Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems

Abstract

In this study, we focus on invariant algebraic curves of generalized Liénard polynomial differential systems $x^{\prime }=y$, $y^{\prime }=-f_m\left(x\right)y-g_n\left(x\right)$, where the degrees of the polynomials f and g are m and n, respectively, and we correct some results previously stated.

1. Introduction and Statement of the Main Results

In this work, we study the generalized Liénard polynomial differential systems of the following form:

$$x^{\prime }=y,y^{\prime }=-f_m\left(x\right)y-g_n\left(x\right),$$

(1)

where the degrees of the polynomials f and g are given by the subscripts m and n, respectively. These generalized Liénard systems are used to model different problems in numerous areas of knowledge and have been intensively studied in the last decades (see for instance [1,2] and references therein).

Consider $F(x,y)=0$ an invariant algebraic curve of the differential system (1) where $F(x,y)$ is a polynomial, then there exists a polynomial $K(x,y)$ such that the following is the case.

$$\frac{\partial F}{\partial x}y+\frac{\partial F}{\partial y}(-f_m\left(x\right)y-g_n\left(x\right))=KF$$

(2)

The knowledge of the algebraic curves of system (1) allows studying modern Darboux and Liouvillian theories of integrability (see [3] and references therein). In fact the existence of invariant algebraic curves is a measure of integrability in such theories. Another problem is finding a bound on the degree of irreducible invariant algebraic curves of system (1). This problem goes back to Poincaré for any differential system and is known as Poincaré problem.

In 1996, Hayashi [4] stated the following result.

Theorem 1.

The generalized Liénard polynomial differential system (1) with $f_m\not\equiv 0$ and $m+1\ge n$ has an invariant algebraic curve if and only if there is an invariant curve $y-P\left(x\right)=0$ satisfying $g_n\left(x\right)=-(f_m\left(x\right)+P{}^{\prime }\left(x\right))P\left(x\right)$, where $P\left(x\right)$ or $P\left(x\right)+F\left(x\right)$ is a polynomial with a degree of at most one, such that $F\left(x\right)={\int }_0^{x}f\left(s\right)ds$.

Given P and Q polynomials, an algebraic curve of the form ${(y+P\left(x\right))}^2-Q\left(x\right)=0$ is called hyperelliptic curve (see for instance [5,6,7,8]). In such works, hyperelliptic curves are used to determine the algebraic limit cycles of generalized Liénard systems (1).

Theorem 1 is also announced in [9], where the author seems to not be aware that the theorem is false. Theorem 1 is not correct as the following proposition shows. More precisely, it shows the existence of hyperelliptic invariant algebraic curves for generalized Liénard systems (1).

Proposition 1.

Under the assumptions of Theorem 1, the generalized Liénard polynomial differential system (1) has the following hyperelliptic invariant algebraic curves:

(a)

$F(x,y)=-(b+ax)\lambda +{(y-b-ax)}^2=0$ for $f_0\left(x\right)=-3a/2$ and $g_1\left(x\right)=a(b+ax-\lambda )/2$ with $a\ne 0$;

(b)

$F(x,y)=-A{x}^2+{(y-ax)}^2$ for $f_0\left(x\right)=-2a$ and $g_1\left(x\right)=(a^2-A)x$ with $aA\ne 0$;

(c)

$F(x,y)=-bc/\left(2a\right)-cx-ac{x}^2/\left(2b\right)+{(b+ax-y)}^2=0$ for $f_0\left(x\right)=-2a$ and $g_1\left(x\right)=(2ab-c)(b+ax)/\left(2b\right)$ with $ab\ne 0$.

Proposition 1 is proved in Section 2.

In fact, the correct statement of Theorem 1 is the following.

Theorem 2.

The generalized Liénard polynomial differential system (1) with $f_m\not\equiv 0$ and $m+1\ge n$ has the invariant algebraic curve $y-P\left(x\right)=0$ if $g_n\left(x\right)=-(f_m\left(x\right)+P{}^{\prime }\left(x\right))P\left(x\right)$, being $P\left(x\right)$ or $P\left(x\right)+F\left(x\right)$ a polynomial of degree at most one, where ${\displaystyle F\left(x\right)={\int }_0^{x}f\left(s\right)ds}$.

Theorem 2 is proved in Section 2.

Note that the mistake in the statement of Theorem 1 is the claim that unique invariant algebraic curves are of the following form $y-P\left(x\right)=0$.

Demina in [10] also detected that Theorem 1 was not correct. She found counterexamples to Theorem 1 with invariant algebraic curves of degree 2 and 3 in the variable y.

Singer in [11] found the characterization of systems that are Liouvillian integrable. Christopher [12] rewrote this result stating that if a polynomial differential system in ${\mathbb{R}}^2$ has an inverse integrating factor of the following form:

$$V=exp\left(\frac{D}{E}\right)\prod _{i=1}^p{F}_i^{{\alpha }_i},$$

(3)

where D, E and $F_i$ are polynomials in $\mathbb{C}[x,y]$ and ${\alpha }_i\in \mathbb{C}$, then this differential system is Liouvillian integrable. For a definition of (inverse) integrating factor, see for instance Section 8.3 of [3].

We say that $exp(g/h)$, with g and $h\in \mathbb{C}[x,y]$, is an exponential factor of the polynomial differential system (1) if there exists a polynomial $L(x,y)$ of a degree with at most d where $d=max\{m,n-1\}$ such that the following is the case.

$$\frac{\partial exp(g/h)}{\partial x}y+\frac{\partial exp(g/h)}{\partial y}(-f_m\left(x\right)y-g_n\left(x\right))=Kexp(g/h).$$

More information on exponential factors can be found in Section 8.5 of [3].

The existence of an inverse integrating factor (3) for a polyomial differential system in ${\mathbb{R}}^2$ is equivalent to the existence of ${\lambda }_i$ and ${\mu }_i\in \mathbb{C}$ is not all zero such that ${\sum }_{i=1}^p{\lambda }_i{K}_i+{\sum }_{j=1}^q{\mu }_j{L}_j=\mathrm{div}(P,Q)$, where $K_i$ and $L_i$ are the cofactors of some invariant algebraic curves and exponential factors of the given polynomial differential system, respectively. See, for more details, statement (iv) of Theorem 8.7 of [3].

We remark that the two kinds of invariant algebraic curves mentioned in Theorem 2 can appear simultaneously in some generalized Liénard polynomial differential systems (1) as the following example shows, which already appeared in [13].

The generalized polynomial Liénard differential system of the following:

$$x^{\prime }=y,y^{\prime }=-e{x}^3-e^2/3x-(3x^2+4e/3)y,$$

(4)

has invariant algebraic curves $f_1=y+ex/3=0$ and $f_2=y+x^3+ex/3=0$. Moreover, system (4) is Liouvillian integrable because it has an inverse integrating factor $V=f_1{f}_2^{1/3}$.

Let U be an open subset of ${\mathbb{R}}^2$. A $C^1$ function $H:U\to \mathbb{R}$ is a first integral of system (1) if it is constant on the orbits of the system contained in U, or equivalently if the following is the case.

$$\frac{dH}{dt}=\frac{\partial H}{\partial x}y+\frac{\partial H}{\partial y}(-f_m\left(x\right)y-g_n\left(x\right))=0on U.$$

(5)

Consider W as an open subset of ${\mathbb{R}}^2\times R$. A $C^1$ function $I:W\to \mathbb{R}$ is a Darboux invariant of system (1) if it is constant on the orbits of the system contained in W, or equivalently if the following is the case.

$$\frac{dI}{dt}+\frac{\partial I}{\partial x}y+\frac{\partial I}{\partial y}(-f_m\left(x\right)y-g_n\left(x\right))=0on W.$$

(6)

Moreover, given ${\lambda }_i$ and ${\mu }_i\in \mathbb{C}$ that is not all zero such that ${\sum }_{i=1}^p{\lambda }_i{K}_i+{\sum }_{j=1}^q{\mu }_j{L}_j=-s$ for some $s\in \mathbb{C}\setminus \left\{0\right\}$, then the (multivalued) function of the following:

$$I=\prod _{i=1}^p{F}_i^{{\alpha }_i}\prod _{j=1}^q{\left(exp\left(\frac{g_j}{h_j}\right)\right)}^{{\mu }_j}exp\left(st\right)$$

(7)

is a Darboux invariant of the differential system (see for more details statement (vi) of Theorem 8.7 of [3]).

Under the assumptions of Theorem 2, there are generalized Liénard polynomial differential systems (1) that are Liouvillian integrable, as it is shown in the next result.

Proposition 2.

Under the assumptions of Theorem 2, if the generalized Liénard polynomial differential system (1) has an invariant algebraic curve $y-P\left(x\right)=0$, then the following statements hold:

(a)

If $P\left(x\right)=-F\left(x\right)+ax+b$, then system (1) has the Darboux invariant $(y-P\left(x\right))e^{at}$;

(b)

If $P\left(x\right)=b$, then system (1) is Liouvillian integrable with the first integral $H=e^{y+F\left(x\right)}{(y-b)}^b$ if $b\ne 0$, and the first integral $H=y+F\left(x\right)$ if $b=0$.

Proposition 2 is proved in Section 2.

We note that Proposition 2 shows that Theorem 2 of [13] and Theorem 4 of [14] are not correct because their proofs are based on the incorrect Theorem 1.

Proposition 3.

Consider the generalized Liénard polynomial differential system (1). Let $P\left(x\right)$ be a polynomial, then $y-P\left(x\right)=0$ is an invariant algebraic curve of system (1) if and only if $g_n\left(x\right)=-(f_m\left(x\right)+P{}^{\prime }\left(x\right))P\left(x\right)$.

Proposition 3 is proved in Section 2. In fact, the statement of Proposition 3 already appears in [8] without proof.

Note that, in Proposition 3, there are no restrictions on the degrees of the polynomials $f_m$, $g_n$ and $P\left(x\right)$.

The Liouvillian integrability of the generalized Liénard polynomial differential system has been studied by several authors. The main result of [15] is that under restriction $2\le n\le m$, system (1) has a Liouvillian first integral if and only if $g_n\left(x\right)=a{f}_m\left(x\right)$, where $a\in \mathbb{C}$ (see also [16] for a shorter proof). Later on, the Liouvillian integrability of differential systems (1) having hyperelliptic curves of the form ${(y+Q\left(x\right)P\left(x\right))}^2-Q{\left(x\right)}^2=0$ was studied (see [17]).

In summary, the Liouvillian integrability in the case where $n>m$ is still open. In fact, the characterization of the invariant algebraic curves of system (1) for this case is not complete. Recently, cases $m=1$ and $n=2$ have been solved (see [18]).

Case $n=m+1$ is the still the objective of several recent works. Thus, for instance in [10,19], some particular cases for $m=2$ and $n=3$ have been solved.

2. Proofs

Proof of Proposition 1.

Assume that system (1) has an hyperelliptic invariant curve $F={(y+P\left(x\right))}^2-Q\left(x\right)=0$. Then, from (2), denoting by $K=K(x,y)$ the cofactor of $F=0$, we obtain the following.

$$\begin{array}{cc}& 2g_n\left(x\right)P\left(x\right)+K(-P{\left(x\right)}^2+Q\left(x\right))\hfill \\ & +y(-2g_n\left(x\right)+2P\left(x\right)(K+f_m\left(x\right)+P{}^{\prime }\left(x\right))-Q{}^{\prime }\left(x\right))\hfill \\ & -y^2(K+2f_m\left(x\right)+2P{}^{\prime }\left(x\right))=0.\hfill \end{array}$$

From this equality, we observe that $K=K\left(x\right)=-2(f_m\left(x\right)+P{}^{\prime }\left(x\right))$:

$$f_m\left(x\right)=-P{}^{\prime }\left(x\right)-\frac{P\left(x\right)Q{}^{\prime }\left(x\right)}{2Q\left(x\right)},and{g}_n\left(x\right)=-\frac{1}{2}Q{}^{\prime }\left(x\right)+\frac{P^2\left(x\right)Q{}^{\prime }\left(x\right)}{2Q\left(x\right)},$$

where $fm\left(x\right)$ and $g_n\left(x\right)$ must be polynomials.

If we assume that $degP=p$ and $degQ=q$, we obtain $deg{f}_m=p-1$ and $deg{g}_n=max\{q-1,p^2-1\}$. Since $m+1\ge n$, we obtain $p\ge max\{q-1,p^2-1\}$, which implies $p=1$. Consequently, $1\ge q-1$, which implies $q=1,2$.

If $q=1$, then $P\left(x\right)=ax+b$ with $a\ne 0$ and $Q\left(x\right)$ must be proportional to $P\left(x\right)$; that is, $Q\left(x\right)=\lambda P\left(x\right)$. Thus, $f_m=-3a/2$ and $g_n=a(ax+b-\lambda )/2$, and $F={(b+ax-y)}^2-(b+ax)\lambda$. Thus, statement (a) follows.

If $q=2$, then we have $P\left(x\right)=ax+b$ and $Q\left(x\right)=A{x}^2+Bx+C$ with $aA\ne 0$, and since $f_m$ must be a polynomial, we obtain $f_m=-2a$; moreover, in order for $g_m$ to be a polynomial, we obtain either $b=B=C=0$ or $A=aB/\left(2b\right)$ and $C=\left(bB\right)/\left(2a\right)$ with $ab\ne 0$.

If $b=B=C=0$, then $g_n=(a^2-A)x$ and $F=-A{x}^2+{(y-ax)}^2$. Therefore, statement (b) is proven.

If $A=aB/\left(2b\right)$ and $C=\left(bB\right)/\left(2a\right)$, then $g\left(x\right)=(2ab-B)(b+ax)/\left(2b\right)$ and $F(x,y)=-bB/\left(2a\right)-Bx-aB{x}^2/\left(2b\right)+{(b+ax-y)}^2$. By renaming B by c, we obtain statement (c). □

Proof of Proposition 2.

We have system (1) with $g_n\left(x\right)=-(f_m\left(x\right)+P^{\prime }\left(x\right))P\left(x\right)$ and the invariant algebraic curve $y-P\left(x\right)=0$ is $P\left(x\right)=-F\left(x\right)+ax+b$. Then, by using Equation (2), we obtain the result where the cofactor of the invariant algebraic curve $y-P\left(x\right)=0$ is $K=-a$. Consequently, system (1) has the Darboux invariant (7), which in our case becomes $I=(y-P\left(x\right))e^{at}$. Hence, statement (a) is proved. Assume now that $P\left(x\right)=b$. Therefore, $g\left(x\right)=-bf\left(x\right)$, and the differential system becomes $\dot{x}=y$ and $\dot{x}=-(y+b)f\left(x\right)$, which has the Darboux first integral $H=e^{y+F\left(x\right)}{(y-b)}^b$ if $b\ne 0$, and the Darboux first integral $H=y+F\left(x\right)$ if $b=0$, as it is easy to verify using (5). □

Proof of Proposition 3.

First, we suppose that $g_n\left(x\right)=-(f_m\left(x\right)+P{}^{\prime }\left(x\right))P\left(x\right)$, and we shall prove that $y-P\left(x\right)=0$ is an invariant algebraic curve. From Equation (7), we have the following.

$$-P{}^{\prime }\left(x\right)y-y{f}_m\left(x\right)-g_n\left(x\right)=K(y-P\left(x\right)).$$

By substituting $g_n\left(x\right)$, we obtain the following.

$$-P{}^{\prime }\left(x\right)(y-P\left(x\right))-f_m\left(x\right)(y-P\left(x\right))=K(y-P\left(x\right)).$$

Dividing the previous equality by $y-P\left(x\right)$, we obtain $K=-P{}^{\prime }\left(x\right)-f_m\left(x\right)$, which is a cofactor of degree $p-1+m$ of system (1). Note that the degree of the polynomial Liénard differential system is the degree of, i.e., the maximum of $m+p$ and $2p-1$.

Now, we assume that $y-P\left(x\right)$ is an invariant algebraic curve of system (1) with cofactor K. Then, from (7), we obtain

$$-P{}^{\prime }\left(x\right)y-y{f}_m\left(x\right)-g_n\left(x\right)=K(y-P\left(x\right)).$$

From this equality, we obtain $K=K\left(x\right)$; then, we have

$$(K\left(x\right)+f_m\left(x\right)+P^{\prime }\left(x\right))y=K\left(x\right)P\left(x\right)-g_n\left(x\right).$$

Therefore, $K\left(x\right)=-(f_m\left(x\right)+P^{\prime }\left(x\right))$ and $g_n\left(x\right)=K\left(x\right)P\left(x\right)$. Hence, $g_n\left(x\right)=-(f_m\left(x\right)$$+P{}^{\prime }\left(x\right))P\left(x\right)$, and the proposition is proved. □

Proof of Theorem 2.

By Proposition 3, we only need to prove that $P\left(x\right)$ or $P\left(x\right)+F\left(x\right)$ are polynomials with a degree of at most one. Since $m+1\ge n$ and n are the maxima of $m+p$ and $2p-1$ where p is the degree of the polynomial $P\left(x\right)$, we have $m+1\ge m+p$; consequently, $p\le 1$, and the theorem is proved. □