Nonintegrability And Partial Integrability For Periodic Systems

As we all known, investigation of the ''integrability" and "nonintegrability" of differential equations is classical work of the field of studying differential equation. In general, integrable systems are characterized by regular, predictable behaviour for all initial conditions and all times. But in many cases, non-integrable systems have regions in the phase space of their dependent variables where the motion is irregular and chaotic.We have already known that a first integral of differential equations equivalents an implicit solution, and n order differential equations should have n - 1 functional independent first integrals. If we can find k first integrals then the original differential equations must be reduced into n - k order differential equations. Moreover, if we are able to find n - 1 functional independent first integrals, then the general solution of differential equations can be expressed by them, the differential equation systems are called complete integration. The past century, through the continuous efforts of mathematicians, there are a lot of methods to determine the integrability of systems, such as Painleve singular analysis, the linear compatibility analysis method, Carleman linearization procedure, Lie group methods and so on.It is also important to prove the nonintegrability of a given system. The general basis for proving the nonintegrability of a system of differential equation is the analysis of the variational equation around a particular solution. If the nonlinear system admits some first integrals so does the variational equation. Therefore, if we can prove that the variational equation does not admit any first integral within a given class of functions, we can conclude that the original system is nonintegrable (with respect to the same class). Early in the 19th century, Poincafe first suggested an easily verifiable criterion of non-existence of nontrivial first integral for general autonomic analytic system. In 1983, Ziglin constituted Ziglin's theory of nonintegrability for Hamilton systems. By considering the relationship of first integrals and the monodromy matrices of variational equations, he presented a necessary condition for non-existence of nontrivial first integral for Hamilton systems. Other results can be seen in [21, 47, 65, 66] and so on. In 2003, Li et al present a verifiable criterion of nonexistence of formal first integrals for periodic systems. In this paper, we give a new proof of this result, furthermore, we also present an easily verifiable criterion of non-existence of rational first integral for periodic systems.Moreover, most equations in mathematics and physics admit a few invariants related to physical conservation laws, but the number of first integrals is less than the number of required for complete integration. So, these systems do not fall in the set of completely integrable or completely nonintegrable systems. Hence, these systems are called partial integrability. In [35], the authors have given some criterions of partial integrabilitv for autonomous differential systems. In this paper, we will give a necessary condition of partial integrabilitv for periodic systems.We consider the following periodic differential systemx = f(t,x), (1)where (t, x)∈S¹Ã—C n with S¹ = R/(NT), f(t, x) = (f₁(t, x),…, f_n(t, x)) is a C^r(r≥1) vector value function and satisfies f(t + T, x) = f(t, x). Suppose that x = 0 is a constant solution of system (1), i.e., f(t, 0) = 0 for all t∈S¹.In this paper, we consider the nonintegrability and partial integrabilitv of formal first integrals and rational first integrals for periodic system (1).In the second part, we firstly give a new proof of the result of Weigu Li, furthermore, we also consider the partial integrabilitv of formal first integrals for periodic system (1). Definition 1 A non-constant functionΦ(t,x) defined on S¹Ã—U with U an open subset of Cⁿ, is called a first integral of system (1) if it is T-periodic with respect to t and constant along every flow defined in U. IfΦ(t,x) is differentiable, then this definition can be written as the conditionwhere〈·,·〉denotes the Euclidean inner product.Remark 1 IfΦ(t, x) is a formal series in x (rational function in x), thenΦ(t,x) is called a formal first integral (rational first integral) of system (1).Assume that x = 0 is a constant solution of system (1), then system (1) can be rewritten as(?) (2)near some neighborhood of the constant solution x = 0, where A(t) = (?), A(t + T,x) = A(t,x), g(t,x) = O(||x||²) and g(t + T,x)= g(t,x).By Floquet's theory[76], there exists T-periodic function Q(t), such that under the change of variablesx = Q(t)_y, system (2) is changed to the following form(?), (3)where B is a constant matrix, h(t, y) = O(||y||²) and h(t + T,y) = h(t, y).Under the transformationy =εu,system (3) becomes(?). (4)where (?) = O(||u||²) and satisfies (?).From the above contents, we have the following lemma. Lemma 1 If system (1) has a nontrivial formal first integralΦ(t, x) in a neighborhood of the constant solution x = 0, then there exists a homogeneous functionΨ_l(t,y) of degree l with respect to y which is an integral of system(?). (5)Remark 2 In fact,Ψ_l(t,y) is the leading term of the formal power series ofΨ(t. y) with respect to y.Remark 3 Expect special explanation, the homogeneous function H(t, x) is homogeneous with respect to x; H₁ (t, x),…, H_s(t, x) are called functional independent which means that they are functional independent in x.Definition 2 The eigenvaluesλ₁,…,λ_n of matrix B are called the characteristic exponents of system(?), (6)and the eigenvaluesμ₁= exp(λ₁T),…,μ_n = exp(λ_nT) of e^BT are the characteristic multipliers of system (6).By using Lemma 1, we can prove the following result. Theorem 1 Assume that x = 0 is a constant solution of system (1). If the characteristic multipliers of system (6) do not satisfy any resonant equality of the type(?) , (7)then system (1) does not have any nontrivial formal first integral in a neighborhood of x = 0.In this section, we are concerned with the partial integrability of system (1). According to the proof of Theorem 1, we know that if system (1) has a nontrivial formal first integralΦ(t,x), then the linear system (5) has a homogeneous integralΨ_l(t, y) of degree l. So in general at least one resonant relationship of type (7) must be satisfied, and the set is a nonempty subset of Z₊ⁿ.Suppose that system (1) admits s nontrivial formal integralsΦ^l(x,y),…,Φ^s(t,y), then according to Lemma 1. linear system (5) has s nontrivial homogeneous formal first integralsΨ_l1^l(t,y),…,Ψ_l5^s(t,y). Here, we assume that B is diagonalizable, just for simplicity, D has already been a diagonal form.Suppose rank (?), and let K₁= (K_1l,…, K_1n),…, K_s= (K_sl,…, K_sn) be the least generating elements of set (?).For the proof of partial intagrability, we firstly give some lemmas.Lemma 2 (?) for i = l,…,s are s first integrals of the linear system (5) which are functionally independent in y.Lemma 3 Any nontrivial homogeneous formal first integralΨ_l(t, y) of (5) is a polynomial function of ρ¹(t, y),…,ρ^s(t, y).Lemma 4 Assume system (1) has s (1≤s < n) nontrivial formal integralsΦ^l(t, x),…,Φ^s(t, x) which are functionally independent. If any nontrivial homogeneous integralΨ_q(t,y) of system (5) is a smooth function of (?),i.e., there exists a smooth function H such thatΨ_q(t,y) (?), then any other nontrivial formal integralΦ(t, x) of system (1) must be a function of Φ^l(t, x).…,Φ^s(t, x), i.e.,where F is a smooth function.By using Lemma 2, Lemma 3, Lemma 4 and Inverse Function Theorem, we can prove the result of partial integrability.Theorem 2 Assume system (1) admits s (1≤s < n) nontrivial formal integralsΦ^l(t, x),…,Φ^s(t, x) in a neighborhood of the constant solution x = 0. If (?) are functionally independent and rank g = s, then any other nontrivial formal integralΦ(t, x) of system (1) must be a function of Φ^l(t, x),…,Φ^s(t,x), i.e., where F is a smooth function.In the third part, we consider the nonintegrability and partial integrability for the periodic systems in the rational function space. Theorem 3 Assume that x = 0 is a constant solution of system (1) andμ_l,…,μ_n are the characteristic multipliers of the linear system (6). If system (1) has a nontrivial rational first integral in a neighborhood of the constant solution x = 0, then there exists a nonzero vector k =(k_l,…, k_n)∈Zⁿ such that(?), (8)According to the proof of above theorem, we can obtain the following corollary.Corollary 1 For any nonzero coefficients (?) and (?), they should satisfyCorollary 2 If system (1) has a nontrivial rational first integralΦ(t,x) in a neighborhood of the constant solution x = 0, then linear system '(?) (9)admits a nontrivial homogeneous rational first integralΨ_p(t,u). In this section we assume that B is diagonalizable, just for simplicity, B has already been a diagonal form. According to the proof of Theorem 3, we know that if system (1) has a nontrivial rational first integralΦ(t, x), then the linear system (9) has a nontrivial homogeneous rational integralΨ_p(t, u). So there must be a nonzero vector k∈Zⁿ satisfying the resonant relationship of the type (8), i.e., is a nonempty subgroup of Zⁿ. Let rank (?) be the number of the least generating elements of group g.To introduce the second result of this paper, we need following lemmas.Lemma 5 Assume rank (?) (1≤s < n), and let (?)be the least generating elements of (?). Thena) (?) are rational first integrals of the linear system (9).b) (?) are functionally independent.c) Any nontrivial homogeneous rational first integralΨ_p(t, u) of (9) is a rational function (?).Lemma 6 Assume system (1) admits s nontrivial rational integralsΦ^l(t, x),…,Φ^s(t, x) which are functionally independent. If any nontrivial homogeneous rational integralΨ_q(t,u) of linear system (9) is a smooth function of (?), then any other nontrivial rational integralΦ(t, x) of system (1) must be a smooth function of (?), i.e.,where D is a smooth function.By using Lemma 5 and Lemma 6, and combining Implicit Function Theorem, we can prove the following result about partial intagrability.Theorem 4 Assume system (1) has a constant solution x = 0 and admits s (l≤s < n) nontrivial rational integrals (?) which are functionally independent in some neighborhood of the constant solution. If (?) are functionally independent and rank (?), then any other nontrivial rational integralΦ(t,x) of system (1) must be a smooth function of (?), i.e.,where F is a smooth function.