Abstract

The goal of this paper is trying to make 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; we will develop several criteria for existence of formal integrals and give some applications to illustrate our results at last.

1. 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,with -dimensional analytic vector-valued function satisfying . A function ( an open set) is said to be a first integral of (1), if 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 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 of (1) is differentiable on , it obviously satisfies where denotes the gradient with respect to and denotes the classical Euclidian inner product. If can be expanded in formal series in a neighborhood of the fixed point , 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 is diagonal and the eigenvalues of it are -independent, then system (1) does not have any formal first integral in a neighborhood of the origin ; that is, they do not satisfy any resonant condition of the form

Following Poincaré’s work, a lot of results have been obtained on the nonintegrability and partial integrability of general nonlinear dynamical systems; see [2, 4–8], 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 [10–12]. 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 systemwhere is a complex matrix and is analytic in . Without loss of generality, we also assume is in Jordan form. Then,(1)if , with not satisfying (3), by the Poincaré’s theorem, system (4) is not formally integrable;(2)if , with satisfying (3) and , 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 is trivial;(3)if with , by the results in [9], system (4) is not formally integrable.

And there remain two cases needing more investigations.

Case 1. Consider .

Case 2. Consider .

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 are analytic vector-valued functions, , . 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.

2. Integrability and Nonintegrability of Planar Systems

2.1. Case 1

By the normal form theory [14], system (4) can be transformed intowith with constants . Then, we have the following statement.

Theorem 2. System (7) is formally integrable if and only if .

Proof. The proof will be divided into three cases.(1)If , then (7) is a classical complex Hamiltonian system with degree of freedom, which implies the integrability of itself.(2)If , then (7) has a nontrivial integral , which also leads to the integrability.(3)If , then there must exist two integers , such that , , and , while , when , . Assume is a first integral of system (7); here is the th order homogeneous terms of , . Then by the definition of first integral Equating the same order terms in , we havewhere complex constants , are coefficients of the polynomials in the right hand side of (10). Furthermore, by direct computation, one can find that there exists a number (≥l) such that and can be represented as a nontrivial polynomial of ; that is, .
Then if the parameters are chosen such that , (10) does not hold; this implies the nonexistence of nontrivial first integrals. In general, making the change of variables with nonzero parameters to be fixed, (7) can be equivalently transformed intowithwith constants , andThen, if , one can conclude from the above two equations that, for any such that , the parameters exist, which implies the nonexistence of first integrals for (11) and (7).
If , then (13) becomes . And we can choose two parameters to make a transcendental number; consequently, and are also transcendental numbers. Now assume system (11) has a first integral ; here is a formal function of , . Then, by the definition of first integral, we haveEquating the same order terms with respect to , we haveFirstly, from the third general equation, it is not difficult to conclude that the lowest order of is at least . Substituting this conclusion into the first equation, we have that the lowest orders of and are at least and , respectively. Repeatedly, considering the third general equation, we can get that, in fact, the lowest orders of and are at least and . Finally, one can easily conclude that . However, from the first equation, in order to make nontrivial, it is necessary to let , which contradicts to the previous discussions. This completes the whole proof.

2.2. Case 2

In this case, it is not difficult to see that if system (4) is formally integrable, then so is the homogeneous subsystemwhere is the second-order homogeneous terms of . Without loss of generality, we expand it as with ; then we have the following results.

Theorem 3. (1) If , (16) is integrable.
(2) If and , (16) is formally integrable if and only if two eigenvalues of the matrix satisfy the resonant condition (3).
(3) If , , and , (16) is not formally integrable.
(4) If , , and , then, (16) is formally integrable if and only if there exist three integers , such that , and .

Remark 4. By the same status of the variables and , Theorem 3 gives a complete illustration of the problem of the integrability for system (16). In fact, if , one can make a change of variableswith a number to be fixed later, such that (16) can be equivalently transformed intowithBy chosing such that and noting the fact that and have the same status, we can, in fact, get a similar situation as .

Proof. (1) If , then ; the result is trivial. If , there exist two constants such that , which means is a first integral.
(2) If and , (16) can be represented as whose integrability is equivalent to that of the system By Poincaré’s result, system (16) is formally integrable if and only if two eigenvalues of the matrix satisfy condition (3).
Now, if and , (16) can be represented as By transformation (18), one can getwith (3) If and , without loss of generality, one can assume that after the same discussion as in Remark 4. Then (4) becomeswith . Assume is a first integral of system (26) with some integers , () being the coefficients to be determined. Then we can get Comparing each term of the above equation, we can get which implies that the first integral is nontrivial if and only if there exists an even integer , such that , which contradicts with the condition .
If , without loss of generality, one can assume that . Then (4) becomeswith . Assume that is a first integral of system (26) with , () being the coefficients to be determined. Then we can get Equating each term of the above equation, one can get these equations imply that the first integral is nontrivial if and only if there exist two integers , such that and .

Then, based on the above discussion, one can summarize several necessary conditions of integrability for the original system (1).

Theorem 5. System (1) is formally integrable, if one of the following cases holds up to a equivalent transformation: (1);(2), , and the eigenvalues of the matrix satisfy the condition (3);(3), , , and there exist three integers , such that , and .

Example 6. Consider systemwhere and are analytic in . This is, in fact, a higher order perturbed system of a homogeneous quadratic vector field (Collins, [15]):Since and have the same status, therefore, (33) is indeed equivalent to a new homogeneous quadratic vector field with Then, by simple computation, one can get that , , and . Further, for any three integers , we have By Theorems 3 and 2, both systems (33) and (32) do not have any nontrivial formal first integral; that is to say, they are not formally integrable.