Nonexistence Of Polynomial First Integrals For Quadratic Planar Vector Systems

Dynamical system is an important component of the nonlinear science. It is a subject which takes research on practical problems of dynamics laws about the state changing with the time.For a long time, integrability and non-integrability has been an important problem in dynamical system's research field, and the relevant problems have been received widely attention by researchers.We consider the following autonomic system of differential equationswhere f(x) is an n-dimensional vector-valued analytic function, and f(0)=0.Definition 1 Let U is an open set. A functionΦ:U→C is called a first integral of system (1),ifΦ(x) along every solution curve of (1) are all constants. IfΦ(x) is analytic function, this condition is equivalent toIfΦ(x) and f(x) are formal series in x and satisfy (2), thenΦ(x) is called a formal first integral of system (1) in a neighborhood of the singular point x = 0. Obviously, if a formal first integral is convergent, it is an analytic first integral.We give the definition of integral of system (1).Definition 2 In general, if differential equations system (1) has a sufficiently rich set of first integrals such that its solutions can be expressed by these integrals, then we say the system is integrable.So, we know that whether we could find the first integral is the key that judge integral of the system. Mathematician and Physicist have developed many methods on studying integrable system, for example, the Painlev(?) analysis[2], the Carleman embedding [3], the lie symmetries[3], Prelle-Singer procedure[4] and compatible vector field method[5] etc. With these methods, we find many integrable system.Recently, polynomial first integrals for the following 3-dimensional quadratic polynomial differential system of Lotka-Volterra kind have been characterized by Moulin-Ollagnier [9] and Labrunie [10]. Cair(?) and Llibre [11] classify the polynomial first integrals for the 2-dimensional quadratic polynomial differential system of Lotka-Volterra kind.So far, there is no such a effective method that constructing first integral for general systems. So, we can prove that systems are nonexistence of first integrals.Early in the 18th century, Poincafe[17] first suggested an easily verifiable criterion of non-existence of nontrivial first integral for general autonomic analytic system.Theorem 1 Let A denote the Jacobi matrix of the vector f(x) at x=0. If detA≠0 and the eigenvaluesλ₁,λ₂,…,λ_n of A are N-independent, i.e. they do not satisfy any resonance conditions of the formThen system (1) does not have any nontrivial analytic first integrals in a neighborhood of x=0.In the second chapter, we will introduce some basic definition, property and relevant result about the quasi-homogeneous and semi-quasi-homogeneous systems.In the first section of the third chapter, by according Kowalevskaya matrix of the semiquasi-homogeneous systems, we will give the main result of this paper.We consider the following planar quadratic system.where a_ij and ba_ij both are real number.By according Kowalevskaya matrix of the semi-quasi-homogeneous systems, we will give some sufficient condition of nonexistence of first integrals for the system(3). The main result is following.Theorem 2 If a₀₂=b₂₀=0, when one of the following conditions satisfies, the system(3) does not have any nontrivial polynomial first integrals.Theorem 3 If a₂₀=b₀₂=b₂₀=0, whenorthe system(3) does not have any nontrivial polynomial first integrals.Theorem 4 If a₁₁=ba₁₁,a₂₀=b₀₂,a₂₀=b₀₂,when one of the following conditions satisfies, the system(3) does not have any nontrivial polynomial first integrals.In the second section, we will give three examples that the system does not have any polynomial first integrals.