The Completely Integrable Differential Systems are Essentially Linear Differential Systems

Let x˙=f(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=f(x)$$\end{document} be a Ck\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^k$$\end{document} autonomous differential system with k∈N∪{∞,ω}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in {\mathbb {N}}\cup \{\infty ,\omega \}$$\end{document} defined in an open subset Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document} of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document}. Assume that the system x˙=f(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=f(x)$$\end{document} is Cr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^r$$\end{document} completely integrable, i.e., there exist n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document} functionally independent first integrals of class Cr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^r$$\end{document} with 2≤r≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le r\le k$$\end{document}. As we shall see, we can assume without loss of generality that the divergence of the system x˙=f(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=f(x)$$\end{document} is not zero in a full Lebesgue measure subset of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document}. Then, any Jacobian multiplier is functionally independent of the n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-1$$\end{document} first integrals. Moreover, the system x˙=f(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{x}=f(x)$$\end{document} is Cr-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{r-1}$$\end{document} orbitally equivalent to the linear differential system y˙=y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{y}=y$$\end{document} in a full Lebesgue measure subset of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document}. Additionally, for integrable polynomial differential systems, we characterize their type of Jacobian multipliers.


Introduction and Statement of Results
Consider a C k autonomous differential systeṁ where k ∈ N ∪ {∞, ω}, the dot denotes derivative with respect to the independent variable t, Ω is an open subset of R n , and f (x) = ( f 1 (x), . . . , f n (x)) ∈ C k (Ω).
Recall that N is the set of positive integers, and C ∞ and C ω are, respectively, the sets of infinitely smooth functions and analytic functions.
The main goal of this paper is to show that a completely integrable differential system defined in an open subset Ω of R n (see below for a precise definition) is essentially linear in the sense that it is diffeomorphic to a linear differential system on a full Lebesgue measure subset of Ω. Later on, we shall see that this result can be seen as a kind of global extension of the well-known flow box theorem.
A function H (x) is a first integral of system (1) if it is continuous and defined in a full Lebesgue measure subset Ω 1 of Ω (i.e., Ω \ Ω 1 has zero Lebesgue measure), and it is not locally constant on any positive Lebesgue measure subset of Ω 1 ; moreover, H (x) is constant along each orbit of system (1) in Ω 1 . System (1) is C r completely integrable, if it has n − 1 functionally independent C r first integrals in Ω with 1 ≤ r ≤ k. See Llibre and Valls (2012) for an example of completely integrable differential systems. Recall that k smooth functions H 1 (x), . . . , H k (x) are functionally independent in Ω if their gradients ∇ H 1 , . . . , ∇ H k have rank k in a full Lebesgue measure subset of Ω. And that the k smooth functions H 1 (x), . . . , H k (x) are functionally dependent in a subset U of Ω if their gradients ∇ H 1 , . . . , ∇ H k have rank less than k in each point of U . We note that from the definitions if the k smooth functions H 1 (x), . . . , H k (x) are not functionally independent in Ω, they must be functionally dependent in a positive Lebesgue measure subset of Ω.
In this paper, we denote by X the vector field associated to system (1). We will use ∂ i to denote the partial derivative with respect to x i for i = 1, · · · , n. By convention C r −1 = C r if r = ∞ or ω; and divX = div f = ∂ 1 f 1 + · · · + ∂ n f n denotes the divergence of the vector field X or f .
A C 1 function J is a Jacobian multiplier of system (1) if it is defined in a full Lebesgue measure subset Ω * ⊂ Ω, and satisfies If system (1) is two dimensional, a Jacobian multiplier is called an integrating factor.
We extend the usual ordering of N to the set N∪{∞, ω} as follows: For all k ∈ N, we have k < ∞ < ω.
where Φ * is the tangent map of Φ and q(y) is a nonvanishing scalar function defined on Ω.
Remark 1 Assume that f (x) is not identically zero in any positive Lebesgue subset of Ω, where f (x) ∈ C k (Ω). If the divergence of the differential system dx/dt =ẋ = f (x) is identically zero in a full Lebesgue measure subset of Ω, doing the change of time dt = h(x)ds being h(x) a convenient positive function we obtain an orbitally equivalent differential system dx/ds = h(x) f (x) whose divergence is not identically zero. Hence, in the rest of this paper, without loss of generality we assume that the divergence of the differential systemẋ = f (x) is not zero in a full Lebesgue measure subset of Ω.
The next is our first main result.
Theorem 2 Letẋ = f (x) be the C k autonomous differential system (1) defined in Ω with k ∈ (N\{1})∪{∞, ω}, and f (x) ≡ 0 in any open subset of Ω. Assume that system (1) is C r completely integrable in Ω with 2 ≤ r ≤ k and that the Lebesgue measure of the set of its singularities is zero. Let H 1 (x), . . . , H n−1 (x) be n − 1 functionally independent C r first integrals. Then, the following statements hold.
(a) System (1) has always a C r −1 Jacobian multiplier defined in a full Lebesgue measure subset of Ω.
There exists a full Lebesgue measure subset Ω 0 ⊂ Ω in which system (1) is C r −1 orbitally equivalent to the linear differential systeṁ We remark that statement (a) is not new and it can be obtained from Zhang (2014b, Theorem 1.1). We include its proof here for completeness. Statement (b) is new. Statement (c) generalizes and improves Theorem 1 of Giné and Llibre (2011) extending it from two-dimensional differential systems to any finite-dimensional differential systems.
The flow box theorem states the existence of n − 1 functionally independent first integrals in a neighborhood of a regular point of the differential systemẋ = f (x) by making it diffeomorphic to the differential system (ẏ 1 ,ẏ 2 , . . . ,ẏ n ) = (1, 0, . . . , 0). Theorem 2 under the assumptions of the existence of n − 1 functionally independent first integrals for the C k differential systemẋ = f (x) defined in an open subset Ω of R n shows that the system is diffeomorphic to the linear differential system (ẏ 1 , . . . ,ẏ n ) = (y 1 , . . . , y n ) in an open and dense subset of Ω.
In the next proposition, we characterize the zero Lebesgue measure subset mentioned in statement (c) of Theorem 2.
Proposition 3 Under the assumptions of Theorem 2, we can assume without loss of generality that Then, the zero Lebesgue measure subset Ω \ Ω 0 of statement (c) of Theorem 2 can be chosen as Theorem 2 shows the existence of Jacobian multipliers for completely integrable differential systems. We now characterize the class of the Jacobian multipliers of these integrable differential systems.
Let C[x] be the ring of all polynomials in the variables (x 1 , . . . , with g and h coprime, and k i ∈ C for i = 1, . . . , r . Recall that the notion of Darboux function was considered by Darboux (1878a, b) in 1878 for studying the existence of first integrals through invariant algebraic curves (or surfaces or hypersurfaces) of polynomial differential systems.
be the field of all rational functions in the variables x with coefficients in C. A function is Liouvillian if it belongs to the Liouvillian field extension of C(x), for more details on the Liouvillian field extension, see, for instance, Singer (1992). A polynomial differential systemẋ = f (x) in R n or C n is Liouvillian integrable if it has n − 1 functionally independent Liouvillian first integrals.
Our next result provides the class of functions which belong to the Jacobian multipliers of an integrable polynomial differential system. Theorem 4 Assume that system (1) with f = ( f 1 , . . . , f n ) is an n-dimensional polynomial differential system with f 1 , . . . , f n relatively prime. Then, the following statements hold.
(a) If system (1) is Liouvillian integrable, then it has a Darboux Jacobian multiplier. (1) is polynomial integrable, then it has a polynomial Jacobian multiplier.
Statement (a) with n = 2 is due to Singer (1992), and Christopher (1999) provided a different proof, see also Christopher and Li (2007) and Pereira (2002). Statement (a) with n > 2 was proved recently by Zhang (2014a). We include this statement here for completeness. Statement (b) was proved in Chavarriga et al. (2003) for n = 2. The proof of statement (c) with n = 2 follows from Chavarriga et al. (2003) and Ferragut et al. (2007).
This paper is organized as follows. In the next section, we prove our results. In Sect. 3, we present an application of Theorem 2.

Proof of the Main Results
For proving Theorem 2, we need the following result, which is due to Olver, see Olver (1993, Theorem 2.16), and it reveals the essential property of functional dependence.
Theorem 5 Assume that M ⊂ R n is a C ∞ manifold, and g 1 , . . . , g k are real C 1 functions on M. Then, g 1 , . . . , g k are functionally dependent on M if and only if for all x ∈ M, there exists a neighborhood U of x and a C 1 real function F(z 1 , . . . , z k ) in k variables such that We must mention that the idea of the proof of statement (a) of Theorem 2 partially comes from Zhang (2014a, b), and the proof of statement (c) partially comes from Giné and Llibre (2011). H 1 (x), . . . , H n−1 (x) are C r first integrals of system (1) in Ω, by definition we have

Proof of Theorem 2 Since
Since H 1 , . . . , H n−1 are functionally independent in Ω, we can assume without loss of generality that where Ω 0 is a full Lebesgue measure subset of Ω, and H := (H 1 , . . . , H n−1 ) T , where T denotes the transpose of a matrix. For i = 1, . . . , n − 1, set Using Cramer's rule to solve (3) with respect to f 1 , . . . , f n−1 , we get It follows that Set It is clear that Q is defined in a full Lebesgue measure subset Ω Q ⊂ Ω 0 ⊂ Ω and is a C r −1 function in Ω Q . Moreover, we get from (5) that We claim that Q(x) is a Jacobian multiplier of system (1) in Ω 0 . We now prove this claim. It follows from (6) and (7) that Next, we only need to prove that the right-hand side of (8) is identically zero. Using the derivative of a determinant, we get easily that Hence, we have  (8) That is Q(x) is a Jacobian multiplier, and consequently, statement (a) follows.
For proving (b), we note that X (J ) = −J divX and divX = 0 in a full Lebesgue measure subset of Ω, so X (J ) = 0 in a full Lebesgue measure subset of Ω. By contrary, if J is not functionally independent of H 1 , . . . , H n−1 in Ω, then they are functionally dependent in a positive Lebesgue measure subset of Ω. Since J, H 1 This is in contradiction to the fact that X (J ) = 0 in a full Lebesgue measure subset of Ω. This contradiction shows that J, H 1 , . . . , H n−1 are functionally independent in a full Lebesgue measure subset of Ω, and so statement (b) follows. By statement (a), system (1) has a C r −1 Jacobian multiplier J (x). Since div X = 0 in a full Lebesgue measure subset of Ω, it follows from (b) that the functions J, H 1 , . . . , H n−1 are functionally independent. So there exists a full Lebesgue measure subset Ω 0 ⊂ Ω such that ∇ J, ∇ H 1 , . . . , ∇ H n−1 have rank n at all points of Ω 0 .
Since divX = 0 in a full Lebesgue measure subset of Ω, there exists a full Lebesgue measure subset Ω 0 ⊂ Ω such that ∇ J, ∇ H 1 , . . ., ∇ H n−1 have rank n at all points of Ω 0 . Taking the invertible change of variables This proves that system (1) is C r −1 orbitally equivalent to the linear system (2). Hence, statement (c) follows. This completes the proof of the theorem.

Proof of Proposition 3
We note that D(x) = D(H)(x). The latter was defined in the proof of Theorem 2. By assumption, it follows that D(x) is a C 1 function and does not vanish on a full Lebesgue measure subset of Ω. From (4) of the proof of Theorem 2, we have that f n (x) does not vanish on a full Lebesgue measure subset of Ω. Otherwise, all f i s are almost zero, and so system (1) has a positive Lebesgue measure subset of singularities, a contradiction.
The proof of Theorem 2 shows that the vector field X has the Jacobian multiplier Q = D(x)/ f n (x). Again from the proof of Theorem 2 and working with the vector field X , we get that This shows that the transformation from systemẋ = h(x) f (x) toẏ = y defined by This proves the proposition.

Proof of Theorem 4
Recall that statement (a) was proved in Singer (1992) and Zhang (2014a). We now prove statements (b) and (c).
We assume without loss of generality that is not zero in R n except perhaps a zero Lebesgue measure subset. Then, we get from the proof of Proposition 3 that f n = 0 in a full Lebesgue measure subset of Ω. Since H i (x) is a Darboux function, we assume that it is of the form where g i j , q i , h i are polynomials, and k i j ∈ C, j = 1, . . . , r i . Computing the partial derivative of G i (x) = log H i (x) with respect to x s for s = 1, . . . , n, we get This is a rational function. So G(x) is also a rational function. By Theorem 2 and its proof, it follows that system (1) has the rational Jacobian multiplier Q(x) = G(x)/ f n (x), because f n is a polynomial. Hence, statement (b) follows.
Finally, we prove statement (c). Let H 1 , . . . , H n−1 be n − 1 functionally independent polynomial first integrals of the polynomial differential system (1). Here, we will use the notations defined in the proof of Theorem 2. We assume that This proves statement (c) and consequently the theorem.

Two Applications of Theorem 2
Consider the differential systeṁ which is the only completely integrable case of the Rössler differential system constructed by Rössler (1987) in 1976. O.E. Rössler inspired by the geometry of threedimensional flows, introduced several systems in the 1970s as prototypes of the simplest autonomous differential equations having chaos, the simplicity is in the sense of minimal dimension, minimal number of parameters, and minimal nonlinearities.
In MathSciNet appear at this moment more than 197 articles about the Rössler's systems. This unique integrable Rössler differential system was first proved in Zhang (2004). Recently, system (9) was studied from the Poisson dynamics point of view, see Tudoran and Gîrban (2012). We can check that system (9) has the two functionally independent first integrals H 1 (x, y, z) = 1 2 (x 2 + y 2 ) + z, H 2 (x, y, z) = e −y z, and the Jacobian multiplier J = e −y . We can check that the transformation of variables is invertible in the region Ω 0 := {(x, y, z) ∈ R 3 | x = 0}, because the Jacobian determinant of this transformation is xe −4y . By Theorem 2, system (9) is transformed tou via the change of variables (10) in Ω 0 . We note that the divergence of system (9) is x. The Lotka-Volterra systems are classical differential systems introduced independently by Lotka and Volterra in the 1920s to model the interaction among species, see Lotka (1920), Volterra (1931), see also Kolmogorov (1936). A particular class of the three-dimensional Lotka-Volterra systems is the so-called May-Leonard models May and Leonard (1975). A completely integrable May and Leonard differential system iṡ See Blé et al. (2013) for the study of the integrability of this system. We can check that system (12) has the two functionally independent first integrals H 1 (x, y, z) = y(x − z) x(y − z) , H 2 (x, y, z) = 2x 2 (y − z) 2 (x − y)y 5/2 (x − z) 3/2 2y(y − x) where F(Φ|m) is the elliptic integral of the first kind. The Jacobian multiplier is We can check that the transformation of variables (10) is invertible in the region Ω 0 := {(x, y, z) ∈ R 3 | x yz(x − y)(x − z)(y − z) = 0}, because the Jacobian determinant of this transformation is 6/(y 6 (x − z) 6 ) and the two first integrals H 1 and H 2 must be well defined. By Theorem 2, system (12) is transformed to system (11) via the change of variables (10) in Ω 0 . We note that the divergence of system (12) is 3.