The following Liénard system \[ \Dot{x}=-y,\quad\Dot{y}=f(x)+yg(x)\tag{1} \] is considered with rational functions \(f\) and \(g\), where the functions \(f\) and \(g\) are linearly independent and holomorphic, and \(f(0)=g(0)=0\) and \(f'(0)=1\). Firstly, the definition of degree of an element \(g(x)/h(x)\) of the field \(k(x)\) and the definition of an algebraic function are given. Then, the theorems of Steinitz, Lüroth, Noether, Cherkas, Otrakov, Amel’kin and Urabe, which are used through the paper, are given, respectively.
In the first section, two theorems are proved. The first theorem includes necessary and sufficient conditions for the System (1) to have a center. The second theorem is about Otrakov functions of (1) as a holomorphic system and a rational system.
In the second section, two theorems, Theorems 3 and 4 are proved such that System (1) has a center. These theorems include necessary and sufficient conditions, when the system (1) can be represented in the form \[ \Dot{x}=-y,\quad \Dot{y}=r_{1}(B(x))B'(x)+yr_{2}(B(x))B'(x),\tag{2} \] in which \(r_{1}(x)\), \(r_{2}(x)\), and \(B(x)\) are irreducible rational functions, \(r_{i}(0)\neq\infty\), \(i=1,2\), and \[ B(0)=B'(0)=0,\quad B''(0)\neq 0.\tag{3} \] Later, the definitions of Lüroth element, algebraic and logarithmic part of a function \(F(x)\), where \(F(x)=\int_{0}^{x}f(t)dt\), are given.
In the third section, an application of Theorem 4 is given as Theorem 5 to construct a rational Liénard system with a center. In the theorem, an example of the construction of a rational Liénard system with a center is given and then it is proved.
In the fourth section, the definition of generating functions for a holomorhic function is given. Then, System (1) is represented in that defined generating function. Next, in the following theorems, some necessary and sufficient conditions for a holomorphic function to be a generic function, the relation formula of any two generating functions of System (1), some necessary and sufficient conditions for a function to be the inverse to a holomorhic function, which is detailed with an example and a lemma, are given.
In the fifth section, the following special case of the Liénard System (1) is considered: \[ \Dot{x}=-y,\quad \Dot{y}=f(x),\tag{4} \] where \(f(x)\in\mathbb{R}(x)\), \(f(0)=0\), and \(f'(0)=1\). The theorem by Amel’kin, which is about the isochronity problem for System (4), is given. Then another theorem by Urabe, which is again about the isochronity problem for the System (4), is given. Both of these theorems are used in the paper. By Lemma 2, a necessary and sufficient condition for the System (4) to be isochronous is given. Then, with a corallary, it is asserted that in the case of the System (4) is isochronous, then the function given before is an algebraic function. Furthermore, the next given theorem is on the existence of a unique isochronous System (4) with a rational function \(f(x)\). Finally, with the last theorem, a rational Liénard system with an isochronous center is constructed.
It should be noted that a serious effort has been done to prepare the result of this paper by the author.