Electronic Journal of Differential Equations, Vol. 217 (217), No. 274, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu HIGHER ORDER CRITERION FOR THE NONEXISTENCE OF FORMAL FIRST INTEGRAL FOR NONLINEAR SYSTEMS ZHIGUO XU, WENLEI LI, SHAOYUN SHI Communicated by Zhaosheng Feng Abstract. The main purpose of this article is to find a criterion for the nonexistence of formal first integrals for nonlinear systems under general resonance. An algorithm illustrates an application to a class of generalized Lokta-Volterra systems. Our result generalize the classical Poincaré s nonintegrability theorem and the existing results in the literature. Let us consider the system 1. Introduction ẋ f(x), x (x 1,..., x n ) T C n, (1.1) where f(x) is an n-dimensional vector-valued analytic function with f(). In 1891, Poincaré [12] provided a criterion on the nonexistence of analytic or formal first integrals for system (1.1). Theorem 1.1 (Poincaré s nonintegrability theorem). If the eigenvalues λ 1,..., λ n of the Jacobian matrix A Df() are N-non-resonant, i.e., they do not satisfy any resonant relations of the form n n k j λ j, k j Z +, k j 1, j1 where Z + N {}, then system (1.1) has neither analytic nor formal first integrals in any neighborhood of the origin x. In 1996, Furta [6] gave another proof to Theorem 1.1 with the additional assumption that the matrix A can be diagonalized, and furthermore, he obtained a result about the nonexistence of formal integral for semi-quasi-homogeneous systems. In 21, Shi and Li [13] presented a different proof to Theorem 1.1 without the assumption of diagonalization of A. In 27, Shi [14] extended Theorem 1.1 and provided a necessary condition for system (1.1) to have a rational first integral. Cong et al [5] in 211 generalized the Shi s result to a more general case. For related information, see [7] and references therein. j1 21 Mathematics Subject Classification. 34C14, 34M25. Key words and phrases. Normal form theory; first integral; resonance. c 217 Texas State University. Submitted April 13 216. Published November 5, 217. 1
2 Z. G. XU, W. L. LI, S. Y. SHI EJDE-217/274 We note that the above results are all obtained under the condition that the eigenvalues of A are non-resonant. How about the case when the eigenvalues of A are resonant? In 23, Li et al [1] studied the case that A has a zero eigenvalue and others are non-resonant. In 21, Liu et al [11] studied the case that several eigenvalues of A are zero and the others are non-resonant, and Li et al [9] studied the case that the eigenvalues of A are pairwise resonant. In this paper, we will extend the above results to the more general case, i.e., the eigenvalues of A are in general resonance. With the help of the theory of normal form [1] and power transformations [4], we give a criterion on the nonexistence of formal first integral for system (1.1) under the case that the eigenvalues of A are in general resonance. This paper is organized as follows. In section 2, we will build a criterion of nonexistence of formal first integral of nonlinear systems under general resonance. In section 3, an algorithm to determine the nonexistence of formal first integrals of nonlinear systems will be illustrated by studying a class of generalized Lotka- Volterra systems. System (1.1) can be rewritten as 2. Main results ẋ Ax + F (x) (2.1) near some neighborhood of x, where A Df(), F (x) o( x ). Let Λ (λ 1,..., λ n ) T, and Θ { n k (k 1,..., k n ) T Z n : k, Λ k i λ i }, then Θ is a subgroup of Z n. Let rank Θ denote the number of the least generating elements of Θ. We assume that A is a diagonalizable matrix, and without loss of generality, A diag(λ 1,..., λ n ). By the Poincaré-Dulac normal form theory [1], there exists a formal transformation x y + h(y), such that system (2.1) can be changed to where y (y 1,..., y n ) T, here k m i1 F i (y) i1 ẏ i λ i y i + F i (y), i 1,..., n, (2.2) r2 k r, k,λ λ i F i k y k, i 1,..., n, (2.3) k i, y k y k1 1... ykn n, F k i are constant coefficients. For convenience, we give the following definition. Definition 2.1. A function Φ(x) is resonant with respect to Λ, if it can be written as Φ(x) Φ k x k, where Φ k are constants. i1 k i, k,λ Lemma 2.2 ([9, 15]). If system (2.1) admits a nontrivial formal first integral Φ(x), then system (2.2) has a nontrivial formal first integral Φ(y) which is resonant with respect to Λ.
EJDE-217/274 NONEXISTENCE OF FIRST INTEGRAL 3 Lemma 2.3. If rank Θ m ( < m < n), then there exist n m eigenvalues which are N-non-resonant. Proof. Assume that any n m eigenvalues of matrix A are N-resonant, then there exist n m constants a 1 1,..., a 1 n m Z + with a 1 1 + + a 1 n m such that a 1 1λ 1 + + a 1 n mλ n m. Thus τ 1 (a 1 1,..., a 1 n m,,..., ) T Θ. Without loss of generality, we assume that a 1 1. Analogously, there exist n m constants a 2 1,..., a 2 n m Z + with a 2 1 + + a 2 n m such that a 2 2λ 2 + + a 2 n m+1λ n m+1. Thus τ 2 (, a 2 2,..., a 2 n m+1,,..., ) T Θ. Similarly, we assume that a 2 2. Repeating the above process, we obtain τ i (,...,, a i i,..., a i n m+i 1,..., ) T Θ, a i i, i 1,..., m + 1. Obviously, τ 1,..., τ m+1 are linear independent. This is contradict to the fact that rank Θ m, the proof is complete. For simplicity of presentation, we make the following assumption. (H1) A diag(λ 1,..., λ n ), rank Θ m ( < m < n), and λ m+1,..., λ n are N-non-resonant. Let Θ { n k Θ : k i 1, and there exists j {1,..., n} such that k + e j }, i1 here e 1,..., e n are the natural bases of R n, and a vector a means every component a i, for i 1,..., n. Lemma 2.4. Assume (H1) holds, then there exist m linear independent vectors τ 1,..., τ m Q n, such that (1) for any k Θ, there exists a (a 1,..., a m ) T Q m, such that k a 1 τ 1 + + a m τ m ; (2) for any k Θ, there exists a (a 1,..., a m ) T Z m +, a 1, such that k a 1 τ 1 + + a m τ m. Proof. Let B (,..., α m ) T, P 1 (α m+1,..., α n ) T, here,..., α m are the least generating elements of Θ, and α m+1,..., α n Z n are fundamental solutions of the linear equations Bx, then P 1 B T. Let 2 1... 1 1 2... 1 R..., P 1 1... 2 ( P2 P 1 ), where P 2 is a m n sub-matrix of R such that P is invertible. Then P 1 (, E n m )P, i.e., P 1 P 1 (, E n m ), where E n m denotes the n m order unit matrix. Let ( ) τ (τ 1,..., τ m ) P 1 Em, (2.4)
4 Z. G. XU, W. L. LI, S. Y. SHI EJDE-217/274 then τ 1,..., τ m Q n, and they are linear independent. Furthermore, ( ) P 1 τ P 1 P 1 Em ( ) ( ) E E m n m, therefore τ 1,..., τ m are fundamental solutions of the linear equations P 1 x. (2.5) (1) For any k Θ, there exist b 1,..., b m R, such that k b 1 + + b m α m. Therefore P 1 k b 1 P 1 + + b m P 1 α m P 1 B T (b 1,..., b m ) T. This means that k is a solution of (2.5), thus there exists a (a 1,..., a m ) T Q m, such that k a 1 τ 1 + + a m τ m. (2) For any k Θ, by (1), there exists a (a 1,..., a m ) T Q m, such that k a 1 τ 1 + + a m τ m. By (2.4), we have ( ) k τa P 1 Em a, thus ( ) a P k ( P2 P 1 ) k ( ) P2 k. According to the choice of P 2, we obtain that (a 1,..., a m ) T Z m +, and a 1. Lemma 2.5. Assume (H1) holds, and τ 1,..., τ m be given in Lemma 2.4. Then under the change of variables system (2.2) becomes z i y τi1 1... y τin n, i 1,..., m, w j y m+j, j 1,..., n m, ż i z i F i (z), i 1,..., m, ẇ j λ m+j w j + w j F m+j (z), j 1,..., n m, (2.6) (2.7) where z (z 1,..., z m ), F i (z) (i 1,..., n) are formal power series with respect to z, F i (). Proof. By Lemma 2.4 and (2.3), we have where ( F 1 (y) F n (y) ) ż i z i τ i1 + + τ in, i 1,..., m, y 1 y n F m+j (y) ẇ j λ m+j w j + w j, j 1,..., n m, y m+j F i (y) y i r2 k r, k,λ λ i i F k y k1 1... yki 1 i (2.8)... y kn n, i 1,..., n. (2.9) Obviously, for every monomial y k1 1... yki 1 i... yn kn in the above expression, the exponents (k 1,..., k i 1,..., k n ) Θ. By Lemma 2.4, there exist a (a 1,..., a m ) T in Z m + with a 1, such that (k 1,..., k i 1,..., k n ) T a 1 τ 1 + + a m τ m.
EJDE-217/274 NONEXISTENCE OF FIRST INTEGRAL 5 Therefore F i (y) y i r2 k r, k,λ λ i r2 k r, k,λ λ i r1 a r i F k y a1τ11+ +amτm1 a1τ1n+ +amτmn i F k (y τ11 τ1n ) a1... (y τm1 τmn F i az a1 1... zam m, i 1,..., n. By (2.8) and (2.1), we can get (2.7), and the lemma is proved. ) am (2.1) Remark 2.6. It should be pointed out that there are similar arguments in [3, 4] as used in Lemma 2.4 and Lemma 2.5. Here, to ensure that F i (z) (i 1,..., n) in (2.7) are formal power series with respect to z, we give a different way to calculate τ 1,..., τ m. Lemma 2.7. Assume that (H1) holds. If system (2.1) has a nontrivial first integral, then system (2.7) has a nontrivial formal first integral which is independent with w 1,..., w n m, and the system ż z F (z) (2.11) has a nontrivial formal first integral, where z F (z) : (z 1 F 1 (z),..., z m F m (z)). Proof. Assume that (2.1) has a nontrivial formal first integral Φ(x), then by Lemma 2.2, Φ(y) Φ(y + h(y)) is a formal first integral of system (2.2), and Φ(y) is resonant with respect to Λ, therefore Φ(y) can be written as Φ(y) i1 k i, k,λ Φ k y k1 1... ykn n, (2.12) where Φ k are nonzero constants. Note that for every monomial y k1 1... ykn n in (2.12), (k 1,..., k n ) T Θ. By Lemma 2.4, there exists a (a 1,..., a m ) T Z m +, a 1, such that Thus Φ(y) (k 1,..., k n ) T a 1 τ 1 + + a m τ m. i1 k i, k,λ i1 k i, k,λ i1 k i, k,λ Φ k y a1τ11+ +amτm1 a1τ1n+ +amτmn Φ k (y τ11 τ1n ) a1... (y τm1 τmn Φ k z a1 1... zam m : Ψ(z). Since (2.2) is changed to (2.7) under the transformation (2.6), Ψ(z) is a first integral of (2.7). It is clearly that Ψ(z) is independent with w 1,..., w n m, so it is also a first integral of (2.11). ) am
6 Z. G. XU, W. L. LI, S. Y. SHI EJDE-217/274 Expand F (z) as F (z) F p (z) + F p+1 (z) +..., where p 1, F k (z)(k p, p + 1,... ) are the k-th order homogeneous polynomial. We can get the following lemma easily. Lemma 2.8. If system (2.11) has a nontrivial formal first integral Ψ(z), then the lowest order homogeneous terms Ψ q (z) is a first integral of system ż z F p (z). (2.13) System (2.13) can be treated as a quasi-homogeneous system of degree p + 1 with the exponents s 1 s m 1 (for more detail, see [6]). Let ξ be a balance of vector field z F p (z), i.e., ξ is a nonzero solution of the algebraic equations 1 p ξ + ξ F p (ξ), then system (2.13) has a particular solution z (t) t 1 p Em ξ. We make the change of variables z(t) t 1 p Em (ξ + u), s ln t, then system (2.13) reads du ds Ku + ˆF (u), where 1 p + F 1 p (ξ) ξ F 1 p (ξ) 1 z K... 1... ξ F 1 p (ξ) 1 z m + 1 p + F..... p m (ξ) ξ F m p (ξ) m z 1... ξ F m p (ξ) m z m is the so-called Kovalevskaya matrix associated to the balance ξ, ˆF (u) 1 p E mξ + (ξ + u) F p (ξ + u) z F p (z) u O( u 2 ). z zξ Now we can state our main result. Theorem 2.9. Assume that (H1) holds, ξ is a balance of vector field z F p (z), µ (µ 1,..., µ m ) is the eigenvalues of Kovalevskaya matrix associated to the balance ξ. Let Ω { k (k 1,..., k m ) T Z m + : k, µ }. (1) If rank Ω, then system (2.1) does not have any nontrivial formal first integrals in the neighborhood of x ; (2) If rank Ω l >, then system (2.1) has at most l functionally independent formal first integrals in the neighborhood of x. Proof. (1) Assume that system (2.1) admits a formal first integral in a neighborhood of x, then by Lemma 2.7, system (2.11) admits a formal first integral, and by Lemma 2.8, system (2.13) has a homogeneous first integral. According to [6, Theorem 1], we know that µ 1,..., µ m are N-resonant, which contradicts with rank Ω. (2) Assume that system (2.1) admits l + 1 functional independent first integrals Φ 1 (x),..., Φ l+1 (x) in a neighborhood of x. By Lemma 2.2, Lemma 2.5 and Lemma 2.7, after a sequence of transformations, we can see that system (2.11) have l+1 formal first integrals Ψ 1 (z),..., Ψ l+1 (z). Since the sequence of transformations which transform (2.1) to (2.11) are local invertible, hence Ψ 1 (z),..., Ψ l+1 (z) are functional independent in the neighborhood of z. According to Ziglin lemma
EJDE-217/274 NONEXISTENCE OF FIRST INTEGRAL 7 in [17], system (2.11) has l + 1 first integrals Ψ 1 (z),..., Ψ l+1 (z) whose lowest order homogeneous terms Ψ 1 l+1 q 1 (z),..., Ψ q l+1 (z) are functionally independent, and by Lemma 2.8, Ψ 1 l+1 q 1 (z),..., Ψ q l+1 (z) are l + 1 first integrals of quasi-homogeneous system (2.13). By [8, Theorem B], we get rank Ω l + 1, which contradicts with the assumption of theorem. The proof is complete. 3. Example Based on the arguments in Section 2, one can give an algorithm to test whether a given system admits formal first integral or not, see the next example. Example 3.1. Consider the generalized Lotka-Volterra system ẋ 1 x 1 + a 1 x 1 x 1 x 2 + a 2 x 1 x 3 x 4 + a 3 x 1 x 5 x 6, ẋ 2 x 2 + b 1 x 2 x 1 x 2 + b 2 x 2 x 3 x 4 + b 3 x 2 x 5 x 6, ẋ 3 2x 3 + c 1 x 3 x 1 x 2 + c 2 x 3 x 3 x 4 + c 3 x 3 x 5 x 6, ẋ 4 2x 4 + d 1 x 4 x 1 x 2 + d 2 x 4 x 3 x 4 + d 3 x 4 x 5 x 6, ẋ 5 3x 5 + e 1 x 5 x 1 x 2 + e 2 x 5 x 3 x 4 + e 3 x 5 x 5 x 6, ẋ 6 3x 6 + f 1 x 6 x 1 x 2 + f 2 x 6 x 3 x 4 + f 3 x 6 x 5 x 6, (3.1) where a i, b i, c i, d i, e i, f i R, i 1, 2, 3. Lotka-Volterra equations can be used to describe cooperation and competition between biological species in ecology. In 1988, Brenig [2] introduced generalized Lotka-Volterra equations. By transforming the original equations into a canonical form, he discussed the integrability of some type generalized Lotka-Volterra equations. As an application, we consider the nonexistence of formal first integral for above Lotka-Volterra system. Obviously, the eigenvalues of Jocabi matrix at x of system (3.1) are λ 1 1, λ 2 1, λ 3 2, λ 4 2, λ 5 3, λ 6 3, and system (3.1) is a normal form. Let Ω { k (k 1,..., k 6 ) T Z 6 + : Θ { k (k 1,..., k 6 ) T Z 6 : 6 k i λ i }, i1 6 k i λ i }, then rank Ω rank Θ 3, and (1, 1,,,, ) T, (,, 1, 1,, ) T, (,,,, 1, 1) T are the least generating elements of Θ. Making the change of variables z 1 x 1 x 2, z 2 x 3 x 4, z 3 x 5 x 6, w 1 x 2, w 2 x 4, w 3 x 6, system (3.1) becomes i1 ż 1 z 1 ( z 1 + α 2 z 2 + α 3 z 3 ), ż 2 z 2 (β 1 z 1 + β 2 z 2 + β 3 z 3 ), ż 3 z 3 (γ 1 z 1 + γ 2 z 2 + γ 3 z 3 ), ẇ 1 w 1 + b 1 w 1 z 1 + b 2 w 1 z 2 + b 3 w 1 z 3, ẇ 2 2w 2 + e 1 w 2 z 1 + e 2 w 2 z 2 + e 3 w 2 z 3, ẇ 3 3w 3 + f 1 w 3 z 1 + f 2 w 3 z 2 + f 3 w 3 z 3, (3.2)
8 Z. G. XU, W. L. LI, S. Y. SHI EJDE-217/274 where α i a i + b i, β i c i + d i, γ i e i + f i, i 1, 2, 3. By Lemma 2.8, we need only to investigate the formal first integral for the system ż 1 z 1 ( z 1 + α 2 z 2 + α 3 z 3 ), ż 2 z 2 (β 1 z 1 + β 2 z 2 + β 3 z 3 ), ż 3 z 3 (γ 1 z 1 + γ 2 z 2 + γ 3 z 3 ). (3.3) System (3.3) is a quasi-homogeneous system of degree 2 with exponents s 1 s 2 s 3 1. Assume, then ξ ( 1,, ) T is a balance of system (3.3). Making the change of variables system (3.3) reads z 1 t 1 ( 1 + u 1 ), z 2 t 1 u 2, z 3 t 1 u 3, t e s, (3.4) u 1 u 1 α 2 u 2 α 3 u 3 + u 1 ( u 1 + α 2 u 2 + α 3 u 3 ), u 2 (1 β 1 )u 2 + u 2 (β 1 u 1 + β 2 u 2 + β 3 u 3 ), (3.5) u 3 (1 γ 1 )u 3 + u 3 (γ 1 u 1 + γ 2 u 2 + γ 3 u 3 ), where means the derivative with respect to s. Here the corresponding Kovalevskaya matrix is Obviously, 1 α2 K 1 β1 α3 1 γ1 µ 1 1, µ 2 1 β 1, µ 3 1 γ 1 are eigenvalues of K. Let Ω 1 { 3 k (k 1, k 2, k 3 ) T Z 3 + : k i µ i }. By Theorem 2.9, we have the following results. Theorem 3.2. Assume. (1) If rank Ω 1, then system (3.1) does not have any nontrivial formal first integrals in the neighborhood of x ; (2) If rank Ω 1 l 1 >, then system (3.1) has at most l 1 functionally independent formal first integrals in the neighborhood of x. It is not difficult to see that if, β 1, γ 1 are Z-non-resonant, and rank Ω 1, by Theorem 3.2, we get nonexistence of formal first integrals in the neighborhood of x. Furthermore, we can obtain following conclusions. Corollary 3.3. (1) If, β 1, γ 1 are Z-non-resonant, then system (3.1) does not have any nontrivial formal first integrals in the neighborhood of x ; (2) If β 1 γ 1, α 2 α 3 and γ3 β 2 Q, then system (3.1) does not have any nontrivial formal first integrals in the neighborhood of x. (3) If β 1 γ 1, α 2 α 3 and γ3 β 2 Q, then system (3.1) has at most one nontrivial formal first integral in the neighborhood of x. i1.
EJDE-217/274 NONEXISTENCE OF FIRST INTEGRAL 9 Proof. The first case is obvious, we omit it. Let us consider the cases when β 1 γ 1, and α 2 α 3. In this situation, we need to consider u 1 u 1 + u 2 1, u 2 u 2 ( u 1 + β 2 u 2 + β 3 u 3 ), u 3 u 3 ( u 1 + γ 2 u 2 + γ 3 u 3 ), u u, (3.6) where u e s. It is clearly that µ 1 1, µ 2, µ 3, and rank Ω 1 2 >. Let Θ 1 { 4 k (k 1, k 2, k 3, k 4 ) T Z 4 : k i µ i }, where µ 4 1. By the Poincaré-Dulac normal form theory, system (3.6) can be reduced to the canonical form u 1 u 1 + u 1 ϕ 1 (u 2, u 3 ), i1 u 2 u 2 (β 2 u 2 + γ 3 u 3 ) + ϕ 2 (u 2, u 3 ), u 3 u 3 (γ 2 u 2 + γ 3 u 3 ) + ϕ 3 (u 2, u 3 ), u u, (3.7) where ϕ 1 (u 2, u 3 ) O( (u 2, u 3 ) ), ϕ 2 (u 2, u 3 ) O( (u 2, u 3 ) 2 ) and ϕ 3 (u 2, u 3 ) O( (u 2, u 3 ) 3 ). The subsystem of system (3.7), u 2 u 2 (β 2 u 2 + β 3 u 3 ) + ϕ 2 (u 2, u 3 ), u 3 u 3 (γ 2 u 2 + γ 3 u 3 ) + ϕ 3 (u 2, u 3 ) is a semi-quasihomogeneous system with exponents s 1 s 2 1, and its quasihomogeneous cut is u 2 u 2 (β 2 u 2 + β 3 u 3 ), u (3.8) 3 u 3 (γ 2 u 2 + γ 3 u 3 ). Note that if β 2, then ( 1 β 2, ) is a balance of (3.8), and the corresponding Kovalevskaya matrix is ( ) K 1 β3 β 2. 1 γ3 β 2 Obviously, µ 1 1, µ 2 1 γ3 β 2 are eigenvalues of K. Let Ω 2 { k (k 1, k 2 ) T Z 2 + : k 1 µ 1 + k 2 µ 2 }. By Theorem 2.9, we know that if rank Ω 2, then (3.1) does not have any nontrivial formal first integrals in the neighborhood of x ; if rank Ω 2 l 2 >, then (3.1) has at most l 2 functionally independent formal first integrals in the neighborhood of x. Therefore we get the proofs of last two cases. Remark 3.4. If Ω 2 l 2 >, one can use the same idea as the change of variables (3.4) to get a new system like (3.5) to do more investigations. While, one particular case should be noted is that, if there exist i N, such that for every j > i, rank Θ j rank Θ i l i 2, we can not get the nonexistence of formal first integral for system (3.1). And this case always implies the partial existence of
1 Z. G. XU, W. L. LI, S. Y. SHI EJDE-217/274 formal first integral for (3.1), i.e., (3.1) may have l i 1 formal first integrals in a neighborhood of x. Acknowledgments. This work is supported by NSFC grants (No. 11771177, 1151242, 113121), by the China Automobile Industry Innovation and Development Joint Fund (No. U1664257), by Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 217TD-2), SF (2175255JH, 2145253JH) of Jilin, China and by the ESF (JJKH216398KJ,JJKH217776KJ) of Jilin, China. References [1] V. I. Arnold; Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983. [2] L. Brenig; Complete factorisation and analytic solutions of generalized Lotka-Volterra equations, Phys. Rev. A, 133 (1988), 378-382. [3] A. D. Bruno; Power asymptotics of solutions of non-linear systems. (in Russian) Izv. Akad. Nauk SSSR Ser. Mat., 29 (1965) 329-364. [4] A. D. Bruno; Local Methods in Nonlinear Differential Equations, Springer-Verlag, Berlin and New York, 1989. [5] W. Cong, J. Llibre, X. Zhang; Generalized rational first integrals of analytic differential systems, J. Differential Equations, 251 (211), 277-2788. [6] S. D. Furta; On non-integrability of general systems of differential equations, Z. Angew. Math. Phys., 47 (1996), 112-131. [7] A. Goriely; Integrability and Nonintegrability of Dynamical Systems, World Scientific, Singapore, 21. [8] K. H. Kwek, Y. Li, S. Y. Shi; Partial integrability for general nonlinear systems, Z. Angew. Math. Phys., 54 (23), 26-47. [9] W. L. Li, Z. G. Xu, S. Y. Shi; Nonexistence of formal first integrals for nonlinear systems under the case of resonance, J. Math. Phys., 51 (21), 2273-11 pp. [1] W. G. Li, J. Llibre, X. Zhang; Local first integrals of differential systems and diffeomorphisms, Z. Angew. Math. Phys., 54 (23), 235-255. [11] F. Liu, S. Y. Shi, Z. G. Xu; Nonexistence of formal first integrals for general nonlinear systems under resonance, J. Math. Anal. Appl., 363 (21), 214-219. [12] H. Poincaré; Sur l intégration des équations différentielles du premier order et du premier degré I and II, Rendiconti del circolo matematico di Palermo, 5 (1891), 161-191; 11 (1897), 193-239. [13] S. Y. Shi, Y. Li; Non-integrability for general nonlinear systems, Z. Angew. Math. Phys., 52 (21), 191-2. [14] S. Y. Shi; On the nonexistence of rational first integrals for nonlinear systems and semiquasihomogeneous systems, J. Math. Anal. Appl., 335 (27), 125-134. [15] S. Walcher; On differential equations in normal form, Math. Ann., 291 (1991), 292-314. [16] H. Yoshida; Necessary condition for the non-existence of algebraic first integral I, II, Celestial Mech., 31 (1983), 363-79, 381-99. [17] S. L. Ziglin; Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics I, Funct. Anal. Appl., 16 (1983), 181-189. [18] X. Zhang; Local first integrals for systems of differential equations, J. Phys. A: Math. Gen., 36 (23), 12243-12253. Zhiguo Xu School of Mathematics, Jilin University, Changchun 1312, China E-mail address: xuzg214@jlu.edu.cn Wenlei Li (corresponding author) School of Mathematics, Jilin University, Changchun 1312, China E-mail address: lwlei@jlu.edu.cn, phone +86-431-8516642
EJDE-217/274 NONEXISTENCE OF FIRST INTEGRAL 11 Shaoyun Shi School of Mathematics & State key laboratory of automotive simulation and control, Changchun 1312, China E-mail address: shisy@jlu.edu.cn