First Integrals Of Some Nonlinear Systems

As well known,investigation of the "integrability" and "noninte-grability" of nonlinear systems is classical work in the field of studying dynamical systems. The relevant problems have been received widely at-tention by researchers.In general integrable systems are characterized by regular, predictable behaviour for initial conditions and all times.And in many cases.non-integrable systems have regions in the phase space of their dependent vari-ables where the motion is irregular and chaotic.Over the years,many scholars have researched the integrability of dynamical systems and developed many effective methods, such as Painleve singularity analysis, Lie group, Lax pair. Ziglin theory and so on. A lot of integrable systems have been found by using these methods. But it is difficult to determine whether a given system is integrable or not,because there is not an effective method to construct first integral of a given system so far.So people turned to research the nonintegrability of systems,namely proving the systems do not admit any first integral in some given function spaces.In this article,we consider the existence and partial existence of first integrals of some nonlinear systems. In the second part, we give a algorithm for the nonexistence and partial existence of formal first integrals under the general resonances.Consider the autonomous system of differential equations x= f(x), (1) where x=(x1,...,xn)T∈Cn,f(x)=(f1(x),...,fn(x))T is a vector-value analytic function, and f(0)=0.Then the system can be rewritten as x=Ax+F(x), (2) where A=Dr(0), F(x)=O(‖x‖2).Early in the 19th century, H. Poincare considered the nonexistence of analytic first integral for system (1), He gave the following result:Theorem 1 Let f be analytic and f(0)=0.If the eigenvalues of Jacobi matrix Df(0) do not satisfy any resonance condition of the type then the system, (1) does not have any analytic first integral in a neighbor hood of the fixed point x=0.Along the Poincare's idea, a lot of work had been expanded. But most of works all considered the cases eigenvalues of Jacobian matrix A are not resonant or simply resonant.we give a algorithm for the nonexistence and partial existence of formal first integrals under the general resonances. Without loss of generality,we assume that A=diag(λ1,...,λn).Step 1: Let If rankΩ=0, then systems (2) does not have any formal first integral in the neighbourhood of x=0. The algorithm is over.If rankΩ=l>0, let. Go into the next step.Step 2: Let rankΘ=s(00.consider system where t1=lnt, u0=e-1/lt1.Let whereμs+1=-1/l.Take (7) andΘ1 instead of (2) andΘ, and return toStep 2.Remark 1 If there exist i0∈N, such that for every j>i0, we have rankΘj=rankΘi0=si0≥2.then we can not get the nonexistence of formal first integral for system (2), while the partil existence of formal first integral for system (2) can obtained, i.e.,the systems (2) at most have si0-1 formal first integrals in a neighbourhood of x=0. For the nonexistence of formal integrals for quasi-periodic system and diffeomorphisms under the general resonant, we have similar results.In the third part,We consider the nonexistence of rational first inte-grals for quasi-periodic system.Consider the quasi-periodic system: where (x,θ)∈Cm×Tn,ω=(ω1,...,ωn) are incommensurable, f(x,θ)= O(‖x‖2),g(x,θ)=O(‖x‖) are analytic functions in x, and 2π-periodic inθ, Assume that x=0 is the fixed point of system (8), i.e. for allθ∈Tn we have f(0,θ)=g(0,θ)=0.Definition 1 Let U C Cm is an open set. A nonconstant functionΦ(x,θ):U×Tn→R is called a first integral of system, (8),if it is 2π-periodic inθ, and a constant along every solution of(8) in U. IfΦ(x,θ) is differentiable, this condition is equivalent to Moreover,ifΦ(x,θ)is analytic(formal)in x, then we callΦ(x,θ)is analytic (formal) first integral of system (8), ifΦ(x,θ) is rational in x, then we callΦ(x,θ)is ratonal first integral system(8).For the nonexistence of rational first for quasi-periodic system, we haveTheorem 2 if the eigenvaluesΛ=(λ1...λn)of A,andω=(ω1,...,ωn) do not satisfy any resonance condition of the type where i denotes the imaginary unit, then the quasi-periodic system (8) does not have any rational first integral at a neighborhood of x=0.In the forth part, we consider the nonexistence and partial existence of rational first integrals for diffeomorphism.Here,we always assume that f(x) is an analytic diffeomorphisms, and f(0) =0.Definition 2 Assume f(x) is an analytic diffeomorphism , U is a connected open subset which contains origin. A functionΦ(x) in U is called a first integral for diffeomorphism f(x), if for any x∈U, f(x)∈U, IfΦ(x) is a rational function and satisfy (10). thenΦ(x) is calked a rational first integral of f(x) in a neighborhood of the fixed point x=0.In some neighborhood of x=0, f(x) can be rewritten as where A=Df(0) is the Jacobi matrix of f at x=0.Λ=(λ1,...,λn) are the eigenvalues of A, F(x)=O(‖x‖2).For the nonexistence of rational first for diffeomorphism, we haveTheorem 3 Assume that diffeomorphism f(x) is analytic.If the eigen-values of A do not satisfy any resonance condition of the type then the diffeomorphism f(x) does not have a rational first integral in the neighborhood of x=0. Moreover we consider the partial integrability for diffeomorphisms. By Theorem 3, if f(x) has a rational first integralΦ(x), thenΦp(x) is homogeneous rational first integral with order p for linear diffeomorphism A,and there exist a nonzero vector k=(k1,...,kn)T∈Zn,such thatΛk=1. Let then F is a subgroup of Zn. We have the following lemmasLemma 1 Assume that matrix A is diagonalizable,η1=(ηl1,...,ηln)T, ...,ηs=(ηs1,...,ηsn)T (0