Formal First Integrals of General Dynamical Systems

The goal of this paper is trying tomake a complete study on the integrability for general analytic nonlinear systems by first integrals. We will firstly give an exhaustive discussion on analytic planar systems. Then a class of higher dimensional systems with invariant manifolds will be considered; wewill develop several criteria for existence of formal integrals and give some applications to illustrate our results at last.


Introduction
Investigating the existence of first integrals is well known as an important problem in the study of nonlinear systems arising in applied mathematics and physics [1].In this paper, we will mainly consider the related topics for the general dynamical system, ż =  () ,  = ( 1 , . . .,   ) ∈ C  , with -dimensional analytic vector-valued function () satisfying (0) = 0.A function Φ :  → C ( an open set) is said to be a first integral of (1), if Φ() is a constant along any solution curve of system (1).In general, if system (1) has sufficient first integrals, such that the general solution can be expressed by quadrature of these integrals, we say it is integrable.

Remark 1.
(1) It is well known that -dimensional autonomous system (1) with  − 1 functionally independent first integrals on  is integrable as the corresponding solutions can be solved by implicit function theorem.Especially for planar systems, the existence of first integral is, in fact, equivalent to integrability.
(2) If a first integral Φ() of ( 1) is differentiable on , it obviously satisfies where ∇  denotes the gradient with respect to  and ⟨⋅, ⋅⟩ denotes the classical Euclidian inner product.If Φ() can be expanded in formal series in a neighborhood of the fixed point  = 0, we call it a formal first integral.
One of the effective approaches to investigate the integrability of system (1) is using normal form theory with the resonance condition of corresponding linear part [2] to find sufficient many necessary conditions for the existence of first integrals.The related idea, as far as we know, can be traced to the work of Poincaré; he found that [3] if the Jacobian matrix  = (0) is diagonal and the eigenvalues  1 , . . .,   of it are N-independent, then system (1) does not have any formal first integral in a neighborhood of the origin  = 0; that is, they do not satisfy any resonant condition of the form (3) Following Poincaré's work, a lot of results have been obtained on the nonintegrability and partial integrability of general nonlinear dynamical systems; see [2,[4][5][6][7][8], for example.In 1996, Furta [2] provided an elementary proof of Poincaré's result; moreover, by using singular analysis, he also gave some valuable study of the nonexistence and partial existence of analytic first integrals for semiquasihomogeneous systems.Recently, Shi and Li [9] extended Furta's results to study the case that  is not necessarily a diagonal matrix and the case that eigenvalues of  are resonant [10][11][12].So far, the above results have been proved to be very useful and effective from a large number of applications [13], while there are very few works using these known results to give a systematic investigation of the integrability of general dynamical systems rather than some specific examples.The main goal of the present paper is trying to make a complete discussion of this topic.As a beginning, let us have a look at the existence of formal integrals to planar system where  is a 2 × 2 complex matrix and () = (|| 2 ) is analytic in .Without loss of generality, we also assume  is in Jordan form.Then, (1) if  = (  1 0 0  2 ), with  1 ,  2 not satisfying (3), by the Poincaré's theorem, system (4) is not formally integrable; (2) if  = (  1 0 0  2 ), with  1 ,  2 satisfying (3) and  1  2 ̸ = 0, by the results in [10], system (4) is formally integrable if and only if one of the equations in the normal form of the transformed system from (4) by transformation (, V) = ( 0  ) with  ̸ = 0, by the results in [9], system (4) is not formally integrable.
And there remain two cases needing more investigations.
Our first main work in the paper is to give an exhaustive discussion of the previous two cases for general planar systems.Then we will go ahead with some discussions for a class of higher dimensional systems, with an invariant manifold expressed as where (, ), (, ), and ℎ() are analytic vector-valued functions, (0, 0) = 0, (0, 0) = 0. We especially will combine the center manifold theory [14] and the known results by Shi and Li [9] to give several necessary conditions of integrability results for general dynamical systems with center manifolds.This paper is organized as follows.In Section 2, we will give an exhaustive discussion of Cases 1 and 2 for general planar systems.In Section 3, we will give a new method to investigate integrability problem of general dynamical systems with center manifolds, and several examples are presented as applications of our results at last.
Proof.The proof will be divided into three cases.
(1) If  1 () ≡ 0, then (7) is a classical complex Hamiltonian system with 1 degree of freedom, which implies the integrability of itself.

Case 2.
In this case, it is not difficult to see that if system (4) is formally integrable, then so is the homogeneous subsystem where  2 () is the second-order homogeneous terms of ().
Without loss of generality, we expand it as with   ,   ,   ∈ C,  = 1, 2; then we have the following results.16) is formally integrable if and only if two eigenvalues of the matrix (  1  1  2  2 ) satisfy the resonant condition (3).
these equations imply that the first integral   () is nontrivial if and only if there exist two integers Then, based on the above discussion, one can summarize several necessary conditions of integrability for the original system (1).
In general, we say a function Φ() in Euclidean space Then the first integrals of (5) can be divided into two classes by the independence from the manifold Z.And it is easy to conclude that any first integral independent from Z also leads to a first integral of ẋ =  (, ℎ ()) .
Proof.The proof is easy and can be found in [11], so we omit it here.
(1) If  = 1 and the eigenvalues  1 , . . .,   of the matrix  are N-independent, then (5) does not have any formal first integral independent from Z.
(2) If  > 1, for any balance  of the vector field   (), the eigenvalues  1 , . . .,   of the Kovalevskaya matrix  associated with the balance  are N-independent.Then (5) does not have any formal first integral independent from Z.
Proof.(1) When  = 1, if the eigenvalues  1 , . . .,   of the matrix  are N-independent, then, by Theorem B in [9], system (37) does not have any formal first integral independent from Z.And (5) does not have any formal first integral independent from Z.
(2) When  > 1, if the eigenvalues  1 , . . .,   of the Kovalevskaya matrix  associated with the balance  are Nindependent, then, by Theorem C in [9], (37) does not have any formal first integral independent from Z. On the other hand, if system (5) has a formal first integral independent from Z, (36) has a nontrivial formal first integral.By Lemma 8, we know that (37) should have a homogeneous first integral.There comes a contradiction.
Example 10.Let us consider a three-dimensional system of Lotka-Volterra type where , , , , ,  are real nonzero constants.
Theorem 11.Assume that  < 0 or  and  are Nindependent.Then system (40) has only one analytic first integral 1 +  1 is an analytic first integral of (40) indeed.Next, we will prove that system (40) does not have any other formal first integral independent from Φ( 1 ,  2 ,  3 ).
One can find that (40) has an invariant manifold ) .
On the other hand, note that (40) has another invariant manifold