Rational Involutions and an Application to Planar Systems of ODE

Abstract

An involution refers to a function that acts as its own inverse. In this paper, our focus lies on exploring two-dimensional involutive maps defined by rational functions. These functions have denominators represented by polynomials of degree one and numerators by polynomials of a degree of, at most, two, depending on parameters. We identify the sets in the parameter space of the maps that correspond to involutions. The investigation relies on leveraging algorithms from computational commutative algebra based on the Groebner basis theory. To expedite the computations, we employ modular arithmetic. Furthermore, we showcase how involution can serve as a valuable tool for identifying reversible and integrable systems within families of planar polynomial ordinary differential equations.

1. Introduction

Let $\mathbb{K}$ be either $\mathbb{R}$ or $\mathbb{C}$ and $\mathbf{x}$ be a vector from ${\mathbb{K}}^n$. An automorphism $\phi \left(\mathbf{x}\right)$ defined on the non-empty domain D in ${\mathbb{K}}^n$ is an involution on D, if

$$\phi \left(\phi \left(\mathbf{x}\right)\right)={\phi }^2\left(\mathbf{x}\right)=\mathbf{x},$$

for all $\mathbf{x}\in D$. Clearly, if the above equality holds, then $\phi \left(\mathbf{x}\right)={\phi }^{-1}\left(\mathbf{x}\right)$. If the coordinate functions of the map $\phi \left(\mathbf{x}\right)$ are rational functions; that is, they can be expressed as a ratio of two polynomials, we are speaking about rational involutions.

Involutions are valuable tools in the study of ordinary differential equations due to their ability to analyze the structure of the equations and provide insights into the behavior of solutions. They play an important role in reducing complexity, finding solutions, and uncovering underlying symmetries, making them essential for researchers in the field of ordinary differential equations (ODEs) and for practitioners (see, e.g., [1] and the references therein). Furthermore, they can help to identify invariant solutions of ODEs—solutions that remain unchanged under certain transformations. Invariant solutions can reveal crucial information about the behavior of ODEs and their geometric properties [2,3]. Involutions are closely related to Lie symmetry, a powerful method for studying ODEs. They are often used to construct these symmetries, which can then be employed to find solutions, conservation laws, and other properties of ODEs (see, e.g., [4]).

In this paper, we first consider a family of rational maps, where both the numerators and denominators are polynomials of degree one. More precisely, let

$$L_1(x,y)=a_{4}x+a_{5}y+a_6,L_2(x,y)=b_{4}y+b_{5}x+b_6,$$

and $\mathcal{L}$ be the set

$$\mathcal{L}=\{(x,y):L_1(x,y)=0\mathrm{or}{L}_2(x,y)=0\}.$$

When $\mathbb{K}=\mathbb{R}$, the rational involutions ${\phi }_{\lambda }:{\mathbb{K}}^2\setminus \mathcal{L}\to {\mathbb{K}}^2\setminus \mathcal{L}$ of the form

$${\phi }_{\lambda }(x,y)=\left(\frac{a_{1}x+a_{2}y+a_3}{a_{4}x+a_{5}y+a_6},\frac{b_{1}y+b_{2}x+b_3}{b_{4}y+b_{5}x+b_6}\right),$$

(1)

where the parameters $a_1,\dots ,a_6,b_1,\dots ,b_6$ are real numbers and $\lambda =(a_1,\dots ,a_6,b_1,\dots ,b_6)$ were studied in [5]. In Theorems 1 and 2 of [5], the authors found all involutions in family (1), in the case where at least one of the coordinate functions of (1) is a polynomial, and in Theorem 3, they claim to have given the list of all involutions in family (1), in the case where $a_4{a}_5{b}_4{b}_5\ne 0.$

It is claimed in Theorem 3 of [5] that the map ${\phi }_{\lambda }$ is an involution if and only if $\lambda \in S_1\cup S_2$, where

$$\begin{gathered} S_1=\{\lambda \in {\mathbb{R}}^{12}\mid a_3=\frac{a_2(a_2{a}_4+a_5{a}_6-a_1{a}_5)}{a_5^2},b_1=\frac{b_4(a_2{a}_4+a_5{a}_6)}{a_5^2},b_2=-\frac{a_4{b}_4(a_1+a_6)}{a_5^2},b_3= \\ \frac{b_4(a_5(a_6^2-a_1^2)+a_2{a}_4(a_1+a_6))}{a_5^3},b_5=\frac{a_4{b}_4}{a_5},b_6=\frac{a_6{b}_4}{a_5}\}, \end{gathered}$$

$$\begin{gathered} S_2=\{\lambda \in {\mathbb{R}}^{12}\mid a_1=\frac{a_2{a}_4}{a_5},a_3=\frac{a_2{a}_6}{a_5},b_1=\frac{b_4(a_2{a}_4+a_5{a}_6)}{a_5^2},b_2=-\frac{b_5(a_4{a}_2+a_6{a}_5)}{a_5^2},b_3= \\ \frac{(a_4{a}_2+a_6{a}_5)(a_2{b}_5{a}_5-a_4{a}_2{b}_4-a_6{a}_5{b}_4)}{a_5^4},b_6=-\frac{a_2{b}_5{a}_5-a_4{a}_2{b}_4-a_6{a}_5{b}_4}{a_5^2}\}. \end{gathered}$$

Theorems 1 and 2 of [5] are correct, but Theorem 3 does not hold. Direct calculations show that substituting either condition $S_1$ or $S_2$ into (1) results in a map, which is not an involution. The main aim of our work is to provide the true set of involutive maps in family (1). Using an approach that is different from the one used in [5], we first find all involutions in family (1) defined on ${\mathbb{R}}^2$ with $\lambda \in {\mathbb{R}}^{12}$, correcting the result of [5]. Then, we search for involutions of form (1) defined on ${\mathbb{C}}^2$ for $\lambda \in {\mathbb{C}}^{12}$.

In Section 4, we consider the family of rational maps

$$\phi (x,y)=\left(\frac{a_{1}x+a_{2}y+a_3{x}^2+a_{4}xy+a_5{y}^2}{1+b_{1}x+b_{2}y},\frac{c_{1}x+c_{2}y+c_3{x}^2+c_{4}xy+c_5{y}^2}{1+d_{1}x+d_{2}y}\right)$$

(2)

and find involutive maps in the family.

We also provide an application of the found involutions to cubic planar differential systems. Such systems, which are two-dimensional systems with polynomials of degree three on the right-hand side, play a significant role in the qualitative theory of ordinary differential equations. They often exhibit rich and diverse dynamical behavior, making them valuable for studying the qualitative aspects of ODEs. Two important open problems in the studies of cubic systems are finding all analytically integrable systems (see, e.g., [6,7,8] and the references given there) in the full family of cubic systems and determining an upper bound for the number of limit cycles in the family. The latter problem is an important part of the still unresolved Hilbert’s 16th problem (see, e.g., [9,10,11]). These two problems are closely related since most methods for estimating the number of limit cycles rely on perturbations of integrable systems. In Section 5, we demonstrate how rational involutions can be utilized to identify integrable systems in families of planar polynomial systems of ODEs and find two new families of analytically integrable planar cubic systems.

The novelty of our paper lies in introducing an efficient computational approach based on algorithms of commutative algebra for determining involutions in parametric families of rational maps, finding new families of rational involutions, and correcting the results of the rational involutions presented in [5]. Using rational involution as a tool to identify time-reversible systems, our work also provides a novel approach to tackle the problem of integrability for polynomial systems of ODEs.

2. Preliminaries

We will reduce our search for involutions to finding the set of solutions to systems of polynomials, so we recall some basics on polynomial ideals and their varieties (see, e.g., [8,12] for more details).

Let $F=\{u_1(z_1,\dots ,z_n),\dots ,u_m(x_1,\dots ,z_n)\}$ be a finite set of polynomials with coefficients in the field $\mathbb{K}$. The ideal, I, of the polynomial ring $\mathbb{K}[z_1,\dots ,z_n]$ generated by the set F is denoted $⟨u_1(z_1,\dots ,z_n),\dots ,u_m(z_1,\dots ,z_n)⟩$. It is defined as the set of all possible linear combinations of polynomials $\{u_1(z_1,\dots ,z_n),\dots ,u_m(z_1,\dots ,z_n)\}$ with the coefficients being polynomials of $\mathbb{K}[z_1,\dots ,z_n]$,

$$\begin{array}{c}I=\{u_1(z_1,\dots ,z_n)v_1(z_1,\dots ,z_n)+\dots \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+u_m(z_1,\dots ,z_n)v_m(z_1,\dots ,z_n):v_1,\dots ,v_m\in \mathbb{K}[z_1,\dots ,z_n]\}.\end{array}$$

The set, F, is called the basis of I.

The variety $\mathbf{V}\left(I\right)$ of the ideal, I, is the set of all common zeros of polynomials of I,

$$\mathbf{V}\left(I\right)=\{(\alpha ,...,{\alpha }_n)\in {\mathbb{K}}^n:f({\alpha }_1,...,{\alpha }_n)=0\mathrm{for}\mathrm{all}f\in I\}.$$

Clearly, $\mathbf{V}\left(I\right)$ is the same as the set of solutions to the system

$$u_1(z_1,\dots ,z_n)=\cdots =u_m(z_1,\dots ,z_n)=0.$$

(3)

Often, the set of solutions to system (3) is infinite, so “to solve” system (3) means to find a decomposition of the variety of the ideal, I, into irreducible components. More precisely, a variety $V\in {\mathbb{K}}^n$ is irreducible if $V=V_1\cup V_2$ for affine varieties $V_1$ and $V_2$, only if either $V_1=V$ or $V_2=V.$ The variety, V, of a given ideal, I, can be represented as a union of irreducible components,

$$V=V_1\cup \cdots \cup V_m,$$

(4)

where each $V_i$ is irreducible. The radical of I, denoted by $\sqrt{I}$, is the set

$$\sqrt{I}=\{f\in \mathbb{K}[z_1,\dots ,z_n]:\mathrm{there}\mathrm{exists}p\in \mathbb{N}\mathrm{such}\mathrm{that}{f}^p\in I\}.$$

It is an intersection of prime ideals,

$$\sqrt{I}={\cap }_{j=1}^s{Q}_j,$$

where $V_i(i=1,\dots ,s)$ in decomposition (4) is the variety of $Q_i$. Ideals $Q_i$ are called the minimal associate primes of I. At present, there exist a few algorithms for computing the minimal associate primes [13,14]. They use Groebner basis computations and are, therefore, extremely time- and memory-consuming when the calculations are performed over the field of rational numbers. In our work, we use the routine minAssGTZ [15] of the computer algebra system Singular [16]. It is based on the algorithm of [13]. To speed up the computation, we apply the modular approach (see [17,18]).

To perform the elimination of variables, we employ a result from the computational commutative algebra called the elimination theorem, which is as follows.

Let I be an ideal in $\mathbb{K}[z_1,\dots ,z_n]$ and ℓ be a fixed number from the set $\{0,1,\dots ,n-1\}$. The ℓth elimination ideal of I is the ideal $I^{(\ell )}=I\cap \mathbb{K}[z_{\ell +1},\dots ,z_n]$. According to the elimination theorem (see, for example, [8,12]), to compute (for any $0⩽\ell ⩽n-1$) the ℓth elimination ideal $I^{(\ell )}$ of an ideal, I, of $\mathbb{K}[z_1,\dots ,z_n]$, one can choose the lexicographic term order on the ring $\mathbb{K}[z_1,\dots ,z_n]$ with $z_1>z_2>\cdots >z_n$ and compute a Groebner basis G for the ideal, I, with respect to this order. Then, by the elimination theorem, the set $G_{\ell }:=G\cap \mathbb{K}[z_{\ell +1},\dots ,z_n]$ is a Groebner basis for the ℓth elimination ideal $I^{(\ell )}$.

In the computer algebra system, Singular, the routine eliminate performs the elimination using an algorithm based on the elimination theorem.

To finish this section, we recall the notion of time-reversible vector fields. Let U be an open set in ${\mathbb{K}}^n$, and

$$\dot{x}=\mathcal{F}\left(x\right),$$

(5)

be a smooth differential system on U. We denote the vector field (5) by $\mathcal{X}$.

Let $\phi :U\to U$ be an involution of class $C^1$, $\mathcal{X}:U\to {\mathbb{K}}^n$ be a vector field of class $C^r$, and $F:U\to \mathbb{K}$ be a continuous function. It is said that $\mathcal{X}$ is time-reversible if

$$D_{\phi }·\mathcal{X}=-\mathcal{X}\circ \phi .$$

In Section 5, we deal with a generalization of the time-reversible systems proposed in [3].

3. Involutions in Family (1)

We note that if ${\phi }_{\lambda }(x,y)=(f(x,y),g(x,y))$ is an involution, then the map obtained after the transformation $R(x,y)=(y,x)$ is also an involution. This implies that if ${\phi }_{\lambda }(x,y)=(f(x,y),g(x,y))$ is an involution, then ${\phi }_{\lambda }(x,y)=(g(y,x),f(y,x))$ is also an involution. We refer to such involutions as conjugate.

Observe that map (1) can be an involution only in the case when both fractions in (1) are irreducible in $\mathbb{K}[x,y]$; that is,

$$\mathrm{Rank}\left(\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\end{array}\right)=\mathrm{Rank}\left(\begin{array}{ccc}{b}_1& b_2& b_3\\ b_4& b_5& b_6\end{array}\right)=2.$$

(6)

A map of form (1) is an involution for a fixed value $\lambda$ of the vector of parameters if

$${\phi }_{\lambda }\left({\phi }_{\lambda }(x,y)\right)-(x,y)=(0,0).$$

(7)

Computing ${\phi }_{\lambda }^2$ for map (1), and then substituting the result into (7), we see that (7) holds for the values of parameters $\lambda$, such that all polynomials

• $f_1=-a_4{a}_5{b}_2-a_1{a}_4{b}_5-a_4{a}_6{b}_5,$
• $f_2=-a_4{a}_5{b}_1-a_5^2{b}_2-a_1{a}_4{b}_4-a_4{a}_6{b}_4-a_2{a}_4{b}_5-a_5{a}_6{b}_5,$
• $f_3=a_2{a}_4{b}_2-a_5{a}_6{b}_2-a_4{a}_5{b}_3+a_1^2{b}_5-a_6^2{b}_5-a_1{a}_4{b}_6-a_4{a}_6{b}_6,$
• $f_4=-a_5^2{b}_1-a_2{a}_4{b}_4-a_5{a}_6{b}_4,$
• $f_5=a_2{a}_4{b}_1-a_5{a}_6{b}_1+a_2{a}_5{b}_2-a_5^2{b}_3+a_1^2{b}_4-a_6^2{b}_4+a_1{a}_2{b}_5+a_3{a}_5{b}_5-a_2{a}_4{b}_6-a_5{a}_6{b}_6,$
• $f_6=a_2{a}_6{b}_2+a_2{a}_4{b}_3-a_5{a}_6{b}_3+a_1{a}_3{b}_5+a_3{a}_6{b}_5+a_1^2{b}_6-a_6^2{b}_6,$
• $f_7=a_2{a}_5{b}_1+a_1{a}_2{b}_4+a_3{a}_5{b}_4,$
• $f_8=a_2{a}_6{b}_1+a_2{a}_5{b}_3+a_1{a}_3{b}_4+a_3{a}_6{b}_4+a_1{a}_2{b}_6+a_3{a}_5{b}_6,$
• $f_9=a_2{a}_6{b}_3+a_1{a}_3{b}_6+a_3{a}_6{b}_6,$
• $f_{10}=-a_4{b}_2{b}_4-a_1{b}_5^2-a_4{b}_5{b}_6,$
• $f_{11}=a_4{b}_1{b}_2+a_1{b}_2{b}_5+a_4{b}_3{b}_5,$
• $f_{12}=-a_4{b}_1{b}_4-a_5{b}_2{b}_4-a_1{b}_4{b}_5-a_2{b}_5^2-a_4{b}_4{b}_6-a_5{b}_5{b}_6,$
• $f_{13}=a_4{b}_1^2+a_5{b}_1{b}_2+a_1{b}_2{b}_4-a_6{b}_2{b}_4+a_2{b}_2{b}_5+a_5{b}_3{b}_5-a_3{b}_5^2-a_1{b}_5{b}_6-a_6{b}_5{b}_6-a_4{b}_6^2,$
• $f_{14}=a_6{b}_1{b}_2+a_4{b}_1{b}_3+a_3{b}_2{b}_5+a_6{b}_3{b}_5+a_1{b}_2{b}_6+a_4{b}_3{b}_6,$
• $f_{15}=-a_5{b}_1{b}_4-a_2{b}_4{b}_5-a_5{b}_4{b}_6,$
• $f_{16}=a_5{b}_1^2-a_6{b}_1{b}_4+a_2{b}_2{b}_4-a_3{b}_4{b}_5-a_6{b}_4{b}_6-a_2{b}_5{b}_6-a_5{b}_6^2,$
• $f_{17}=a_6{b}_1^2+a_5{b}_1{b}_3+a_3{b}_2{b}_4+a_2{b}_2{b}_6+a_5{b}_3{b}_6-a_3{b}_5{b}_6-a_6{b}_6^2,$
• $f_{18}=a_6{b}_1{b}_3+a_3{b}_2{b}_6+a_6{b}_3{b}_6$
• vanish on these values.

That is, map (1) is an involution if the parameters $a_1,\dots ,a_6,b_1,\dots ,b_6$ satisfy the system

$$f_1=f_2=\cdots =f_{18}=0$$

(8)

and condition (6) holds. As pointed out in [5], in the case when all parameters are real, condition (6) can be written in the polynomial form as follows:

$$g_1:=1-c_1{h}_1=0,g_2:=1-c_2{h}_2=0,$$

(9)

where $c_1$ and $c_2$ are new variables, and $h_1$ and $h_2$ are defined as follows:

$$\begin{array}{cc}\hfill h_1& :={(a_1{a}_5-a_2{a}_4)}^2+{(a_1{a}_6-a_3{a}_4)}^2+{(a_2{a}_6-a_3{a}_5)}^2,\hfill \\ \hfill h_2& :={(b_1{b}_5-b_2{b}_4)}^2+{(b_1{b}_6-b_3{b}_4)}^2+{(b_2{b}_6-b_3{b}_5)}^2.\hfill \end{array}$$

We denote $H=h_1{h}_2.$

Theorem 1.

Real map (1) is an involution if and only if its parameters $(a_1,\dots ,a_6,b_1,\dots ,b_6)$ belong to one of the following sets:

(1) 

$\mathbf{V}\left(I_1\right)\setminus \mathbf{V}\left((a_1^2+a_3{a}_4)(b_1^2+b_3{b}_4)\right),$

(2) 

$\mathbf{V}\left(I_2\right)\setminus \mathbf{V}\left((a_1^4+(a_2^2+a_3^2)a_4^2+a_1^2(a_2^2+2a_3{a}_4))b_6\right),$

(3) 

$\mathbf{V}\left(I_3\right)\setminus \mathbf{V}\left((b_1^4+(b_2^2+b_3^2)b_4^2+b_1^2(b_2^2+2b_3{b}_4))a_1\right),$

(4) 

$\mathbf{V}\left(I_4\right)\setminus \mathbf{V}\left(a_6{b}_6\right),$

(5) 

$\mathbf{V}\left(I_5\right)\setminus \mathbf{V}\left((a_3{a}_5-a_2{a}_6)(b_3{b}_5-b_2{b}_6)\right),$

(6) 

$\mathbf{V}\left(I_6\right)\setminus \mathbf{V}\left(a_6{b}_6\right),$

where

• $I_1=⟨b_5,b_2,b_1+b_6,a_5,a_2,a_1+a_6⟩,$
• $I_2=⟨b_5,b_4,b_3,b_2,b_1-b_6,a_5,a_1+a_6⟩,$
• $I_3=⟨b_5,b_1+b_6,a_5,a_4,a_3,a_2,a_1-a_6⟩,$
• $I_4=⟨b_5,b_4,b_3,b_2,b_1-b_6,a_5,a_4,a_3,a_2,a_1-a_6⟩,$
• $$\begin{gathered} I_5=⟨b_4,b_1,a_4,a_1,a_2{b}_5+a_5{b}_6,a_5{b}_3-a_3{b}_5,a_2{b}_3+a_3{b}_6,a_5{b}_2+a_6{b}_5,a_3{b}_2+a_6{b}_3,a_2{b}_2- \\ {a}_6{b}_6⟩, \end{gathered}$$
• $$\begin{gathered} I_6=⟨a_6{b}_5-a_4{b}_6, a_6{b}_4-a_5{b}_6, a_4{b}_4-a_5{b}_5, a_5{b}_2+a_1{b}_5+a_4{b}_6, a_5{b}_1+a_2{b}_5+a_5{b}_6, a_1{b}_1+ \\ {a}_6{b}_1-a_2{b}_2+a_1{b}_6+a_6{b}_6, b_2^2{b}_4-b_1{b}_2{b}_5-b_3{b}_5^2+b_2{b}_5{b}_6, a_3{b}_2{b}_4-a_2{b}_3{b}_5, a_2{b}_2{b}_4- \\ {a}_2{b}_1{b}_5-a_3{b}_4{b}_5+a_2{b}_5{b}_6, a_1{b}_2{b}_4-a_2{b}_2{b}_5+a_5{b}_3{b}_5+a_1{b}_5{b}_6+a_4{b}_6^2, a_1{a}_2{b}_4+a_3{a}_5{b}_4- \\ {a}_2^2{b}_5-a_2{a}_5{b}_6, a_6{b}_1{b}_3+a_3{b}_2{b}_6+a_6{b}_3{b}_6, a_4{b}_1{b}_3+a_3{b}_2{b}_5+a_4{b}_3{b}_6, a_2{a}_6{b}_3+a_1{a}_3{b}_6+ \\ {a}_3{a}_6{b}_6, a_5^2{b}_3-a_1^2{b}_4+a_1{a}_2{b}_5+a_2{a}_4{b}_6+a_5{a}_6{b}_6, a_2{a}_5{b}_3+a_1{a}_3{b}_4+a_3{a}_5{b}_6, \\ {a}_2{a}_4{b}_3+a_1{a}_3{b}_5+a_3{a}_4{b}_6, a_3{b}_2^2-a_1{b}_2{b}_3+a_6{b}_2{b}_3-a_4{b}_3^2, a_6{b}_1{b}_2+a_1{b}_2{b}_6+a_4{b}_3{b}_6, \\ {a}_4{b}_1{b}_2+a_1{b}_2{b}_5+a_4{b}_3{b}_5, a_3{b}_1{b}_2-a_2{b}_2{b}_3+a_3{b}_3{b}_5-a_3{b}_2{b}_6, a_2{a}_6{b}_2-a_5{a}_6{b}_3+a_1^2{b}_6- \\ {a}_6^2{b}_6, a_2{a}_4{b}_2-a_4{a}_5{b}_3+a_1^2{b}_5-a_4{a}_6{b}_6, a_2{a}_3{b}_2-a_1{a}_2{b}_3-a_3{a}_5{b}_3-a_1{a}_3{b}_6-a_3{a}_6{b}_6, a_6{b}_1^2+ \\ {a}_2{b}_2{b}_6-a_3{b}_5{b}_6-a_6{b}_6^2, a_4{b}_1^2+a_2{b}_2{b}_5-a_3{b}_5^2-a_4{b}_6^2, a_2{a}_6{b}_1+a_1{a}_2{b}_6+a_3{a}_5{b}_6, a_2{a}_4{b}_1+ \\ {a}_1{a}_2{b}_5+a_3{a}_5{b}_5, a_2{a}_3{b}_1-a_2^2{b}_3+a_3^2{b}_4-a_2{a}_3{b}_6, a_2^2{a}_4-a_1{a}_2{a}_5-a_3{a}_5^2+a_2{a}_5{a}_6, \\ {a}_5{a}_6{b}_3^2+a_1{a}_3{b}_2{b}_6+a_3{a}_6{b}_2{b}_6-a_1^2{b}_3{b}_6+a_6^2{b}_3{b}_6, a_4{a}_5{b}_3^2+a_1{a}_3{b}_2{b}_5-a_1^2{b}_3{b}_5+a_3{a}_4{b}_2{b}_6+ \\ {a}_4{a}_6{b}_3{b}_6⟩. \end{gathered}$$

Proof. 

Let G be the ideal of the polynomial ring $\mathbb{C}[a,b]:=\mathbb{C}[c_1,c_2,a_1,\dots ,a_6,b_1,\dots ,b_6]$ generated by polynomials $g_1,g_2,f_1,\dots ,f_{18}$,

$$G=⟨g_1,g_2,f_1,f_2,\dots ,f_{18}⟩.$$

(10)

Using the routine eliminate of the computer algebra system Singular [16], we compute the second elimination ideal of J, which we denote by I,

$$I=J\cap \mathbb{R}[a_1,\dots ,a_6,b_1,\dots ,b_6].$$

Employing the routine minAssGTZ [15], we look for the minimal associate primes of I obtaining the ideals $I_1,\dots ,I_6$ given in the statement of the theorem. It leads to the conclusion that

$$\sqrt{I}=\bigcap _{j=1}^6{I}_j.$$

However, the projection $\pi \left(\mathbf{V}\right(G\left)\right)$ of the variety $\mathbf{V}\left(G\right)$ on the affine space

$$(a_1,\dots ,a_6,b_1,\dots ,b_6)$$

is not necessarily the variety $\mathbf{V}\left(I\right)$. Generally speaking, $\pi \left(\mathbf{V}\right(G\left)\right)\subset \mathbf{V}\left(I\right)$. Thus, to prove the theorem, we will show that all points from $\mathbf{V}\left(I\right)\setminus \mathbf{V}\left(H\right)$ correspond to maps (1) that are involutions.

For each $k=1,\dots ,6$, the variety after the setminus sign in the k-th condition of the statement of Theorem 1 is the restriction of $\mathbf{V}\left(H\right)$ on $\mathbf{V}\left(I_k\right)$. To prove the theorem, we have to check that the parameters of map (1) belonging to one of the six sets given in the statement of the theorem corresponding maps (1) are involutions.

(1) The variety of $I_1$ is the set defined by the equations

$$b_5=b_2=b_1+b_6=a_5=a_2=a_1+a_6=0.$$

(11)

Solving this system for $a_6$ and $b_6$, we obtain the involution

$$\phi (x,y)=\left(\frac{a_{1}x+a_3}{a_{4}x-a_1},\frac{b_{1}y+b_3}{b_{4}y-b_1}\right).$$

Under condition (11) the expression $h_1{h}_2=0$ is equivalent to the expression $(a_1^2+a_3{a}_4)(b_1^2+b_3{b}_4)=0$. Therefore, if condition (1) of the theorem holds, then the corresponding maps are involutions.

Similarly, we see that conditions (2) and (3) of the theorem yield the involutions

$$\phi (x,y)=\left(\frac{a_{1}x+a_{2}y+a_3}{a_{4}x-a_1},y\right)$$

and

$$\phi (x,y)=\left(x,\frac{b_{1}y+b_{2}x+b_3}{b_{4}y-b_1}\right),$$

respectively. Clearly, these involutions are conjugate. Condition (4) gives the identity map.

(5) To obtain the involutions related to the ideal, $I_5$, we have to look for parametrizations of the set of solutions of the polynomial system

$$\begin{array}{c}{b}_4=b_1=a_4=a_1=a_2{b}_5+a_5{b}_6=a_5{b}_3-a_3{b}_5=a_2{b}_3+a_3{b}_6=\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{a}_5{b}_2+a_6{b}_5=a_3{b}_2+a_6{b}_3=a_2{b}_2-a_6{b}_6=0.\end{array}$$

If $a_6\ne 0,$ we obtain the involution

$$\phi (x,y)=\left(\frac{a_{2}y+a_3}{a_{5}y+a_6},\frac{a_{6}x-a_3}{a_2-a_{5}x}\right).$$

(12)

If $a_6=0,$ we have equations $a_5{b}_2=0$, $a_3{b}_2=0$, and $a_2{b}_2=0$. From these equations, we see that $b_2$ must be zero; otherwise, we have $a_5=a_3=a_2=0,$ which is not possible because $\phi (x,y)$ is not defined for such values of the parameters. Substituting $b_2=0$ into the equations, we proceed further by expressing the remaining parameters. If $a_2\ne 0,$ then we have the expressions $b_5=-\frac{a_5{b}_6}{a_2}$ and $b_3=-\frac{a_3{b}_6}{a_2},$ giving us an involution of form (12), where $a_6=0.$ If $a_2=0,$ then it follows that $b_6=0$, so we obtain involution (12) with $a_2=a_6=0$.

(6) Let $b_5\ne 0$. Using the fifth polynomial of the list defining the ideal, $I_6$, and solving the corresponding equation for $a_2,$ we have $a_2=-\frac{a_5{b}_1+a_5{b}_6}{b_5}$. If $b_4\ne 0$, we have

$$\begin{array}{c}{a}_6=\frac{a_5{b}_6}{b_4},a_4=\frac{a_5{b}_5}{b_4},b_2=\frac{-a_1{b}_4{b}_5-a_5{b}_5{b}_6}{a_5{b}_4},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{a}_3=\frac{(a_5{b}_1+a_1{b}_4)(b_1+b_6)}{b_4{b}_5},b_3=\frac{a_1{a}_5{b}_1{b}_4+a_1^2{b}_4^2+a_5^2{b}_1{b}_6+a_1{a}_5{b}_4{b}_6}{a_5^2{b}_4}.\end{array}$$

The parametrization yields the involution

$$\phi (x,y)=\left(\frac{a_1{b}_4(b_1+b_6+b_{5}x)+a_5(b_1+b_6)(b_1-b_{4}y)}{a_5{b}_5(b_6+b_{5}x+b_{4}y)},\frac{(a_1{b}_4+a_5{b}_6)(a_1{b}_4+a_5(b_1-b_{5}x))+a_5^2{b}_1{b}_{4}y}{a_5^2{b}_4(b_6+b_{5}x+b_{4}y)}\right).$$

(13)

Using the rational implicitization theorem (see, e.g., [12]), we can easily verify that this parametrization covers almost all variety $\mathbf{V}\left(I_6\right)-$ in the sense that the Zariski closure of the set covered by the parametrization is precisely $\mathbf{V}\left(I_6\right)$. For the points not covered by the parametrization, we now show that the corresponding maps are involutions as well.

When $b_4=0$, we have $a_5{b}_5=0$, yielding $a_5=0$. Now, we have two cases: $a_1=-a_6$ or $b_1=-b_6$. In $a_1=-a_6,$ we obtain the involution

$$\phi (x,y)=\left(\frac{b_1^2-b_6^2-b_5{b}_{6}x}{b_5{b}_6+b_5^{2}x},\frac{-b_1{b}_2+b_2{b}_6+b_2{b}_{5}x+b_1{b}_{5}y}{b_5{b}_6+b_5^{2}x}\right).$$

(14)

For the other cases, we either have involutions that are conjugate to the ones given above or polynomial involutions provided by Theorem 1 of [5].    □

It is clear that all involutions listed in the proof of Theorem 1 can also be considered as involutions in ${\mathbb{C}}^2$ with complex coefficients.

Theorem 2.

The complex involutions of form (1) have the same expressions as real involutions (1).

Proof. 

We only need to prove that there are no additional complex involutions beyond those given by the expressions in Theorem 1 (which are considered as maps $\phi :{\mathbb{C}}^2\to {\mathbb{C}}^2$ with complex parameters). The involutions of Theorem 1 were obtained using the claim that condition (6) is equivalent to the condition $g_1=g_2=0.$ However, this claim is valid only for real maps. Assuming that the ideal $F=⟨f_1,\dots ,f_{18}⟩$ is from the ring $\mathbb{C}[a,b]$ and using minAssGTZ of Singular, we compute the minimal associate primes of F and obtain the ideals $I_1,\dots ,I_6$ listed in the statement of Theorem 1 and, additionally, the ideals

• $I_7=⟨b_6,b_5,b_4,b_3,b_2,b_1⟩,$
• $I_8=⟨b_6,b_5,b_4,a_6,a_5⟩,$
• $I_9=⟨b_6,b_5,b_4,b_1,a_5,a_2⟩,$
• $I_{10}=⟨a_6,a_5,a_4,a_3,a_2,a_1⟩,$
• $I_{11}=⟨b_5{b}_2,a_6,a_5,a_4,a_1⟩,$
• $$\begin{gathered} I_{12}=⟨-b_2{b}_6+b_3{b}_5,-b_1{b}_6+b_3{b}_4,-b_1{b}_5+b_2{b}_4,a_1{b}_4-a_2{b}_5-a_5{b}_6+a_6{b}_4,a_1{b}_6+a_5{b}_3+ \\ {a}_6{b}_6,a_1{b}_5+a_5{b}_2+a_6{b}_5,a_3{b}_5+a_6{b}_1+a_6{b}_6,a_2{b}_5+a_5{b}_1+a_5{b}_6,a_1{b}_5+a_4{b}_1+a_4{b}_6,a_1{b}_1+a_1{b}_6 \\ -a_2{b}_2-a_3{b}_5,-a_2{a}_6+a_3{a}_5,-a_1{a}_6+a_3{a}_4,-a_1{a}_5+a_2{a}_4,a_1{a}_3{b}_6+a_2{a}_6{b}_3+a_3{a}_6{b}_6,a_1{a}_3{b}_5+ \\ {a}_2{a}_6{b}_2+a_3{a}_6{b}_5⟩. \end{gathered}$$

For the values of parameters from $\mathbf{V}\left(I_7\right),\dots ,\mathbf{V}\left(I_{11}\right),$ corresponding maps (1) are either undefined or not involutions. To establish the same for the parameters in $\mathbf{V}\left(I_{12}\right),$ it is sufficient to show that (6) does not hold for such parameter values. In this case, we show that the vectors ${\mathbf{a}}_1=(a_1,a_2,a_3)$ and ${\mathbf{a}}_2=(a_4,a_5,a_6)$ are linearly dependent; in other words, for each point of $\mathbf{V}\left(I_{12}\right)$, there exists a number k such that  

$$\begin{array}{cc}\hfill a_1-a_{4}k=& 0,\hfill \\ \hfill a_2-a_{5}k=& 0,\hfill \\ \hfill a_3-a_{6}k=& 0.\hfill \end{array}$$

(15)

In the case when $a_4\ne 0,$ solving the first equation in (15) for k yields $k=\frac{a_1}{a_4}$. Substituting this into the other two equations of (15), we obtain

$$a_2{a}_4-a_1{a}_5=a_3{a}_4-a_1{a}_6=0,$$

which holds on $\mathbf{V}\left(I_{12}\right)$, as both polynomials written above are in the ideal, $I_{12}$. This implies that vectors ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are linearly dependent for the values of parameters in $\mathbf{V}\left(I_{12}\right)$. If $a_4=0,$ then the ideal, $I_{12}$, contains the polynomials $a_1{a}_6$ and $a_1{a}_5$. However, this yields $a_1=0$, making the first equation in (15) trivially true ($0=0$). Solving the second equation of (15) in the case when $a_5\ne 0$, we have $k=\frac{a_2}{a_5}$. The last equation of (15) gives $a_3{a}_5-a_2{a}_6=0,$ which is true on $\mathbf{V}\left(I_{12}\right)$. Thus, ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ are linearly dependent on $\mathbf{V}\left(I_{12}\right)$. If $a_5=0$, then $I_{12}$ contains polynomial $a_2{a}_6$, implying $a_2=0$. But if $a_1=a_2=a_4=a_5=0$, then map (1) cannot be an involution.    □

4. Involutions with Quadratic Numerators

Theorem 1 gives all involutions in family (1). In a similar way, we search for rational involutions with a polynomial of degree two in the numerator, which are of form (2) (the case of rational maps where both numerators and denominators are polynomials of degree two is computationally unfeasible).

We denote by $(a,b,c,d)$ the vector of parameters of map (2), so

$$(a,b,c,d)=(a_1,\dots a_5,b_1,b_2,c_1,\dots ,c_5,d_1,d_2),$$

and by $\mathbb{C}[a,b,c,d]$, the ring of polynomial in parameters of (2) with the coefficients in $\mathbb{C}$.

Theorem 3.

The maps listed below and their conjugates are involutions from family (2):

1. 

$\phi (x,y)=\left(\frac{-x+a_{2}y-b_{2}xy+a_5{y}^2}{1+b_{1}x+b_{2}y},y\right),$

2. 

$\phi (x,y)=\left(\frac{-x+\frac{b_1{d}_2-d_1{d}_2}{2d_1-b_1}xy}{1+b_{1}x+\frac{d_1{d}_2}{2d_1-b_1}y},\frac{-y+(d_1-b_1)xy}{1+d_{1}x+d_{2}y}\right),$

3. 

$\phi (x,y)=\left(\frac{-x+b_{2}xy+a_5{y}^2}{1+b_{1}x+b_{2}y},-y\right),$

4. 

$\phi (x,y)=\left(-x,\frac{-y+d_{1}xy}{1+d_{1}x+d_{2}y}\right),$

5. 

$\phi (x,y)=\left(\frac{x+a_{2}y-b_{2}xy}{1+b_{2}y},-y\right),$

6. 

$\phi (x,y)=\left(-x,\frac{c_{1}x+y-d_{1}xy}{1+d_{1}x}\right),$

7. 

$\phi (x,y)=\left(x,\frac{c_{1}x-y+c_3{x}^2-d_{1}xy}{1+d_{1}x+d_{2}y}\right),$

8. 

$\phi (x,y)=\left(\frac{-c_{2}x+\frac{1-c_2^2}{c1}y-b_{2}xy}{1+b_{2}y},\frac{c_{1}x+c_{2}y+\frac{c_1{b}_2}{1+c_2}xy}{1-\frac{c_1{b}_2}{1+c_2}x}\right),$

9. 

$\phi (x,y)=\left(\frac{-x-c_{5}xy+a_5{y}^2}{1+d_{1}x-c_{5}y},\frac{-y-d_{1}xy+c_5{y}^2}{1+d_{1}x-c_{5}y}\right),$

10. 

$\phi (x,y)=\left(\frac{-d_{1}x-d_1^2{x}^2-d_1(2c_5+d_2)xy-(c_5^2+c_5{d}_2)y^2}{d_1+d_1^{2}x+d_1{d}_{2}y},\frac{-y+d_{1}xy+c_5{y}^2}{1+d_{1}x+d_{2}y}\right),$

11. 

$\phi (x,y)=\left(\frac{-x-d_{2}xy+a_5{y}^2}{1+d_{1}x+d_{2}y},\frac{-a_{5}y+d_1{d}_2{x}^2-a_5{d}_{1}xy}{a_5+a_5{d}_{1}x+a_5{d}_{2}y}\right),$

12. 

$\phi (x,y)=\left(\frac{-x-d_1{x}^2-d_{2}xy}{1+d_{1}x+d_{2}y},\frac{-y+c_3{x}^2+d_{1}xy}{1+d_{1}x+d_{2}y}\right),$

13. 

$\phi (x,y)=\left(\frac{-2x-(c_4+d_1)x^2-2c_{5}xy}{2+2d_{1}x-2c_{5}y},\frac{-4c_{5}y+(c_4^2-d_1^2)x^2+4c_4{c}_{5}xy+4c_5^2{y}^2}{4c_5+4c_5{d}_{1}x-4c_5^{2}y}\right),$

14. 

$\phi (x,y)=\left(\frac{-c_{2}x-\frac{b_2(1+c_2)}{d_1}y+(d_2-b_2)xy}{1+\frac{d_1(2b_2-d_2)}{b_2}x+b_{2}y},\frac{\frac{c_2{d}_1-d_1}{b_2}x+c_{2}y+\frac{d_1(b_2-d_2)}{b_2}xy}{1+d_{1}x+d_{2}y}\right),$

15. 

$\phi (x,y)=\left(\frac{-c_{2}x+\frac{d_2(1+c_2)}{b_1}y+d_{2}xy}{1+b_{1}x},\frac{\frac{b_1-b_1{c}_2}{d_2}x+c_{2}y+b_{1}xy}{1+d_{2}y}\right),$

16. 

$\phi (x,y)=\left(\frac{x+a_{2}y+(d_2-b_2)xy}{1+b_{2}y},\frac{-y}{1+d_{2}y}\right),$

17. 

$\phi (x,y)=\left(\frac{-x}{1+b_{1}x},\frac{c_{1}x+y+(b_1-d_1)xy}{1+d_{1}x}\right),$

18. 

$\phi (x,y)=\left(\frac{x+d_{2}xy}{1+b_{1}x},\frac{2b_{1}x-d_{2}y+b_1{d}_{2}xy}{d_2+d_2^{2}y}\right),$

19. 

$\phi (x,y)=\left(\frac{-c_2{d}_{1}x-d_2(1+c_2)y-d_1^2{x}^2-d_1{d}_{2}xy}{d_1+d_1^{2}x+d_1{d}_{2}y},\frac{d_1(c_2-1)x+c_2{d}_{2}y-d_1{d}_{2}xy-d_2^2{y}^2}{d_2+d_1{d}_{2}x+d_2^{2}y}\right),$

20. 

$\phi (x,y)=\left(\frac{x+a_{2}y-d_{2}xy}{1+d_{2}y},-y\right),$

21. 

$$\begin{gathered} \phi (x,y)= \\ \left(\frac{-c_{2}x-\frac{2(c_2{c}_5-c_5)}{c_4-d_1}y+\frac{d_1-c_4}{2}{x}^2+(d_2-2c_5)xy-\frac{2(c_5^2-c_5{d}_2)}{c_4-d_1}{y}^2}{1+d_{1}x+d_{2}y},\frac{\frac{(1+c_2)(c_4-d_1)}{2c_5}x+\frac{c_4^2-d_1^2}{4c_5}{x}^2+c_{2}y+c_{4}xy+c_5{y}^2}{1+d_{1}x+d_{2}y}\right), \end{gathered}$$

22. 

$\phi (x,y)=\left(\frac{-x+a_{2}y-c_{5}xy}{1+d_{1}x+c_{5}y},y\right),$

23. 

$\phi (x,y)=\left(\frac{-c_{2}x+a_{2}y+d_{2}xy+\frac{a_2{d}_2}{1-c_2}{y}^2}{1+d_{1}x+d_{2}y},\frac{\frac{1-c_2^2}{a_2}x+c_{2}y+\frac{d_1-c_2{d}_1}{a_2}{x}^2+d_{1}xy}{1+d_{1}x+d_{2}y}\right),$

24. 

$\phi (x,y)=\left(\frac{-x+d_{2}xy+a_5{y}^2}{1+d_{1}x+d_{2}y},\frac{\frac{2d_2}{a_5}x+y+\frac{d_1{d}_2}{a_5}{x}^2+d_{1}xy}{1+d_{1}x+d_{2}y}\right),$

25. 

$\phi (x,y)=\left(\frac{-x}{1+d_{1}x},\frac{2c_{1}x+2y+c_1{d}_1{x}^2+2d_{1}xy}{2+2d_{1}x}\right).$

Proof. 

Taking the coefficients of powers of $x^k{y}^m$ in the numerator of

$$\phi \left(\phi \right(x,y\left)\right)-(x,y)$$

(16)

we obtain a set of polynomials dependent on $a_1,\dots a_5,b_1,b_2,c_1,\dots ,c_5,d_1,d_2.$ Let I be the ideal in $\mathbb{C}[a,b,c,d]$ generated by these polynomials.

We were unable to compute the irreducible decomposition of the variety $\mathbf{V}\left(I\right)$ of the ideal, I, working in the field of rational numbers. Instead, we computed the minimal associate primes of I using minAssGTZ in the field of characteristic 32003, and obtained 17 ideals listed in Appendix A. Performing the rational reconstruction with the algorithm of [19], we obtain ideals from which we derive the corresponding involutions listed in the statement of the theorem.

For instance, consider the 8th ideal in Appendix A ($K_8$). Using the algorithm for the rational reconstruction from [19] we lift the numbers $8001,-8001,16,001,-16,001$ of ${\mathbb{Z}}_{32003}$ to the rational numbers $\frac{1}{4},-\frac{1}{4},-\frac{1}{2},\frac{1}{2}$ and obtain the ideal

• $$\begin{gathered} {\widehat{K}}_8=⟨-\frac{1}{4}{c}_4^2+c_3{c}_5+\frac{1}{4}{d}_1^2,a_5{c}_4+2c_5^2+a_5{d}_1+2c_5{d}_2,a_5{c}_3+\frac{1}{2}{c}_4{c}_5-\frac{1}{2}{c}_5{d}_1+\frac{1}{2}{c}_4{d}_2- \\ \frac{1}{2}{d}_1{d}_2,b_2-d_2,b_1-d_1,c_2+1,c_1,a_4+2c_5+d_2,a_3+\frac{1}{2}{c}_4+\frac{1}{2}{d}_1,a_2,a_1+1⟩. \end{gathered}$$

It is easily seen that the variety of ${\widehat{K}}_8$ is parameterized by $a_1=-1,a_2=0,a_3=0,a_4=-d_2,b_1=d_1,b_2=d_2,c_1=0,c_2=-1,c_3=\frac{d_1{d}_2}{a_5},c_4=-d_1,c_5=0,$ where $a_5$ is not equal to zero. It yields the map

$$\phi (x,y)=\left(\frac{-x-d_{2}xy+a_5{y}^2}{1+d_{1}x+d_{2}y},\frac{-a_{5}y+d_1{d}_2{x}^2-a_5{d}_{1}xy}{a_5+a_5{d}_{1}x+a_5{d}_{2}y}\right),$$

which is the 11th map listed in the statement of the theorem. A direct check shows that it is an involution.

We lift the ideal $K_{17}$ of Appendix A to the ideal over the field of rational numbers similarly.

In the statement of theorem, we list only involutions with quadratic polynomials in the numerator. From the analysis of $\mathbf{V}\left(I\right),$ we also obtain involutions for which the numerator is linear or a constant. However, we do not write them down here since they are among the ones presented in Theorem 1 or in Theorems 1 and 2 of [5].    □

5. Reversible Systems

We recalled the definition of time-reversibility in Section 2. Recently, the following generalization of the notion of time-reversibility was introduced in [3].

Definition 1.

Let $\phi :U\to U$ be an involution of class $C^1$, $\mathcal{X}:U\to {\mathbb{K}}^n$ be a vector field of class $C^r$, and $F:U\to \mathbb{K}$ be a continuous function. It is said that $\mathcal{X}$ is F-factor φ-reversible if

$$D_{\phi }·\mathcal{X}=F\mathcal{X}\circ \phi .$$

The case $F\equiv -1$ corresponds to the classical notion of time-reversibility.

Consider the following cubic system with a $1:-1$ resonant singular point at the origin

$$\begin{array}{cc}\hfill \dot{x}=& x-a_{10}{x}^2-a_{20}{x}^3-a_{01}xy-a_{11}{x}^{2}y-a_{12}{y}^2-a_{02}x{y}^2-a_{13}{y}^3=P(x,y),\hfill \\ \hfill \dot{y}=& -y+b_{21}{x}^2+b_{31}{x}^3+b_{10}xy+b_{20}{x}^{2}y+b_{01}{y}^2+b_{11}x{y}^2+b_{02}{y}^3=Q(x,y),\hfill \end{array}$$

(17)

where $a_{jk},b_{kj}$ are complex parameters.

An important problem in the qualitative theory of polynomial differential systems is the classification of analytically integrable systems in family (17). It is known that analytic integrability is closely related to reversibility (see, e.g., [3,8,20,21]).

To look for systems in family (17), which are F-factor $\phi$-reversible with respect to a given involution $\phi (x,y)$, in (17), we perform the substitution

$$(x_1,y_1)=\phi (x,y)$$

(18)

obtaining the system

$${\dot{x}}_1=P_1(x_1,y_1),{\dot{y}}_1=Q_1(x_1,y_1).$$

System (17) is F-factor $\phi$-reversible if

$$P{Q}_1-P_{1}Q\equiv 0.$$

As an example of applying involutions to identifying reversible and integrable systems in a given polynomial family of ODEs, we consider involution (13). To complete the computations, we set $a_1=-\frac{a_5{b}_1}{b_4},$ so the involution is

$$\phi (x,y)=\left(-\frac{b_1{b}_{5}x+b_4(b_1+b_6)y}{b_5(b_{4}y+b_{5}x+b_6)},\frac{b_5(b_1-b_6)x+b_1{b}_{4}y}{b_4(b_{4}y+b_{5}x+b_6)}\right),$$

(19)

and the fixed points set Fix $\phi$ of the involution contains the origin.

By performing in (17) the substitution (18) with the map $\phi$ given by (19), we obtain an ideal J in $\mathbb{C}[a,b,b_1,b_4,b_5,b_6]$. Since all polynomials of the generating set of J contain $b_6$ as a factor, we divide them by $b_6$ and, in order to complete the computations, we set $b_5=1$ and $b_4=1$. By computing the minimal associate primes of the obtained ideal in the field of characteristic 32003 (since computations in characteristic zero are infeasible), we obtain five ideals. Then, we perform the rational reconstruction using the algorithm proposed in [19]. It results in ideals $J_1,\dots ,J_4$, with their varieties parametrized as follows:

• $$\begin{gathered} \mathbf{V}\left(J_1\right):a_{02}=\frac{b_{01}^2-a_{01}^2}{4},a_{10}=b_{10},a_{11}=\frac{b_{10}(b_{01}-a_{01})}{2},a_{12}=a_{13}=a_{20}=0, \\ {b}_{02}=\frac{a_{01}^2-b_{01}^2}{4},b_{11}=\frac{b_{01}(a_{01}-b_{01})}{2},b_{20}=b_{21}=b_{31}=0,b_1=0,b_6=\frac{2}{a_{01}-b_{01}}, \end{gathered}$$
• $$\begin{gathered} \mathbf{V}\left(J_2\right):a_{01}=-b_{10},a_{02}=-b_{02},a_{11}=\frac{b_{10}(a_{10}+b_{01})}{2},a_{13}=0,a_{20}=\frac{b_{01}^2+4b_{02}-2a_{12}{b}_{01}-2a_{10}{a}_{12}-a_{10}^2}{4}, \\ {b}_{11}=-\frac{b_{10}(a_{10}+b_{01})}{2},b_{20}=-\frac{b_{01}^2+4b_{02}-2a_{12}{b}_{01}-2a_{10}{a}_{12}-a_{10}^2}{4},b_{21}=-a_{12},b_{31}=0,b_1=0, \\ {b}_6=-\frac{2}{a_{10}+b_{01}}, \end{gathered}$$
• $$\begin{gathered} \mathbf{V}\left(J_3\right):a_{02}=-2a_{01}^2+a_{01}{b}_{01}+b_{20}-a_{01}{b}_{21}+b_{01}{b}_{21}+b_{21}^2,a_{10}=3a_{01}-b_{01}+3b_{21},a_{11}= \\ -3a_{01}^2+b_{20}+a_{01}(b_{01}-3b_{21})+b_{01}{b}_{21},a_{12}=-b_{21},a_{13}=a_{01}{b}_{21}+b_{21}^2+b_{31},a_{20}=-2a_{01}^2+ \\ {a}_{01}{b}_{01}-3a_{01}{b}_{21}+b_{01}{b}_{21}-b_{21}^2+b_{31},b_{02}=a_{01}^2-a_{01}{b}_{01}+2a_{01}{b}_{21}-b_{01}{b}_{21}+b_{21}^2+b_{31},b_{10}= \\ {b}_{21},b_{11}=b_{20},b_1=0,b_6=-\frac{1}{a_{01}+b_{21}}, \end{gathered}$$
• $$\begin{gathered} \mathbf{V}\left(J_4\right):a_{01}=-a_{10}+b_{01}+b_{10}+2b_{21},a_{02}=b_{10}{b}_{21},a_{11}=(a_{10}-b_{10})(a_{10}-b_{01}-b_{10}- \\ 2b_{21}),a_{12}=-b_{21},a_{13}=b_{21}(b_{01}-a_{10}+b_{10}+2b_{21}),b_{10}(b_{10}-a_{10}),b_{02}=(a_{10}-b_{10}- \\ 2b_{21})(a_{10}-b_{01}-b_{10}-2b_{21}),b_{11}=b_{10}(b_{10}-a_{10}+2b_{21}),b_{20}=b_{21}(a_{10}-b_{01}-b_{10}-2b_{21}), \\ {b}_{31}=-b_{10}{b}_{21}. \end{gathered}$$

The corresponding systems are F-factor $\phi$-reversible.

As shown in [3], the generalized reversibility can lead to the local analytic integrability of the system (the existence of a center in the real case; see [8]). We discuss the analytic integrability within the found families.

The systems corresponding to $\mathbf{V}\left(J_1\right)$ and $\mathbf{V}\left(J_4\right)$ become quadratic after a rescaling of time. Therefore, we do not consider them here, as the family of quadratic systems is classified in [22].

The system corresponding to $\mathbf{V}\left(J_2\right)$ is

$$\begin{array}{cc}\hfill \dot{x}=& x-a_{10}{x}^2+\frac{1}{4}\left(a_{10}^2+2a_{10}{a}_{12}-b_{01}^2+2b_{01}{a}_{12}-4b_{02}\right)x^3\hfill \\ & -\frac{1}{2}(a_{10}{b}_{10}+b_{01}{b}_{10})x^{2}y+b_{02}x{y}^2+b_{10}xy-a_{12}{y}^2,\hfill \\ \hfill \dot{y}=& -y+\frac{1}{4}\left(a_{10}^2+2a_{10}{a}_{12}-b_{01}^2+2b_{01}{a}_{12}-4b_{02}\right)x^{2}y\hfill \\ & -\frac{1}{2}(a_{10}{b}_{10}+b_{01}{b}_{10})x{y}^2+b_{01}{y}^2+b_{02}{y}^3+b_{10}xy-a_{12}{x}^2.\hfill \end{array}$$

(20)

Using the computational procedure described above, we found that

$$F(x,y)=-1+\frac{1}{2}(a_{10}+b_{01})x+\frac{1}{2}(a_{10}+b_{01})y$$

and involution (19) has the following expression:

$$\phi (x,y)=\left(\frac{2y}{(a_{10}+b_{01})x+(a_{10}+b_{01})y-2},\frac{2x}{(a_{10}+b_{01})x+(a_{10}+b_{01})y-2}\right).$$

(21)

System (20) has two invariant lines:

$$l_1=\frac{x\left(-\sqrt{b_{10}}\sqrt{A}+B\right)}{2b_{10}}+\frac{y\left(\sqrt{b_{10}}\sqrt{A}-C\right)}{2b_{10}}+1$$

and

$$l_2=\frac{x\left(\sqrt{b_{10}}\sqrt{A}+B\right)}{2b_{10}}-\frac{y\left(\sqrt{b_{10}}\sqrt{A}+C\right)}{2b_{10}}+1,$$

where

$$\begin{array}{cc}\hfill A=& a_{12}^2{b}_{10}+2a_{12}{b}_{01}{b}_{10}+4a_{12}{b}_{11}+b_{01}^2{b}_{10}+4b_{02}{b}_{10},\hfill \\ \hfill B=& a_{12}{b}_{10}+b_{01}{b}_{10}+2b_{11},\hfill \\ \hfill C=& a_{12}{b}_{10}+b_{01}{b}_{10}.\hfill \end{array}$$

Using the lines, we obtain the Darboux integrating factor:

$$\mu =l_1^{{\alpha }_1}{l}_2^{{\alpha }_2},$$

where

$$\begin{array}{cc}\hfill {\alpha }_1=& \frac{\sqrt{b_{10}}(b_{10}-2a_{12})}{\sqrt{b_{10}\left({(a_{12}+b_{01})}^2+4b_{02}\right)+4a_{12}{b}_{11}}}-2,\hfill \\ \hfill {\alpha }_2=& -\frac{\sqrt{b_{10}}(b_{10}-2a_{12})}{\sqrt{b_{10}\left({(a_{12}+b_{01})}^2+4b_{02}\right)+4a_{12}{b}_{11}}}-2,\hfill \end{array}$$

yielding that system (20) is Darboux-integrable. Since the integrating factor is analytic in a neighborhood of the origin, system (20) has an analytic first integral near the origin.

For involution (21), the set of fixed points is

$$\mathrm{Fix}\phi =\{(x,y)\in {\mathbb{R}}^2|y=-x\}\cup \left\{(\frac{2}{a_{10}+b_{01}},\frac{2}{a_{10}+b_{01}})\right\}.$$

For points p on the line $y=-x$, we see that $F\left(p\right)=-1$. This means that $F(0,0)=-1,$ and calculating $D\mathcal{X}\left(\right(0,0\left)\right)$, one obtains $-1$ for any value of the parameters. Theorem 1.2 from [3] tells us that the origin, which is in Fix $\phi$, is a saddle point of the system.

In the case of $\mathbf{V}\left(J_3\right)$, we have the system

$$\begin{array}{cc}\hfill \dot{x}=& x-(3a_{01}-b_{01}+3b_{21})x^2-a_{01}xy+b_{21}{y}^2\hfill \\ & +(2a_{01}^2-a_{01}{b}_{01}+3a_{01}{b}_{21}-b_{01}{b}_{21}+b_{21}^2-b_{31})x^3\hfill \\ & +(3a_{01}^2-a_{01}{b}_{01}-b_{20}+3a_{01}{b}_{21}-b_{01}{b}_{21})x^{2}y\hfill \\ & +(2a_{01}^2-a_{01}{b}_{01}-b_{20}+a_{01}{b}_{21}-b_{01}{b}_{21}-b_{21}^2)x{y}^2-(a_{01}{b}_{21}+b_{21}^2+b_{31})y^3,\hfill \\ \hfill \dot{y}=& -y+b_{21}{x}^2+b_{31}{x}^3+b_{21}xy+b_{20}{x}^{2}y+b_{01}{y}^2+b_{20}x{y}^2\hfill \\ & +(a_{01}^2-a_{01}{b}_{01}+2a_{01}{b}_{21}-b_{01}{b}_{21}+b_{21}^2+b_{31})y^3.\hfill \end{array}$$

(22)

Computations give

$$F(x,y)=-{(1-(a_{01}+b_{21})x-(a_{01}+b_{21})y)}^2$$

and

$$\phi (x,y)=\left(\frac{y}{-1+(a_{01}+b_{21})x+(a_{01}+b_{21})y},\frac{x}{-1+(a_{01}+b_{21})x+(a_{01}+b_{21})y}\right).$$

The system has two invariant lines

$$l_1=1-(a_{01}+b_{21})x-(a_{01}+b_{21})y$$

and

$$l_2=1-2(a_{01}+b_{21})x+{(a_{01}+b_{21})}^2{x}^2-2(a_{01}+b_{21})y+2{(a_{01}+b_{21})}^{2}xy+{(a_{01}+b_{21})}^2{y}^2,$$

yielding the Darboux first integral $\Psi =l_1{l}_2^{-1/2}$, which is analytic in a neighborhood of the origin.

Thus, using involution (21) and looking for the generalized reversibility, we find two new families of locally analytically integrable cubic systems.

We provide two examples of phase portraits of the generalized reversible systems.

Example 1.

Using the P4 program [23] we draw the phase portrait of system (20) for fixed values of parameters. We set $a_{12}=-2,$ $b_{10}=1,b_{01}=-4,b_{02}=2,a_{10}=6$. Then, the system is

$$\begin{array}{cc}\hfill \dot{x}=& x-6x^2+x^3+xy-x^{2}y+2y^2+2x{y}^2,\hfill \\ \hfill \dot{y}=& -y+2x^2+xy+x^{2}y-4y^2-x{y}^2+2y^3\hfill \end{array}$$

(23)

This system is orbitally F-factor φ-reversible with

$$\phi (x,y)=\left(\frac{y}{-1+x+y},\frac{x}{-1+x+y}\right)$$

and

$$F(x,y)=-1+x+y.$$

The curve of fixed point Fix φ of the map $\phi (x,y)$ is the line $y=-x$. The only points that belong to Fix φ and are stationary are $(0,0),(\frac{1}{4},-\frac{1}{4})$ and $(1,1).$ We already know that $D\mathcal{X}\left(\right(0,0\left)\right)=-1<0,$ whereas straightforward calculations show that $D\mathcal{X}\left((\frac{1}{4},-\frac{1}{4})\right)=-\frac{43}{16}<0$. Using Theorem 1.2 in [3] and looking at the phase portrait of the system, we see that both the origin and the point $\left(\frac{1}{4},-\frac{1}{4}\right)$, which are on the line $y=-x$ (the dashed line), are saddle points (see Figure 1). It is evident from the picture that two other singular points of the system are nodes.

Figure 1. Phase portrait of (23).

In Example 1, we treated a real system from family (20). In the next example, we deal with a complex system from the family.

Example 2.

We can consider (20) as the complexification of a real system ([8], Chapter 3). To recover the real system embedded into (20), we multiply both equations of (20) by $i=\sqrt{-1}$ and make the substitution

$$a_{jk}=A_{jk}+i{B}_{jk},b_{kj}=\overline{a_{jk}},\mathrm{where}{A}_{jk},B_{jk}\in \mathbb{R},x=u+iv,y=u-iv.$$

(24)

Then, from (20), we obtain the real system

$$\begin{array}{cc}\hfill \dot{u}=& -v+(B_{01}+B_{10}+B_{21})u^2-A_{10}(B_{01}+B_{10}+B_{21})u^3\hfill \\ & +2A_{10}uv+4A_{20}{u}^{2}v+A_{10}(B_{01}-B_{10}+B_{21})u{v}^2+(-B_{01}+B_{10}-B_{21})v^2\hfill \\ \hfill \dot{v}=& \phantom{-}u-A_{10}{u}^2+(2B_{01}-2B_{21})uv-A_{10}(B_{01}+B_{10}+B_{21})u^{2}v\hfill \\ & +A_{10}{v}^2+4A_{20}u{v}^2+A_{10}(B_{01}-B_{10}+B_{21})v^3.\hfill \end{array}$$

(25)

Under condition (24), the complex line $\Pi :=\{(x,y):y=\overline{x}\}$ is invariant for system (20). Viewing $\Pi$ as a two-dimensional hyperplane in ${\mathbb{R}}^4$, the flow on $\Pi$ is the flow of system (25) on ${\mathbb{R}}^2$; that is, the phase portrait of the real system (25) is embedded in an invariant set in the phase portrait of the complex system obtained from (20) after the multiplication by i.

The choice of the parameters

$$A_{10}=-1,A_{20}=-1,B_{01}=-3,B_{10}=0,B_{21}=1$$

gives us the system

$$\begin{array}{cc}\hfill \dot{u}=& -v-2u^2-2u^3-2uv-4u^{2}v+2v^2+2u{v}^2,\hfill \\ \hfill \dot{v}=& u+u^2-8uv-2u^{2}v-v^2-4u{v}^2+2v^3\hfill \end{array}$$

(26)

It has two centers (which correspond to the saddles of the complex system). One center is located at the origin and the other one is at the point $(0,\frac{1}{2}).$ The point $(-1,0)$ is a saddle point (see Figure 2).

Figure 2. Phase portrait of (26).

6. Conclusions

Our study delves into the intriguing realm of rational involutions, which are ratios of polynomials of a degree of, at most, two. For two families of such rational maps, we successfully identified the sets in the parameter space that give rise to the involutive maps. This exploration has been made possible through the application of algorithms from computational commutative algebra, particularly the ones based on the Groebner basis theory. To enhance computational efficiency, modular arithmetic was employed.

Our findings not only contribute to the understanding of rational involutions but also unveil their potential as a powerful tool for identifying reversible and integrable systems within families of planar polynomial ordinary differential equations. Our work underscores the significance of rational involutions in shedding light on the intricate interconnections between algebraic properties and the qualitative aspects of differential equations.

We note that studying rational involutions $\phi :{\mathbb{K}}^n\mapsto {\mathbb{K}}^n$, where $n>2$ and their relationship to the integrability of higher-dimensional systems of ODEs, represents an interesting and important problem.