Advances in Mathematical Physics
Volume 2016 (2016), Article ID 1036089, 9 pages
http://dx.doi.org/10.1155/2016/1036089

Research Article

Formal First Integrals of General Dynamical Systems

Jia Jiao,¹ Wenlei Li,² and Qingjian Zhou¹

¹College of Science, Dalian Minzu University, Dalian 116600, China
²School of Mathematics, Jilin University, Changchun 130012, China

Received 22 February 2016; Accepted 4 July 2016

Academic Editor: Andrei D. Mironov

Copyright © 2016 Jia Jiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 system

where 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 into

with

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 have

where 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 into

with

with constants , and

Then, 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 have

Equating the same order terms with respect to , we have

Firstly, 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 subsystem

where 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 variables

with a number to be fixed later, such that (16) can be equivalently transformed into

with

By 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 get

with

(3) If and , without loss of generality, one can assume that after the same discussion as in Remark 4. Then (4) becomes

with . 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) becomes

with . 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 system

where 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.

3. Higher Dimensional Systems with Invariant Manifolds

Consider system (5), where and are analytic, . Suppose (5) has an invariant manifold expressed as (6), where is an analytic vector-valued function.

In general, we say a function in Euclidean space is independent from -dimensional invariant manifold , if on . Then the first integrals of (5) can be divided into two classes by the independence from the manifold . And it is easy to conclude that any first integral independent from also leads to a first integral of

Lemma 7. If is a nontrivial formal first integral of system (5), then is a formal first integral of the subsystem (36). Furthermore, if is independent from , is also a nontrivial formal first integral of (36).

Expanding with the lowest order terms , we have the following.

Lemma 8. If system (5) has a nontrivial formal first integral independent from , then

has a nontrivial homogeneous first integral.

Proof. The proof is easy and can be found in [11], so we omit it here.

If , can be represented as ( is matrix). If , (37) can be treated as a quasihomogeneous system of degree with exponents (for more details, see [2, 9]), we look for a balance of the given vector field by solving the algebraic equations . Then one can make the change of variables , such that (37) reads

where the prime means the derivative with respect to and

is the so-called Kovalevskaya matrix associated with the balance

Theorem 9. Assume with the lowest order terms .

(1)If and the eigenvalues of the matrix are -independent, then (5) does not have any formal first integral independent from .(2)If , for any balance of the vector field , the eigenvalues of the Kovalevskaya matrix associated with the balance are -independent. Then (5) does not have any formal first integral independent from .

Proof. (1) When , if the eigenvalues of the matrix are -independent, then, by Theorem in [9], system (37) does not have any formal first integral independent from . And (5) does not have any formal first integral independent from .
(2) When , if the eigenvalues of the Kovalevskaya matrix associated with the balance are -independent, then, by Theorem in [9], (37) does not have any formal first integral independent from . On the other hand, if system (5) has a formal first integral independent from , (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 or and are -independent. Then system (40) has only one analytic first integral , in sense of functional independence.

Proof. Firstly, it is easily shown that 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 .
One can find that (40) has an invariant manifold

then it can be reduced to

Denote

Since or and are -independent, then the eigenvalues of are -independent. From the case of of Theorem 9, we know that system (40) does not have any other formal first integral independent from , which implies that if (40) has another nontrivial formal first integral functionally independent from , without loss of generality, can be expressed as with some integer and formal function which can not be divided exactly by .
On the other hand, note that (40) has another invariant manifold

then (40) can be reduced to

By simple calculation, it is not difficult to get that (45) does not have any formal first integrals, which means can be further expressed as with some integer and formal function which can not be divided exactly by .
Without loss of generality, let ; then

this implies that is also a formal first integral of (40) and independent from . However, from the above discussions, one can easily find the contradiction. The proof is completed.

We remark that a large number of systems in applications can be viewed as the objects with center manifolds. In what follows, we consider this class of systems

where , , , the eigenvalues of the matrix have zero real parts, and the eigenvalues of the matrix have nonzero real parts.

We are now going to investigate some new criteria for nonexistence of formal integrals for system (47) by combining Theorem 9 with the center manifold theory [14]. In fact, by center manifold theorem, (47) has a center manifold in a neighborhood of the original point, which can be locally represented as

with being sufficiently small, and satisfies equation

or

The following theorem allows us to compute the local center manifold to any desired degree of accuracy by solving (49) to the same degree of accuracy. Therefore it also guarantees the effectiveness of our discussions in the latter part of this section.

Theorem 12. Let be a mapping with such that as for some . Then

Proof. See [16].

On the center manifold , system (47) can be reduced to

Expanding with the lowest order terms , then, for the case , with the same idea as in the above section, one can consider the corresponding homogeneous system

And we also treat it as a quasihomogeneous system of degree with exponents .

Assuming the algebraic equation has a nonzero solution , one can get the corresponding Kovalevskaya matrix

Then we have the following results.

Theorem 13. (1) If the eigenvalues of the matrix are -independent, then system (47) does not have any formal first integral independent from .
(2) If and, for any balance of the vector field , the eigenvalues of the Kovalevskaya matrix associated with are -independent, then system (47) does not have any formal first integral independent from .
(3) If the eigenvalues of the matrix are -independent, and one of the above two cases holds, then system (47) does not have any nontrivial formal first integral.

Proof. (1) From Theorem in [9], if the eigenvalues of are -independent, then system (52) does not have any formal first integrals. Hence it is easy to prove that system (47) does not have any formal first integrals independent from ; we get the first result.
(2) If and the eigenvalues of the Kovalevskaya matrix associated with the balance are -independent, from Theorem in [9], we can get the nonexistence of formal first integrals for system (53); then (47) does not have any formal first integrals independent from ; the second result is obtained.
(3) If both eigenvalues of the matrix and the ones of the matrix are -independent, respectively, by the assumptions for system (47), all the eigenvalues are, in fact, -independent; then, by Theorem in [9], it is obvious that (47) does not have any formal first integrals.
At last, assume the eigenvalues of the matrix are -independent and the second condition holds. If is a formal first integral of (47), we claim that is trivial. Otherwise, expanding with the lowest order terms of order , we have

where denotes the higher order terms. Without loss of generality, we can assume that . Equating the same order of in (55), we have

Then, on the one hand, since are -independent, it is not difficult to see that for all nonzero . On the other hand, from the discussions in the second case, must be dependent on ; that is,

which implies that must contain ; there is a contradiction. The proof is completed.

Example 14. Consider the following system:

where are parameters.
By simple computation, we can get that system (58) has a center manifold which can locally be represented as follows:

with sufficiently small.
Restricting system (58) to the center manifold , we have

which can be treated as a quasihomogeneous system of degree with exponents . So , , . Suppose (60) has a particular solution of the form

Then has to satisfy the algebraic equation as follows:

which has four nontrivial solutions,

with . And the corresponding Kovalevskaya matrices are, respectively,

and the eigenvalues are

respectively; here

Corollary 15. System (58) does not have any formal first integral, if

(1) is irrational;(2);(3) and are -independent.

Proof. We only prove Cases (2) and (3), the proof for case (1) is similar. Denote

Then, by the given conditions, for any ,

This implies that the eigenvalues of the Kovalevskaya matrix associated with the balance are -independent. By Theorem 13, system (58) does not have any formal first integral.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by NSF (11301210 and 11226157) of China, National 973 project (2012CB821200) of China, NSF (20140520053JH) of Jilin, China, Postdoctoral Science Foundation (2014M560038) of China, Education Department Talents Project of Liaoning (LJQ2015029), Science and Technology Department Foundation of Liaoning (2015020021), Education Department Foundation of Liaoning (L2014549), and Fundamental Research Funds for the Central Universities (DC201502050304, DC201502050409, and DC201501043).

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