\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 78, pp. 1--6.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/78\hfil Explicit non-algebraic limit cycles] {Polynomial differential systems with explicit non-algebraic limit cycles} \author[R. Benterki, J. Llibre \hfil EJDE-2012/78\hfilneg] {Rebiha Benterki, Jaume Llibre} % in alphabetical order \address{Rebiha Benterki \newline D\'{e}partement de Math\'{e}matiques, Centre Universitaire de Bordj Bou Arr\'{e}ridj, Bordj Bou Arr\'{e}ridj 34265, El anasser, Algeria} \email{r\_benterki@yahoo.fr} \address{Jaume Llibre \newline Departament de Matematiques, Universitat Aut\`{o}noma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain} \email{jllibre@mat.uab.cat} \thanks{Submitted February 13, 2012. Published May 15, 2012.} \subjclass[2000]{34C29, 34C25} \keywords{Non-algebraic limit cycle; polynomial vector field} \begin{abstract} Up to now all the examples of polynomial differential systems for which non-algebraic limit cycles are known explicitly have degree at most 5. Here we show that already there are polynomial differential systems of degree at least exhibiting explicit non-algebraic limit cycles. It is well known that polynomial differential systems of degree 1 (i.e. linear differential systems) has no limit cycles. It remains the open question to determine if the polynomial differential systems of degree 2 can exhibit explicit non-algebraic limit cycles. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \allowdisplaybreaks \section{Introduction and statement of the main results}\label{s1} Probably the existence of limit cycles is one of the more difficult objects to study in the qualitative theory of differential equations in the plane. There is a huge literature dedicated to this topic, see for instance the book of Ye et al \cite{Ye}, or the famous Hilbert 16th problem \cite{Hi} and \cite{Il}. Publications more closely related to the problem in this article are \cite{GGT, GG, Al, DL, LL, LM}. A \emph{polynomial differential system} is a system of the form \begin{equation}\label{e1} \begin{gathered} \dot{x}=P(x,y), \\ \dot{y}=Q(x,y), \end{gathered} \end{equation} where $P(x, y)$ and $Q(x, y)$ are real polynomials in the variables $x$ and $y$. The \emph{degree} of the system is the maximum of the degrees of the polynomials $P$ and $Q$. As usual the dot denotes derivative with respect to the independent variable $t$. A \emph{limit cycle} of system \eqref{e1} is an isolated periodic solution in the set of all periodic solutions of system \eqref{e1}. If a limit cycle is contained in an algebraic curve of the plane, then we say that it is \emph{algebraic}, otherwise it is called \emph{non-algebraic}. In other words a limit cycle is algebraic if there exists a real polynomial $f(x,y)$ such that the algebraic curve $f(x,y)=0$ contains the limit cycle. In general, the orbits of a polynomial differential system \eqref{e1} are contained in analytic curves which are not algebraic. To distinguish when a limit cycle is algebraic or not, usually, it is not easy. Thus, the well-known limit cycle of the van der Pol differential system exhibited in 1926 (see \cite{Po}) was not proved until 1995 by Odani \cite{Od} that it was non-algebraic. The van der Pol system can be written as a polynomial differential system \eqref{e1} of degree $3$, but its limit cycle is not known explicitly. These previous years (from 2006 up to now) several papers have been published exhibiting polynomial differential systems for which non-algebraic limit cycles are known explicitly. This means that in some coordinates we have an explicit analytic expression of the curve containing the non-algebraic limit cycle. The first explicit non-algebraic limit cycle, due to Gasull, Giacomini and Torregrosa \cite{GGT}, was for a polynomial differential system of degree $5$. Of course, multiplying the right hand part of this polynomial differential system of degree $5$ by $(a x+b y+c)^n$ with $n$ an arbitrary positive integer, where the straight line $a x+b y+c=0$ must be chosen in such a way that it does not intersect the explicit limit cycle of the system, we get a polynomial differential system of degree $5+n$ exhibiting an explicit non-algebraic limit cycle. Immediately after this first paper appeared the paper of Al-Dosary \cite{Al} inspired by \cite{GGT} (note that this reference is quoted in \cite{Al}), providing a similar polynomial differential system of degree $5$ exhibiting an explicit non-algebraic limit cycle. Gin\'{e} and Grau \cite{GG} provide a polynomial differential system of degree $9$ exhibiting simultaneously two explicit limit cycles one algebraic and another non-algebraic. Note that the paper \cite{GGT} is also quoted in \cite{GG}. The aim of this paper is to show that there exist polynomial differential systems of degree $3$ exhibiting explicit non-algebraic limit cycles. Thus, our main result is the following one. \begin{theorem}\label{thm1} The differential polynomial system of degree $3$, \begin{equation}\label{e2} \begin{gathered} \dot{x}= x+(y-x)(x^2-x y+y^2), \\ \dot{y}= y-(y+x)(x^2-x y+y^2), \end{gathered} \end{equation} has a unique non-algebraic limit cycle whose expression in polar coordinates $(r,\theta)$, defined by $x= r \cos \theta$ and $y= r \sin \theta$, is \begin{equation}\label{e3} r(\theta)= e^{\theta } \sqrt{r_*^2-f(\theta)}, \end{equation} where \begin{gather*} r_*= e^{2 \pi} \sqrt{ \frac{f(2\pi)}{e^{4 \pi}-1}}\approx 1.1911644871948721\dots, \\ f(\theta)= 4 \int_0^{\theta} \frac{e^{-2 s}}{2-\sin (2s)}\, ds. \end{gather*} Moreover, this limit cycle is a stable hyperbolic limit cycle. \end{theorem} The above theorem is proved in section \ref{s2}. In short, since it is well known that the linear differential systems (or polynomial differential systems of degree $1$) have no limit cycles, it remains the following open question: \noindent\textbf{Open question}. \emph{Are there or not polynomial differential systems of degree $2$ exhibiting explicit non-algebraic limit cycles.} \section{Proof of Theorem \ref{thm1}}\label{s2} The polynomial differential system \eqref{e2} in polar coordinates becomes \begin{equation}\label{e4} \begin{gathered} \dot r= r+ \frac{1}{2} (\sin (2 \theta )-2) r^3, \\ \dot \theta= \frac{1}{2} r^2 (\sin (2 \theta )-2). \end{gathered} \end{equation} Taking as independent variable the coordinate $\theta$, this differential system writes \begin{equation}\label{e5} \frac{d r}{d\theta}= r+\frac{2}{r(\sin (2 \theta )-2)}. \end{equation} Note that since $\dot \theta<0$, the orbits $r(\theta)$ of the differential equation \eqref{e5} has reversed their orientation with respect to the orbits $(r(t),\theta(t))$ or $(x(t),y(t))$ of the differential systems \eqref{e4} and \eqref{e2}, respectively. It is easy to check that the solution $r(\theta;r_0)$ of the differential equation \eqref{e5} such that $r(0;r_0)= r_0$ is \begin{equation}\label{e6} r(\theta;r_0)= e^{\theta } \sqrt{r_0^2-f(\theta)}, \end{equation} where $f(\theta)$ is the function defined in the statement of Theorem \ref{thm1}. Clearly the unique equilibrium point of the differential system \eqref{e2} is the origin of coordinates, which is an unstable node because its eigenvalues are $1$ with multiplicity two, for more details see for instance \cite[Theorem 2.15]{DLA}. This equilibrium point in polar coordinates become $r=0$. This is the unique point of the plane where the differential equation \eqref{e5} is not defined. But we can extend the flow of this differential equation to $r=0$, assuming that at the origin of the plane in polar coordinates we have an unstable node. \begin{figure}[ht] \begin{center} \includegraphics[width=0.5\textwidth]{fig1} \end{center} \caption{The phase portrait in the Poincar\'{e} disc of the polynomial differential system \eqref{e2}} \label{aa} \end{figure} The periodic orbits $r(\theta;r_0)$ of \eqref{e5} must satisfy $r(2\pi;r_0)= r_0$. A solution of this equation is $r_0= r_*$, where $r_*$ is defined in the statement of Theorem \ref{thm1}. So, if $r(\theta;r_*)>0$ for all $\theta\in \mathbb{R}$, we shall have $r(\theta;r_*)>0$ would be a periodic orbit, and consequently a limit cycle. In what follows it is proved that $r(\theta;r_*)>0$ for all $\theta\in \mathbb{R}$. Indeed \begin{align*} r(\theta;r_*)&= e^{\theta} \sqrt{\frac{e^{4\pi}}{e^{4\pi}-1}f(2\pi)- f(\theta)} \\ &\geq e^{\theta} \sqrt{f(2\pi)-f(\theta)} \\ &= 2e^{\theta} \sqrt{ \int_\theta^{2\pi} \frac{e^{-2 s}}{2-\sin (2s)}\, ds }>0, \end{align*} because $e^{-2 s}/(2-\sin (2s))>0$ for all $s\in \mathbb{R}$. An easy computation shows that \[ \frac{d\, r(2\pi;r_0)}{d\, r_0}\big|_{r_0= r_*} =e^{4\pi}>1. \] Therefore the limit cycle of the differential equation \eqref{e5} is unstable and hyperbolic, for more details see \cite[section 1.6]{DLA}. Consequently, this is a stable and hyperbolic limit cycle for the differential system \eqref{e2}. Clearly the curve $(r(\theta)\cos \theta, r(\theta)\sin \theta)$ in the $(x,y)$ plane with \[ r(\theta)^2= e^{2\theta } (r_*^2-f(\theta)), \] is not algebraic, due to the expression $e^{2\theta } r_*^2$. More precisely, in cartesian coordinates the curve defined by this limit cycle is \[ f(x,y)= x^2+y^2-e^{2 \arctan(y/x)} \Big(r_*^2-4 \int_0^{\arctan(y/x)} \frac{e^{-2 s}}{2-\sin (2s)} \, ds\Big)=0. \] If the limit cycle is algebraic this curve must be given by a polynomial, but a polynomial $f(x,y)$ in the variables $x$ and $y$ satisfies that there is a positive integer $n$ such that $\partial^n f/(\partial x)^n =0$, and this is not the case because in the derivative \begin{align*} \frac{\partial f}{\partial x}&= 2 x+\frac{2y e^{2 \arctan(y/x)} }{x^2+y^2} \Big(r_*^2-4 \int_0^{\arctan (y/x)} \frac{e^{-2 s}}{2-\sin(2 s)} \, ds\Big) \\ &\quad -\frac{4 y}{(x^2+y^2) \big(2-\sin \big(2\arctan(y/x)\big)\big)} \end{align*} it appears again the expression \[ e^{2 \arctan(y/x)} \Big(r_*^2-4 \int_0^{\arctan(y/x)} \frac{e^{-2 s}}{2-\sin (2 s)} \, ds\Big), \] which already appears in $f(x,y)$, and this expression will appear in the partial derivative at any order. Now we shall prove that the limit cycle given by $r(\theta;r_*)$ is the unique periodic orbit of the differential system, and consequently the unique limit cycle. We recall the so called Generalized Dulac's Theorem, for a proof of it see \cite[Theorem 7.12]{DLA}. \begin{theorem}\label{thm2} Let $R$ be an $n$-multiply connected region of $\mathbb{R}^2$ (i.e. $R$ has one outer boundary curve, and $n-1$ inner boundary curves). Assume that the divergence function $\partial P/\partial x+\partial Q/\partial y$ of the $C^1$ differential system $\dot x= P(x,y)$, $\dot y= Q(x,y)$ has constant sign in the region $R$, and is not identically zero on any subregion of $R$. Then this differential system has at most $n-1$ periodic orbits which lie entirely in $R$. \end{theorem} We take as new independent variable the variable $\tau$ defined by $d\tau= (x^2 + y^2) (x^2 - x y + y^2) dt$. Since $(x^2 + y^2) (x^2 - x y + y^2)$ only vanishes at the origin of coordinates the differential system \eqref{e2} and the differential system \begin{equation}\label{e222} \begin{gathered} x'= \frac{x+(y-x)(x^2-x y+y^2)}{(x^2 + y^2) (x^2 - x y + y^2)}, \\ y'= \frac{y-(y+x)(x^2-x y+y^2)}{(x^2 + y^2) (x^2 - x y + y^2)}, \end{gathered} \end{equation} where the prime denotes derivative with respect to the variable $\tau$, have the same phase portrait in $R= \mathbb{R}^2\setminus \{(0,0)\}$. An easy computation shows that the divergence of the differential system \eqref{e222} is \[ -\frac{2}{(x^2 + y^2) (x^2 - x y + y^2)}<0 \quad \text{in } R. \] So, by Theorem \ref{thm2}, and since $R$ is $2$-multiply connected region of $\mathbb{R}^2$ it follows that the differential system \eqref{e222} and consequently the differential system \eqref{e2} has at most one periodic solution. In short, the unique periodic solution of system \eqref{e2} is $r(\theta;r_*)$. This completes the proof of Theorem \ref{thm1}. Now we shall present the phase portrait of the differential system \eqref{e2} in the Poincar\'{e} disc, see the Poincar\'{e} compactification in \cite[Chapter 5]{DLA}. Since the polynomial $\dot x y-\dot y x= (x^2 + y^2) (x^2 - x y + y^2)$ has no real linear factors, the compactification of Poincar\'{e} of the differential system \eqref{e2} has no equilibrium points at infinity, i.e. the infinity is a periodic orbit. Doing the change of variables $r=1/\rho$, the infinity of the differential equation \eqref{e5} passes at the origin, and equation \eqref{e5} becomes \[ \frac{d \rho}{d\theta}= -\rho -\frac{2 \rho^3}{\sin(2\theta)-2}. \] Hence, clearly $\rho=0$ is an stable equilibrium point of this differential equation, consequently the periodic orbit at infinity of the differential equation \eqref{e3} is an unstable limit cycle. 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