Some Methods Of Finding The First Integral For Ordinary Differential Equations

It is well known that integrability has been an old butsignificant topic in the field of differential equations. In general,if a system of differential equations has a sufficiently rich set offirst integrals such that its solutions can be expressed by theseintegrals, then we call the system is integrable. Thererfore it is animportant problem to find a simple test for the existence ofnontrivial first integrals in considering the for differentialequations. But till now there is no decision procedure to built thefirst integral for general differential equations, even we considerthe simple polynomial first integrals for polynomial differentialequations.In this paper , we describe three methods for finding thepolynomial first integral or rational first integral for polynomialsystems: the method of invariant measure, theLagutinskii-Levelt procedure and the method of compatiblevector fields. And we will apply these methods to some system,such as Lorenz system,homogeneous Lotka-Volterra system,homogeneous May-Leonard system , Halphen system andOhyama system。In Chapter 2, we present the method of invariant measurre.Let us consider a system of ODEsddt xi = Fi( x1 ,x2,x3),i=1,2,3, (1)with smooth homogenous right-hand side functions Fi ,i =1,2,3of the same degree m。Let3123321232111231112233,(,,),,(,,),,(,,),detxFxxxdxxFxxxdxxFxxxdxω = ωdx +ωdx+ωdxdef (2)Generally, the from ω is not closed. However, if we find anintegrating factor of ω , i.e., a function f ≠0,such that thefrom ? = ωfis closed,i.e., d ?=0,then,at least locally,? = dφ,and, as it is easy to see that φ is an integral of the system (1). S.I. Popov shows that when the integrating factor of ω is ahomogeneous function, then it is the density of the invariantmeasure(the Last Jacobi Multiplier)Theorem 2.1 Let ω ≠0(i.e. at least one of ω i≠0). A smoothhomogeneous function f = f( x1 ,x2,x3) of degree l is anintegrating factor of ω if and only if l = ?2 ?m, and f is theLast Jacobi Multiplier for the system (1).The following result shows that the Last Jacobi Multiplier ofquadratic factorizable system can be found easily.ddt xi = xiLi( x1 ,...,xn),i=1,...,n, (3)where (,...,)1,...,.1L x1 xnLxjinji n= ∑ij==Let [ ]L = Lij。Theorem 2.2 If det( L )≠0, then there exist the uniquelydetermined real numbers α 1 ,...,αn such that the measuredxxn dxdxnnμ = 1α 1 ... α1... (4)is invariant for the (local) flow induced by the system (3).For n =3, we have l + m+2 =α 1+α2+α3.当 n =3时, l +m+2 =α 1+α2+α3. And as a conclusion fromtheorem 2.2, we derive the following procedure for obtaining thefirst integral of system (3).Algorithm 1: Let n = 3, detL≠0, and α 1 , α2,α3definded by(1)0,1.1nLi jLiiinj∑j ++=≤≤=αsatisfy the following conditionα 1 + α2+α3=?4.1. Take the following 1-form∏== 311i? ωxα, (5)whereω is defined by (2) withFi ( x1 ,x2,x3)= xiLi(x1,x2,x3),i=1,2,3。2. Form ? , at least locally, is an exact differential, i.e? = dφ, for some function φ . If this function is defined globally,then it is a first integral of the factorizable system (3).As we will see below, when this procedure is applicable, italways allows one to obtain a globally defined first integral. Moreprecisely, this procedure leads to an explicit formula for globallyfirst integral of a class of three-dimensional factorizable system(3).φ = {[ Li j ]det[Lij]≠0,α1 +α2+α3=?4}The system (3) has globally first integral of the form.Theorem 2.3 Let matrix L ∈φ. If α i? {? 1,?2},1≤i≤3. Thenthe system (3) has globally first integral of the form:( 1 ,2,3)111221331(112233)I xxx= xα + xα+xα+Ix+Ix+Ixwhere)I 1 = 13 (Lα31 1 ? +L221+Lα112? +L131+Lα213? +L111.The Lagutinskii-Levelt procedure introduced in Chapter 3is an algebraic tool for proving the integrability of thehomogeneous polynomial systems of ODEs [14], see[15,16] fortheir applications. This method was presented in the book byJouanolou [9]. In [9], Levelt proved, using this method, thatJouanolou system= , =,z=ys∈N,z≥2,dtyxddtxzddtd ssshas no polynomial partial first integral(see Definition 3.2 below).In fact, the basic ideas of the method were already introduced byLagutinskii [10,11], more details on Lagutinskii and his workscan be found in [2]。Consider the following systems of ODEsxVxxindtdi = i( 1 ,...,n),=1,..., (6)where V i ∈ C[x 1 ,...,xn] ,i=1,...,n.Therorem 3.1 (1) An element F = A/ B∈C(x1 ,...,xn), withrelatively prime polynomials is a first integral of (6) if and only ifA and B are Darboux polynomial with the same "eigenvalue" P,i.e. dV ( A)=PA and dV ( B)=PB.(2) If F ∈ C( x1 ,...,xn)is a Darboux polynomial of (6), then allits irreducible factors are also Darboux polynomials.(3) The finite product of Darboux polynomials. Moreprecisely,if d V ( Fi)= PiFi,i=1,...,s,then ( 1)(1)(1).∏∑∏d V is = Fi= is =Piis =Fi(4) For homogeneous derivations d V. If F is a Darbouxpolynomial of (6), with eigenvalue P such that dV (F )= PF,then Pis a homogeneous polynomial and all homogeneous componentsof F are also Darboux polynomials of (6) corresponding to thesame P.(5) Let us assume that d Vis homogeneous derivations and letd V ( Fi)= PiFi,1≤i≤s,where Fi 1 ≤i≤s are irreduciblehomogeneous polynomial. If the polynomials Pi , for 1 ≤i ≤s, arelinearly independent over Z, then any Darboux polynomial of d Vis of form sii Fi α∏=1 ,where α i,1 ≤i ≤s,are non-negative integers.(6) If derivations d Vsatisfies all assumptions stated in (5),then it has no rational first integral different from constant.Algorithm 2:1. For a given homogeneous system (6) find all Darbouxpoints.2. Find Lagutinskii-Levelt exponents for all Darboux points.3. If for a chosen Darboux point∑?=1=1.nki kρ kχ , ∑?=1=≤1deg..nki k hmhas no solution, then system (6) has no a polynomial firstintegral(if P =0) or a rational first integral (if P ≠0).We introduce the method of compatible vector fields inChapter 4. This method was presented by Strelcyn andWojciechowski in 1988[18] and, later on, was appliedsuccessfully to a study of the Lotka-Volterra system[5] .Althoughthis method is algorithmic and effective, but it was not used tostudy systems other than those mentioned in [18, 5].With a given vector fieldX = X1 ( x)?1+X2(x)?2+X3(x)?3considered equationsddt xi = Xi( x),i=1,2,3, (7)If vector field X and Y = Y1 ( x)?1+Y2(x)?2+Y3(x)?3 arecompatible, i.e[ X ,Y]= a(x)X+b(x)Y?x∈R3,where a and b are two funcitions, and [? , ?], denotes the Liebracket. And Y possesses two independent first integrals u andv . We can deduce from the Frobenius integrability theorem, in aneighbornood of a point where they are linearly independent,there exists a local two-dimensional foliation tangent to X and Y .Let us assume that invariant foliation tangent to X and Y is levelsurfaces of a function G . Then G is a first integral ofY (and X ), and is a function of (u ,v)only since for avector field inR 3 can have two functionally independent first integrals. We caneasily show that G is a first integral of the following singleequation:()(,),(,)()XvGuvGuvXudvdu = =(8)If f (u ,v)is a first integral of the above equation thenF ( x1 ,x2,x3)= f(u(x1,x2,x3),v(x1,x2,x3)) is a first integral of(7).For the above ideas, we can show the following algorithms.Algorithms 3:1. Let system[7]depend on several parameters. Find valuesof parameters such that there exists a linear vector fieldcompatible with X .2. For values of parameters write down equation (8) and findits general solution.3. For values of parameters, system (7) possesses a firstintegral that is explicitly given.