\
\dot{y}=-1-y-uy(1+y)+L(1+xy),\end{cases}$ (14)

where $L$ and $u$ are real parameters. It has the invariant algebraic cubic
$f(x,y)=-1+2x+x^{2}y=0,$ with cofactor $k(x,y)=u-x,$ and for some values of
$L$ and $u$ it also has a limit cycle. We prove:

######  Proposition 6.7.

System (14), with $(1-u+Lu)(6-5u+9Lu)\geq 0,$ does not have periodic orbits.

###### Proof.

The proof follows exactly the same steps that the proof of Proposition 6.5,
but the computations are much easier and explicit. By taking in Theorem 2.1

$g(x,y)=-(u\alpha-u+1)(u\alpha-u-3)+8\alpha(Lu-u+1)x+4\alpha u^{2}y,$

with $\alpha$ real and such that

$-{u}^{2}{\alpha}^{2}+2u(6Lu-5u+5)\alpha+4L{u}^{2}-5{u}^{2}+6u-1=0,$

it holds that $N(x,y)=rw^{2}(x,y),$ for some linear polynomial $w.$ The
discriminant with respect to $\alpha$ of the above quadratic equation is
$16u^{2}(1-u+Lu)(6-5u+9Lu)\geq 0,$ and hence the proposition follows. ∎

##  7\. Other families of planar systems

This final section is devoted to present some results on non existence of
limit cycles for other families of polynomial vector fields, most of them
having an invariant algebraic curve. Again we apply Theorem 2.1 and it
includes cubic and Liénard systems.

###  7.1. Some cubic systems

We start with the cubic system

$\begin{cases}\dot{x}=ax+y-(ax^{2}+xy+ay^{2})(x+y)=P(x,y),\\\
\dot{y}=ay-(ax^{2}+xy+ay^{2})(y-x)=Q(x,y),\end{cases}$ (15)

with the invariant algebraic curve $f(x,y)=x^{2}+y^{2}-1$ and cofactor
$k(x,y)=-2(ax^{2}+xy+ay^{2}).$ This system is introduced in [26] and there it
is proved that it has exactly two limit cycles, counted with their own
multiplicities, when $a\in I=(-1/2,(1-\sqrt{2})/2).$ Moreover, one of these
limit cycles is always $f(x,y)=0,$ and it is the only one (and it is double)
when $a=-1/4.$ Furthermore, it is not difficult to prove that for $a\in
J:=(-\infty,(1-\sqrt{2})/2)\cup((1+\sqrt{2})/2,\infty)$ the invariant circle
$f(x,y)=0$ is always a limit cycle.

By using our approach we will show that it does not have periodic orbits
different of $f(x,y)=0,$ for most of the values $a\not\in I.$

######  Proposition 7.1.

System (15) does not have periodic orbits different of $x^{2}+y^{2}-1=0,$ for
$a\leq-1/2$ and for $a\geq 0.$

###### Proof.

Notice first that $x^{2}+y^{2}-1=0$ is a periodic orbit if and only if the
system does not have critical points on it, and this happens if and only if
$4a^{2}-4a-1>0,$ which is equivalent to $a\in J.$

First we apply Theorem 2.1 with $g=1$ and $\alpha=1.$ The
$N_{1,1}(x,y)=k(x,y)=-2(ax^{2}+xy+ay^{2}).$ Hence when $1\leq 4a^{2},$ that is
when $|a|\geq 1/2,$ the result follows.

To prove the non existence of periodic orbits different of $f(x,y)=0,$ when
$a\geq 0$ we take in Theorem 2.1, $\alpha=-1-2a$ and

$g(x,y)=4a\big{(}2a^{2}x^{2}+2axy+(2a^{2}+1)y^{2}\big{)}.$

After several computations we get that

$N_{\alpha,g}(x,y)=16a^{2}(a^{2}x^{2}+2axy+(a^{2}+1)y^{2})+8a(4a^{2}+1)(ax^{2}+xy+ay^{2})^{2}.$

Since the discriminant of $a^{2}x^{2}+2axy+(a^{2}+1)y^{2}$ with respect to $x$
is $-4a^{4}y^{2}\leq 0$ and $a\geq 0,$ it holds that $N_{\alpha,g}(x,y)\geq 0$
and the result follows. ∎

The upper bound for the number of limit cycles of system (15) given in [26] is
obtained by transforming it into an Abel differential equation. Here, we add
some comments of another approach that also allows to study in detail the
exact number of limit cycles of system (15). First we observe that if we
introduce the function

$V(x,y)=(x^{2}+y^{2}-1)(ax^{4}+x^{3}y+2ax^{2}y^{2}+xy^{3}+ay^{4}-y^{2})$

it holds that

$\operatorname{div}\left(\frac{P(x,y)}{V(x,y)},\frac{Q(x,y)}{V(x,y)}\right)=\frac{2(x^{2}+y^{2}-1)^{2}(ax^{2}+xy+ay^{2})^{2}}{V^{2}(x,y)}\geq
0.$

Hence by using the Bendixson-Dulac Theorem, we know that the number of limit
cycles of (15) is controlled by the number of holes of the set
$\mathbb{R}^{2}\setminus\\{V(x,y)=0\\},$ see [25, Sec 3.1] for more details on
the method and also for an application to a similar system. In particular it
can be proved that two is an upper bound of the number of limit cycles for all
values of $a$ and also that limit cycles different of $f(x,y)=0,$ do exist if
and only if $a\in I.$

Our second example is extracted from [34, Sec. 4] and writes as

$\begin{cases}\dot{x}=-2h-y+x^{2}-4xy-y^{2}-\frac{2}{3}x^{3}-3x^{2}y-5xy^{2}+\frac{7}{3}y^{3},\\\
\dot{y}=-h+x+\frac{1}{2}x^{2}+3xy+2y^{2}-\frac{4}{3}x^{3}-3x^{2}y+3xy^{2}+\frac{25}{6}y^{3},\end{cases}$
(16)

with $h\in\mathbb{R}.$ It has the cubic invariant algebraic curve
$f(x,y)=6h-3(x^{2}+y^{2})+2x^{3}-9xy^{2}+2y^{3}=0,$ with cofactor
$k(x,y)=(1-x+2y)(2x+y).$

######  Proposition 7.2.

System (16) does not have periodic orbits.

###### Proof.

By applying Theorem 2.1 with $\alpha=-2$ and $g(x,y)=(1-x+2y)^{2},$ we get
that $N_{\alpha,g}(x,y)=2k^{2}(x,y)\geq 0$ and the proof follows. ∎

###  7.2. Liénard systems

In this section, instead of using invariant algebraic curves we will use
exponential factors, that are a particular case of generalized invariant
curves. More concretely, an exponential factor will be a function
$f(x,y)=\exp(h(x,y)),$ with $h$ a polynomial and such that it holds that
$\dot{f}(x,y)=k(x,y)f(x,y),$ as in (2) with $k$ being also a polynomial,
called its cofactor.

Recall that Theorem 2.1 also holds if the $f$ of its statement is an
exponential factor, instead of an invariant algebraic curve, see Remark 2.3. A
simple example of exponential factor for the Liénard system

$\begin{cases}\dot{x}=y-xH(x),\\\ \dot{y}=-x\end{cases}$ (17)

is $f(x,y)=\exp(y),$ because $\dot{f}(x,y)=-xf(x,y)$ and $k(x,y)=-x.$ We
prove:

######  Proposition 7.3.

If there exists $\alpha\in\mathbb{R}$ such that $\alpha x+2H(x)\not\equiv 0$
does not change sign then system (17) does not have periodic orbits.

###### Proof.

From the above discussion we know that we can apply Theorem 2.1 by taking $f$
as the exponential factor $f(x,y)=\exp(y).$ We also take $g(x,y)=2-2\alpha
y-\alpha^{2}x^{2}.$ Then some computations give that
$N_{\alpha,g}(x,y)=\alpha^{2}x^{2}\big{(}\alpha x+2H(x)\big{)},$ and the
result follows. ∎

Next we will apply the above result to get effective conditions for the non
existence of periodic solutions for two concrete families, that have limit
cycles for some values of the parameters. Concretely, we will consider Liénard
systems of degrees 3 and 5.

######  Corollary 7.4.

Liénard system

$\begin{cases}\dot{x}=y-a_{1}x-a_{2}x^{2}-x^{3},\\\ \dot{y}=-x\end{cases}$

does not have periodic solutions when $a_{1}\geq 0.$

###### Proof.

By using Proposition 7.3 with $\alpha=4\sqrt{a_{1}}-2a_{2}$ and
$H(x)=a_{1}+a_{2}x+x^{2}$ we get that

$\alpha
x+2H(x)=2a_{1}+(\alpha+2a_{2})x+2x^{2}=2\big{(}\sqrt{a_{1}}+x\big{)}^{2}\geq
0,$

and the result follows. ∎

To study the degree 5 case we will use the next result that follows from [38].
In fact, in that paper more elaborated conditions are given to characterize
the number of real and complex roots of quartic polynomials. Next lemma only
presents a couple of useful and well known simple consequences.

######  Lemma 7.5.

For the quartic real equation

$P(x)=ax^{4}+bx^{3}+cx^{2}+dx+e=0,\quad a\neq 0,$

define $\Delta=\operatorname{disc}(P),$ that is

$\displaystyle\Delta=\operatorname{disc}_{x}(P)$
$\displaystyle=256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e-27a^{2}d^{4}$
$\displaystyle\quad+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de+18abcd^{3}+16ac^{4}e$
$\displaystyle\quad-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2},$

and $R=R(P)=3b^{2}-8ac.$ It holds that:

  1. (i)

If $\Delta<0$ then the polynomial $P$ has two real roots and two complex
conjugated real roots.

  2. (ii)

If $\Delta>0$ and $R<0,$ then the polynomial $P$ does not have real roots.

######  Corollary 7.6.

Liénard system

$\begin{cases}\dot{x}=y-a_{1}x-a_{2}x^{2}-a_{3}x^{3}-a_{4}x^{4}-x^{5},\\\
\dot{y}=-x\end{cases}$

does not have periodic solutions when $U\neq 0,$ $V>0$ and $a_{1}W>0,$ where

$\displaystyle U$
$\displaystyle=\big{(}-{a_{{4}}}^{4}+8a_{{3}}{a_{{4}}}^{2}-16{a_{{3}}}^{2}+64a_{{1}})^{2}a_{1},\quad
V=8a_{3}-3a_{4}^{2},$ $\displaystyle W$
$\displaystyle=27a_{{1}}{a_{{4}}}^{4}-{a_{{3}}}^{3}{a_{{4}}}^{2}-108a_{{1}}a_{{3}}{a_{{4}}}^{2}+3{a_{{3}}}^{4}+72a_{{1}}{a_{{3}}}^{2}+432{a_{{1}}}^{2}.$

###### Proof.

For the sake of shortness we will write $\mathbf{A}$ to refer to the set of
parameters $a_{i},i=1,2,3,4.$ By Proposition 7.3 we know that the result will
follow if we can choose parameters $\mathbf{A}$ and a value of $\alpha$ such
that

$P_{\alpha}(x)=2a_{1}+(\alpha+2a_{2})x+2a_{3}x^{2}+2a_{4}x^{3}+2x^{4}\geq 0,$

for all $x\in\mathbb{R}.$ We will study the sign of $P_{\alpha}$ by applying
Lemma 7.5. With this aim, we compute $R=R(\alpha,{\mathbf{A}})=-4V<0$ and

$\Delta(\alpha,{\mathbf{A}})=\operatorname{disc}_{x}\big{(}P_{\alpha}\big{)}=\sum_{j=0}^{4}q_{j}({\mathbf{A}})\alpha^{j}=:Q(\alpha),$

where $q_{4}=-108$ and the $q_{i},i=0,1,2,3,$ are polynomials that we do not
detail. We claim that the inequalities of the statement imply that there exist
$\mathbf{A}$ and $\alpha$ such that $\Delta(\alpha,{\mathbf{A}})>0.$ From this
claim the result follows by item (ii) of the Lemma applied to $P_{\alpha},$
because $\Delta(\alpha,{\mathbf{A}})>0,$ $R<0$ and then $P_{\alpha}(x)>0.$

To prove the claim we apply again the lemma, but in this case to the
polynomial $Q(\alpha)$ and we will use item (i). The claim follows because

$\Delta(Q)=-2^{32}UW^{3}<0$

and hence the polynomial $Q(\alpha)$ has exactly two real simple roots and, as
a consequence, it takes positive (and also negative) values for suitable
values of $\alpha.$ ∎

## Acknowledgements

The first author is partially supported by the Ministerio de Ciencia e
Innovación (PID2019-104658GB-I00 grant), by the grant Severo Ochoa and María
de Maeztu Program for Centers and Units of Excellence in R&D
(CEX2020-001084-M) and also by the Agència de Gestió d’Ajuts Universitaris i
de Recerca (2017 SGR 1617 grant).

## References

  * [1] M. Alberich-Carramiñana, A. Ferragut, J. Llibre, Quadratic planar differential systems with algebraic limit cycles via quadratic plane Cremona maps, Adv. Math. 389 (2021), 107924:1–38.
  * [2] R. Bamón, Quadratic vector fields in the plane have a finite number of limit cycles, Inst. Hautes Études Sci. Publ. Math. 64 (1986), 111–142.
  * [3] N. N. Bautin, On periodic solutions of a system of differential equations. (in Russian), Akad. Nauk SSSR. Prikl. Mat. Meh. 18 (1954), 128.
  * [4] M. M. Carnicer, The Poincaré problem in the nondicritical case, Ann. Math. (2) 140 (1994) 289–294.
  * [5] J. Chavarriga, I. A. García, Existence of limit cycles for real quadratic differential systems with an invariant cubic, Pacific J. Math. 223 (2006), 201–218.
  * [6] J. Chavarriga, I. A. García, Some criteria for the existence of limit cycles for quadratic vector fields, J. Math. Anal. Appl. 282 (2003), 296–304.
  * [7] J. Chavarriga, I. A. García, J. Sorolla, Non-nested configuration of algebraic limit cycles in quadratic systems, J. Differential Equations 225 (2006), 513-–527.
  * [8] J. Chavarriga, H. Giacomini, J. Llibre, Uniqueness of algebraic limit cycles for quadratic systems, J. Math. Anal. Appl. 261 (2001), 85–99.
  * [9] J. Chavarriga, M. Grau, A family of non-Darboux-integrable quadratic polynomial differential systems with algebraic solutions of arbitrarily high degree, Appl. Math. Lett. 16 (2003) 833–837.
  * [10] J. Chavarriga, J. Llibre, J. Sorolla, Algebraic limit cycles of degree 4 for quadratic systems, J. Differential Equations 200 (2004), 206–244.
  * [11] L. A. Cherkas, Methods for estimating the number of limit cycles of autonomous systems (in Russian), Diff. Uravneniya 13 (1977), 779–802, translated in Differential Equations 13 (1978), 529–547.
  * [12] L. A. Cherkas, L. I. Zhilevich, Some tests for the absence or uniqueness of limit cycles (in Russian), Diff. Uravneniya 6 (1970), 1170–1178, translated in Differential Equations 6 (1970), 891–897.
  * [13] L. A. Cherkas, L. I. Zhilevich, The limit cycles of certain differential equations (in Russian), Diff. Uravneniya 8 (1972), 1207–1213, translated in Differential Equations 8 (1972), 924–929.
  * [14] C. J. Christopher, Quadratic systems having a parabola as an integral curve, Porceedings of the Royal Society of Edinburg 112A (1989), 113–134.
  * [15] C. Christopher, J. Llibre, A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals, Proc. Am. Math. Soc. 130 (2002), 2025–2030.
  * [16] C. J. Christopher, J. Llibre, G. Świrszcz, Invariant algebraic curves of large degree for quadratic systems, J. Math. Anal. Appl. 303(2) (2005) 450–461.
  * [17] B. Coll, J. Llibre, Limit cycles for a quadratic system with an invariant straight line and some evolution of phase portraits, Colloquia Mathematica Societatis Janos Bolyai 53, Qualitative Theory of Differential Equations, Bolyai Institut, Szeged, Hungary (1988), 111–123.
  * [18] W. A. Coppel, W. A., A survey of quadratic systems, J. Differential Equations 2 (1966), 293–304.
  * [19] W. A. Coppel, Some quadratic systems with at most one limit cycle, in: Dynam. Report. Ser. Dynam. Systems Appl., Vol. 2, Wiley, Chichester, 1989, pp. 61–88.
  * [20] V. F. Filipstov, Algebraic limit cycles (in Rusian), Diff. Uravneniya 9 (1973), 1281–1288, translated in Differential Equations 9 (1973), 983–988.
  * [21] I. A. García, J. Giné, Non-algebraic invariant curves for polynomial planar vector fields, Discrete Contin. Dyn. Syst. 10 (2004), 755–768.
  * [22] I. A. García, J. Llibre, Classical planar algebraic curves realizable by quadratic polynomial differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750141 (12 pages).
  * [23] A. Gasull, On polynomial systems with invariant algebraic curves. International Conference on Differential Equations, Vol. 1, 2 (Barcelona, 1991), 531–537, World Sci. Publ., River Edge, NJ, 1993.
  * [24] A. Gasull, H. Giacomini, Some applications of the extended Bendixson-Dulac theorem. In Progress and Challenges in Dynamical Systems, 233–252. Springer, 2013.
  * [25] A. Gasull, H. Giacomini, Effectiveness of the Bendixson-Dulac theorem J. Differential Equations 305 (2021), 347–367.
  * [26] A. Gasull, J. Llibre, J. Sotomayor, Limit cycles of vector fields of the form $X(v)=Av+f(v)Bv$, J. Differential Equations 67 (1987), 90–110.
  * [27] H. Giacomini, M. Grau, On the stability of limit cycles for planar differential systems, J. Differential Equations 213 (2005), 368–388.
  * [28] C. A. Holmes, Some quadratic systems with a separatrix cycle surrounding a limit cycle, J. Lond. Math. Soc., II. Ser. 37 (1988) 545–551.
  * [29] J. Llibre, Open problems on the algebraic limit cycles of planar polynomial vector fields, Bul. Acad. Ştiinţe Repub. Mold. Mat. 56 (2008), 19–26.
  * [30] J. Llibre, G. Świrszcz, Relationships between limit cycles and algebraic invariant curves for quadratic systems, J. Differential Equations 229 (2006), 529–537.
  * [31] J. Llibre, G. Świrszcz, Classification of quadratic systems admitting the existence of an algebraic limit cycle, Bull. Sci. Math. 131 (2007), 405–421.
  * [32] J. Llibre, C. Valls, Algebraic limit cycles on quadratic polynomial differential systems, Proc. Edinb. Math. Soc., II. Ser. 61 (2018) 499–512.
  * [33] J. Llibre, C. Valls, Algebraic limit cycles for quadratic polynomial differential systems, Discrete Contin. Dyn. Syst., Ser. B 23 (2018), 2475–2485.
  * [34] J. Llibre, Y. Zhao, Algebraic limit cycles in polynomial systems of differential equations, J. Phys. A 40 (2007), 14207–14222.
  * [35] J. Moulin Ollagnier, About a conjecture on quadratic vector fields, J. Pure Appl. Algebra 165 (2001), 227–234.
  * [36] Z. Pingguang, On the distribution and number of limit cycles for quadratic systems with two foci, Qual. Theory Dyn. Syst. 3 (2002), 437–463.
  * [37] Y.-X. Qin, On the algebraic limit cycles of second degree of the differential equation $dy/dx=\sum_{0\leq i+j\leq 2}a_{ij}x_{i}y_{j}/\sum_{0\leq i+j\leq 2}b_{ij}x_{i}y_{j}$, Acta Math. Sinica 8 (1958), 23–35.
  * [38] E. L. Rees, Graphical Discussion of the Roots of a Quartic Equation, Amer. Math. Monthly 29 (1922), 51–55.
  * [39] G. S. Rychkov, The limit cycles of the equation $u(x+1)du=(-x+ax^{2}+bxu+cu+du^{2})dx$ (in Russian), Diff. Uravneniya 8 (1972), 2257–2259, translated in Differential Equations 8 (1972), 1748–1750.
  * [40] S. Shui, A planar quadratic system with an invariant cubic curve has at most one limit cycle (in Chinese), Acta Math. Appl. Sinica 24 (2001), 590–595.
  * [41] A. I. Yablonskii, On the limit cycles of a certain differential equation (in Russian), Diff. Uravneniya 2 (1966), 335–344, translated in Differential Equations 2 (1966), 164–168.
  * [42] Yan Qian Ye; et alt., Theory of limit cycles. Second edition. Translations of Mathematical Monographs, 66. American Mathematical Society, Providence, RI, 1986.
  * [43] A. Zegeling, R. E. Kooij, Uniqueness of limit cycles in polynomial systems with algebraic invariants, Bull. Austral. Math. Soc. 49 (1994), 7–20.
  * [44] A. Zegeling, R. E. Kooij, Quadratic systems with a symmetrical solution, Electron. J. Qual. Theory Differ. Equ. 2018 Paper No. 32, 18 pp.