Nonexistence Of Formal First Integrals For Nonlinear Systems Under Simple Resonances

Nonexistence of Formal First Integrals for Nonlinear Systems under Simple ResonancesDynamical 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.At present,dynamical system has been taken widely applications on many natural science fields like classical mechanics,engineering design,signal transmission,astronomy,geography,meteorology,biology, chemistry,statistics,physics,information and computing sciences and social science fields and so on.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 equations (?)=f(x),x∈U(?)Cⁿ,x=(x¹,…,xⁿ).(1) where 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. For a long time,because of the needs of solving equations,the integrability, partial integrability and non-integrability for differential equations received wide attention by researchers.Our research mainly bases on the following definition.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.And if system(1)does not admit any nontrivial integral,then we say it is non-integrable.It is an important problem to find a simple test for the existence or nonexistence of nontrivial first integrals in given function spaces(such as those of polynomials,rational,analytic or algebraic functions and so on)in considering integrability and non-integrability.Indeed,it is very difficult to find out all the first integrals the system admits,therefore,proving the non-integrability of system is very meaningful.Early in the 18th century,Poinca(?)e[20]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 form Then system(1)does not have any nontrivial analytic first integrals in a neighborhood of x=0.Many previous works all considered the cases the eigenvalues of Jacobian matrix A are N-independent.And so far people have achieved comprehensive and satisfactory results.But when the eigenvalues of A are N-dependent,there is not a satisfying result yet.In the present paper,we will give a verifiable criterion of nonexistence of formal first integral for nonlinear system(1)under simple resonance using Poincaré-Dulac Normal Form Theorem.Here the simple resonance means the Jacobi matrix of vector field at some fixed point has some single multiply zero eigenvalues,and other nonzero eigenvalues which are N-independent.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 third chapter,by using Poincaré-Dulac Normal Form Theorem,we will give the main result of this paper.We consider the following system wherew=(u,v),u=(u₁,…,u_m),v=(v₁,…,v_n),The eigenvalues of A isλ₁,λ₂,…,λ_n,and(?)(u,v)=O(|(u,v)|²),(?)(u,v)=O(|(u,v)|²).Now we make the following hypothesis.(H)The n eigenvaluesλ₁,λ₂,…,λ_n of A is N-independence.Lemma 1 Assume that(H)holds.After a sequence of coordinate transformations, system(3)can be reduced to the following canonical form wherex=(x₁,…,x_m)∈C^m,y=(y₁,…,y_n)∈Cⁿ,f(x)=O(|x|²),g(x,y)= O(|(x,y)|²),g(x,0)≡0,f(x)and g(x,y)are all vector-valued formal series.Lemma 2 Assume that(H)holds.If system(4)have a formal first integral, then it must be independent on y.That is(?)Φ/(?)y=0. Let us expand f(x)as f(x)=f₂(x)+f₃(x)+…+f_x(x)+…, where f_k(x)are vectors of homogeneous polynomial of degree k.Then we have the following lemma.Lemma 3 Assume that(H)holds and f₂(x)(?)0.If system(4)has a formal first integral,then system (?)= f₂(x)(5) has a homogeneous first integral.System(5)can be treated as quasi-homogeneous system of degree 2 with exponents s₁=s₂=…=s_m=1.Letζbe a balance of it,and the relevant Kowalevskaya matrix is K=E+(?)f₂/(?)x(ζ).The main result of this paper as followsTheorem 2 Assume that(H)holds.If the eigenvaluesμ₁,…,μ_m,of Kowalevskaya matrix K associated to the balanceζare N-independent,i.e.,they do not satisfy any resonance condition of the form then system(3)does not have any nontrivial formal first integrals at a neighborhood of(u,v)=0.In the fourth chapter,we will cite two examples to show the validness of our result.