Inverse Jacobi multipliers and first integrals for nonautonomous differential systems

In this paper, we consider nonautonomous differential systems of arbitrary dimension and first find expressions for their inverse Jacobi multipliers and first integrals in some nonautonomous invariant set in terms of the solutions of the differential system. Given an inverse Jacobi multiplier V, we find a relation between the Poincaré translation map Π at time T that extends to arbitrary dimensions the fundamental relation for scalar equations, V(T,Π(x))=V(0,x)Π′(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V(T, \Pi(x)) = V(0,x)\Pi'(x)}$$\end{document}, found in García et al. (Trans Am Math Soc 362:3591-3612, 2010). The main result guarantees the existence of continua of T-periodic solutions for T-periodic systems in the presence of T-periodic first integrals and inverse Jacobi multipliers.


Introduction
We consider a nonautonomous differential systeṁ where f : I ×Ũ → R n is a C 1  . . , f n ). We associate with system (1) the vector field The divergence of X is divX = n i=1 ∂f i /∂x i . Definition 1. A function V : Ω → R is said to be an inverse Jacobi multiplier for system (1) in the open set Ω ⊂ I ×Ũ if V is of class C 1 (Ω), it is not locally null and it satisfies the following linear first order partial differential equation: 574 A. Buicȃ, I. A. García ZAMP factors) have been used in order to study autonomous planar systems resulting interesting and powerful properties (see, e.g., the survey [8]). For the use of inverse Jacobi multipliers in the study of higherdimensional systems (mainly autonomous), we remind here only the survey [2] and the recent work [3], and we emphasize that the literature is not so rich as in the case of planar systems. One of the specific subjects where inverse Jacobi multipliers have been proved to be especially useful is related to closed orbits of autonomous systems: limit cycles or continua of periodic solutions. When studying planar autonomous systems via inverse integrating factors, sometimes the authors need to transform the system and the inverse integrating factor into a nonautonomous scalar equation and, respectively its corresponding nonautonomous inverse integrating factor. This is the case in [6,9,10]. We were motivated by these works to study nonautonomous scalar equations and their inverse integrating factors, and in fact, we gave results for nonautonomous systems of arbitrary dimension.
In Theorem 3 of [9], passing to curvilinear coordinates, a planar autonomous system is transformed in a neighborhood of a regular orbit into some scalar (thus with n = 1) nonautonomous equation of the form (1). The authors prove a fundamental relation which is a differential equation for the Poincaré map written in terms of an inverse integrating factor. This relation proved to be useful in analyzing the bifurcation of limit cycles from a multiple limit cycle as well as from a homoclinic orbit in [9]. In this work, we find an analogous of this fundamental relation for nonautonomous systems of arbitrary dimension. The result is stated below.
First, we introduce some notation used throughout this work. For each (t 0 , y) ∈ I ×Ũ , we denote by ψ(·; t 0 , y) the solution of (1) satisfying ψ(t 0 ; t 0 , y) = y and by I (t0,y) its maximal interval of existence. Moreover, we recall that the Poincaré translation map at time τ > 0 is the map Π : U ⊂Ũ →Ũ defined by Π(x) = ψ(τ ; 0, x) where U is such that [0, τ] ⊂ I (0,x) for any x ∈ U . Theorem 2. Let V : Ω → R be an inverse Jacobi multiplier of (1). If there exists some τ > 0 and some A key ingredient in the proof of one of the main results (Theorem 1.2) of [6] uses this fundamental relation (3) when n = 1. In [10], the same relation was used to study multiple Hopf bifurcation in a neighborhood of certain singular point of focus type of a planar autonomous system by means of generalized polar coordinates. Moreover, in [4,14], the above Theorem 2 was used to study multiple Hopf bifurcation in a neighborhood of a Hopf point of systems in arbitrary dimensions. We use this result here in order to prove our main result Theorem 3).
Both in [9] and [10] appears in a natural way that the corresponding scalar differential equation of the form (1) is also periodic and the inverse integrating factor of the planar system is transformed into a nonautonomous inverse integrating factor which is periodic.
We were motivated by these facts to study periodic inverse Jacobi multipliers for periodic systems (1) of arbitrary dimension. In this context, it is natural to assume what we will call Hypothesis * in the rest of the paper.
Hypotheses *: (a) System (1) is well defined for any time, that is, (1) is T -periodic for some fixed period T > 0. This means that the function Our main result is stated below. Our contribution is part (ii), and for completeness, we state also (i) which is known by specialists.

Vol. 66 (2015)
Inverse Jacobi multipliers and first integrals 575 (i) If there exist n independent first integrals in R ×Ũ of (1) which are T -periodic, then there exists an open neighborhood U 0 ofx such that ψ(·; 0, x) is T -periodic for any initial condition x ∈ U 0 . (ii) If there exist n − 1 independent first integrals, and an inverse Jacobi multiplierṼ in R ×Ũ which are T -periodic and such thatṼ (0, x) = 0 for all x ∈Ũ , then the T -periodic solution ψ(·; 0,x) is included into a 1-parameter family of T -periodic solutions ψ(t; 0, x * (μ)), where x * is a C 1 function in some open interval of reals.
The notion of independence for first integrals is the usual one (can be found also in Section 2). A key step in the proof of Theorem 3 (ii) is the fundamental relation (3). We also prove in the forthcoming Theorem 24 the T -periodicity of any first integral and any inverse Jacobi multiplier of a system whose solutions are T -periodic. We note here that in [7] are given sufficient conditions to ensure the existence of periodic first integrals for Hamiltonian systems of Lie type.
Our paper is organized as follows. In Section 2, we present the proof of Theorem 2, and we also find in Proposition 7, using the characteristics method, the expressions for an inverse Jacobi multiplier and a first integral in some invariant nonautonomous set, as well as a relation between these two objects. Section 3 contains the above-mentioned Theorems 3 and 24 on T -periodic systems together with some useful examples.
In this paper, det denotes the determinant, D is the symbol for the derivative (the Jacobian matrix if applied to a vector function of various variables), D x is the symbol for the derivative with respect to x, and ∇ is the symbol for the gradient of a real function of various variables.

Inverse Jacobi multipliers and first integrals
Let U ⊂Ũ be open. We fix some t * ∈ I, but for simplicity, we write t * = 0. Let This set is an invariant nonautonomous set in the sense of [13] for the differential equation (1).
In the next lemma, we state some properties of a process ψ(t; t 0 , x), and we also write an equivalent definition of the set Ω * defined in (4). (ii) Ω * = (t, y) ∈ I ×Ũ : 0 ∈ I (t,y) , ψ(0; t, y) ∈ U and is open.
Proof. Statement (i) is Theorem 9.5 from [1]. Statement (ii) follows directly from the definition of Ω * . Statements (iii) and (iv) express the cocycle property of a process of a nonautonomous differential equation that is a consequence of the fact that the solutions are determined uniquely by their initial values. For insight in these topic, one can see [11,13].
Remark 5. Relation (iv) in Lemma 4, which can be written as is a key ingredient in the proof of many statements in this paper.
On the other hand, the matrix function D x ψ(·; 0, x) is the principal fundamental matrix solution of the linear variational systemu = D x f (t, ψ(t; 0, x))u. Liouville's formula (see for example page 152 of [5]) written for this linear system is Using (5) and (6), we obtain Relation (3) is obtained taking t = τ in (7).
We remind the definition of a first integral.
, it is not locally constant and it satisfies the following linear first order partial differential equation: In the next proposition, it is shown how, with a given initial condition, one can build the explicit expression of an inverse Jacobi multiplier and a first integral of (1) in the nonautonomous invariant set Ω * in terms of the associated process ψ(t; t 0 , y). Parts (i) and (ii) of this result are well known, but we present it here for completeness.

Proposition 7. The following statements hold.
(i) Let Ω ⊂ I ×Ũ be open and assume that there exists an inverse Jacobi multiplierṼ : Ω → R of (1) such thatṼ (t, x) = 0 for any (t, x) ∈ Ω. We have that V : Ω → R is an inverse Jacobi multiplier of (1) if and only if there exists a first integral H (or a constant) of (1) in Ω such that where V * : Ω * → R has the expressions (1) and, moreover, given F ∈ C 1 (U ), V : Ω * → R is an inverse Jacobi multiplier of system (1) with V (0, ·) = F if and only if (9) holds.
∈ Ω * and using (iv) of Lemma 4 we obtain the conclusion. Conversely, let now H(t, y) = F (ψ(0; t, y)) for all (t, y) ∈ Ω * . Using again Lemma 4 one can see that H satisfies for any x ∈ U and t ∈ I (0,x) , H(t, ψ(t; 0, x)) = F (x). Taking the derivative with respect to t we obtain that H satisfies X H = 0 in Ω * . (iii) Like in the proof of Theorem 2 one can show that V satisfies (5) for any x ∈ U and t ∈ I (0,x) . In , and x = ψ(0; t, y) for (t, y) ∈ Ω * such that finally we obtain (9) with V * having the first expression given in the statement. In order to obtain the second expression of V * we replace x = ψ(0; t, y) for (t, y) ∈ Ω * in (6). Finally assume that V * ∈ C 1 (Ω * ) and note that, taking in its expression y = ψ(t; 0, x), for any x ∈ U and t ∈ I (0,x) we obtain Taking the derivative with respect to the variable t in (10), one obtains the relation Since, by (ii), F (ψ(0; t, y)) is a first integral, and V * is an inverse Jacobi multiplier in Ω * , applying (i) we obtain that V given by (9) is indeed an inverse Jacobi multiplier.
In the next example, we see the expression of V * (given in Proposition 7 (iii)) for a linear system.
have that the inverse Jacobi multiplier V * is given by V * (t, x) = exp t 0 Tr(A(s)) ds , for all (t, x) ∈ R × R n where Tr(A(s)) is the trace of A(s). We can choose here U = R n and find that Ω * = R × R n . Now, for the simple scalar equationẋ = x 3 , we present its family of time-dependent inverse Jacobi multipliers. This is just to see how Proposition 7 works. Anyway, in some problems, it is important to know the expression of an inverse Jacobi multiplier of some system in order to study perturbations of that system (see [4,9,10,14]).
Example 9. We consider the differential equationẋ = If we take, for example, F 1 (x) = x or F 2 (x) = x 3 for all x ∈ R, we obtain two inverse Jacobi multipliers V 1 (t, x) = x + 2tx 3 and V 2 (x) = x 3 , respectively. Note that they are defined in the whole R 2 , and they satisfy (2) in the whole R 2 . Here, The following is the usual definition of a set of C 1 (functionally) independent real functions, see for example page 436 of [5]. We consider it for a set of first integrals of (1), and we will use it in the next Section.  (1) are independent in Ω if the gradient vectors ∇H 1 (t, y), . . . , ∇H k (t, y) are linearly independent for each (t, y) ∈ Ω.
In the next proposition, we will see that n first integrals H 1 , . . . , H n of (1) are independent in Ω * if and only if the mapF : U → R n defined byF = (H 1 (0, ·), . . . , H n (0, ·)) is a local diffeomorphism. This result must be known, and we state and prove it for completeness and also for further use.
Proof. We take n first integrals H 1 , . . . , H n which are independent in Ω * and considerF andH like in the statement of this Proposition. Using Proposition 7 (ii), we must havẽ In order to prove that the set of vectors ∇H 1 (t, y), . . . , ∇H n (t, y) from R n+1 are linearly independent for each (t, y) ∈ Ω * , it is necessary and sufficient to prove that the n × (n + 1) matrix DH(t, y) has constant rank n in Ω * . Since using (8)  For our next result, we will need the following stronger notion of independence for n first integrals of (1). This notion essentially requires that the local diffeomorphismF : U →F (U ) defined in Proposition 11 to be actually a global diffeomorphism. This justifies the use of the term globally in the following definition.
As is well known, independent first integrals can be used to describe any other first integral or inverse Jacobi multiplier. We state below this idea for the n globally independent first integrals given by Corollary 13. Let H 1 , . . . , H n be C m first integrals of (1) in Ω * which are globally independent. Define the vector functionH = (H 1 , . . . , H n ). Then we have. Proof. First, we recall that the inverse Jacobi multiplier V * defined in Proposition 7 (iii) is of class C m−1 at least and satisfies V * (t, y) = 0 for any (t, y) ∈ Ω * . Second, since H 1 , . . . , H n are C m first integrals of (1) in Ω * by hypothesis, we can define according to Definition 12 the diffeomorphismF : U →F (U ) of class C m asF (x) =H(0, x). (i) Let H : Ω * → R be a C m first integral of (1), take F = H(0, ·) and φ = F •F −1 . Then, φ is C m and F = φ(F ) on U and, further, F (ψ(0; t, y)) = φ(F (ψ(0; t, y))) for all (t, y) ∈ Ω * . But, using Proposition 7, this last relation reads as H = φ(H) on Ω * . The fact that, given some C m function φ, φ(H) is a C m first integral follows using Definition 6 of a first integral. Statement (ii) follows using the previous one and Proposition 7.
Since the first n − 1 components ofH(·, x) are T -periodic for any x ∈Ũ , we have that the first n − 1 components ofF (y) and ofF (ψ(0; T, y)) are the same for any y ∈ U 0 . Then, for each h = (h 1 , . . . , h n ) ∈ F (U 0 ), the first n − 1 components ofF • ψ(0; T, ·) •F −1 (h) are h 1 , . . . , h n−1 . From here, we deduce that the inverse of this map shares the same property. Denote nowŨ 0 =F (ψ(0; T, U 0 )) and note that it is an open neighborhood ofF (x). But, using properties of the inverses of compositions and recalling that the inverse of ψ(0; T, ·) is ψ(T ; 0, ·), one can see that the inverse ofF • ψ(0; T, ·) After all these, taking into account the definition of the Poincaré map Π = ψ(T ; 0, ·), we obtain that for each h ∈Ũ 0 the first n − 1 components ofF • Π •F −1 (h) are h 1 , . . . , h n−1 . We denote the last component of this function by p(h). In this way, we obtain p :Ũ 0 → R, which is C 1 , such that Since ψ(·; 0,x) is T -periodic, we have that Π(x) =x and further, denoting This gives p(h) =h n . We define the C 1 scalar functionp of a scalar variable μ →p(μ) = p(h 1 , . . . ,h n−1 , μ) where the variable μ ranges in an open neighborhood ofh n . Hence, we havẽ We observe thatp We claim thatp is the identity map in an open neighborhood ofh n . For any μ in that neighborhood, we denote Then, Π(x * (μ)) = x * (μ) finishing the proof. It remains to prove the claim. For this, we need to use the existence of an inverse Jacobi multiplier. The hypotheses of Theorem 2 are fulfilled for the inverse Jacobi multiplierṼ : R ×Ũ → R; hence, (3) is satisfied. Using thatṼ (0, x) = 0 for all x ∈Ũ and that, by T -periodicity ofṼ ,Ṽ (T, Π(x)) =Ṽ (0, Π(x)) for any x ∈ U , (3) gives This is a fully nonlinear first-order partial differential equation for the Poincaré map Π, which has as solution the identity map. Of course, when n = 1, this reduces to a first-order scalar ordinary differential Vol. 66 (2015) Inverse Jacobi multipliers and first integrals 581 equation for Π. In arbitrary dimensions, using (13), we will find thatp(μ) satisfies a first-order scalar ordinary differential equation. For this, we compute the determinant of the Jacobian matrix ofF •Π•F −1 in two ways. First, we take into account the structure of this map emphasized before, so that where I n denotes the identity matrix of order n. Therefore, we obtain that Second, we use the chain rule and properties of determinants so that Using the two relations above and (13), we obtain that holds for every variable h ∈Ũ 0 . If we write (14) only for the vectors using (12), we obtain dp dμ Hence,p is the unique solution of the Cauchy problem for the ordinary differential equation (15) with the initial conditionp(h n ) =h n . It is not difficult to see that the identity functionp(μ) = μ, with μ in a neighborhood ofh n , is the solution of this initial value problem. The claim is proved.
Remark 16. The assumption that there exists n−1 independent first integrals which are T -periodic in Theorem 3 (ii) is essential when n ≥ 2 as the following example shows. We consider the linear 2-dimensional systemẋ = A(t)x with the continuous T -periodic in R diagonal matrix A(t) = diag{a 1 (t), a 2 (t)} such that T 0 a i (s)ds = 0 for i = 1, 2 but T 0 (a 1 (s) + a 2 (s))ds = 0. This system has the T -periodic inverse Jacobi multiplier V * (t, x) = exp t 0 [a 1 (s) + a 2 (s)]ds with V * (0, x) = 1, but the only T -periodic solution is the trivial one ψ(t; 0, (0, 0)) = (0, 0) as can be easily shown using the theory of linear systems under the conditions imposed to the coefficients of the system.
Example 17. Let us consider the following family of systems defined in R × R 2 : where k i ∈ N, and a i ∈ C(R) are T -periodic functions satisfying T 0 a i (s)ds = 0. The system admits the T -periodic in R × R 2 inverse Jacobi multiplier Hence,Ṽ (0, Notice that the axis of coordinates are invariant under the process and we have In particular, the origin is an equilibrium and ψ(t; 0, (x 1 , 0)) and ψ(t; 0, (0, x 2 )) are (non-isolated) Tperiodic solutions for each x 1 ∈ R and x 2 ∈ R. Now, the question is: What about the T -periodicity of ψ(t; 0, (x 1 , x 2 )) with x 1 x 2 = 0? Can the system have isolated T -periodic solutions?
To give a partial answer to this question, we consider the particular case of (16), with m i ∈ N and b ∈ C(R). We found for (17) the T -periodic first integral One can see that ∇H(t, x 1 , x 2 ) = (0, 0, 0) if (t, x 1 , x 2 ) ∈ R ×Ũ . We intend to apply Theorem 3 (ii) in order to deduce that (17) does not have isolated T -periodic solutions. Let ψ(t; 0, (x 1 ,x 2 )) be a T -periodic solution withx 1x2 = 0 (whenx 1x2 = 0 we already know that it is not isolated). Take U an open neighborhood of (x 1 ,x 2 ) with the properties guaranteed in Remark 15. We can assume that U ⊂Ũ . We conclude now that all the hypotheses of Theorem 3 (ii) are fulfilled with n = 2, the first integral H and the inverse Jacobi multiplierṼ given above. Hence, indeed, the T -periodic solution ψ(t; 0, (x 1 ,x 2 )) (if it exists) is not isolated in the set of T -periodic solutions of (17).
An interesting particular case of system (1) is when it is divergence free, i.e., divX = n i=1 ∂f i /∂x i = 0 in I ×Ũ . In this case, the inverse Jacobi multiplier V * given in Proposition 7 (ii) is simply V * = 1 and hence always T -periodic and non-null. Theorem 3 (ii) becomes in this case.
Corollary 18. Assume Hypothesis * and that there existsx ∈ U such that ψ(·; 0,x) is a T -periodic solution of (1). Assume, in addition, that div X = 0 in I ×Ũ . If there exist n − 1 independent first integrals in R ×Ũ which are T -periodic then the T -periodic solution ψ(·; 0,x) is included into a 1-parameter family of T -periodic solutions ψ(t; 0, x * (μ)), where x * is a C 1 function in some open interval of reals.
We restate now Theorem 3 in the particular case n = 1, but introducing the small improvement that the conclusion holds globally. The proof does not introduce any new idea.
(i) If there exists a first integral H in R ×Ũ of (1) which is T -periodic and such that H(0, ·) : U → R is a diffeomorphism onto its image, then ψ(·; 0, x) is T -periodic for any initial condition x ∈ U ∩ ψ(T ; 0, U). (ii) If there exist an inverse Jacobi multiplierṼ in R ×Ũ which is T -periodic and such thatṼ (0, x) = 0 for all x ∈Ũ , then ψ(·; 0, x) is T -periodic for any x ∈ U .
Proof. Part (i) can be easily seen following the proof of Theorem 3 (i).
Vol. 66 (2015) Inverse Jacobi multipliers and first integrals 583 (ii) In this particular case, we need only relation (13) from the proof of Theorem 3 (ii) that gives that the Poincaré map Π : U →Ũ is a solution of the ordinary differential equation In addition, we have that Π(x) =x. We deduce using the uniqueness of the solution of an ordinary differential equation with C 1 right-hand side that we must have Π(x) = x for any x ∈ U .
Remark 21. The assumption that there exists a T -periodic solution ψ(·; 0,x) withx ∈ U is essential in Theorem 19 (ii) as the following example shows. We consider the scalar linear ordinary differential equationẋ = a Remark 22. The assumption thatṼ (0, x) = 0 for all x ∈Ũ is essential in Theorem 19 (ii) as the following example shows. We consider again the scalar ordinary differential equationẋ = x, this time with I =Ũ = U = R and T > 0 arbitrary fixed. We have that the solution ψ(t; 0, 0) = 0 is T -periodic, and the inverse Jacobi multiplier V (t, x) = x is also T -periodic in R×R. But there are no other T -periodic solutions ofẋ = x.
We are interested now to study the reciprocal of Theorem 3 (i). The next Lemma and Theorem state in what conditions is valid.
If there exists an open set U 0 ⊂ U such that R × U 0 ⊂ Ω * then both H and V are T -periodic in R × U 0 .
The conclusion follows using the expressions of H and V given in Proposition 7 and relations (18) and (19).
Proof. We consider first the particular case that the fixed solution is the null solution, i.e., ψ(t; 0, 0) = 0 for all t ∈ R. In this case, for any t ∈ [0, T ], we have that ψ(t; 0, U) is an open neighborhood of 0, which implies that is an open neighborhood of 0. Since Ω * = {(t, ψ(t; 0, x)) : t ∈ R, x ∈ U )} and ψ(·; 0, x) is T -periodic for all x ∈ U , from the definition of U 0 it is clear that R × U 0 ⊂ Ω * . The conclusion follows by Lemma 23. Consider now the general case and let H and V like in the hypothesis. We introduce the new variable u = x − ϕ(t) so that Eq. (1) is transformed into equatioṅ u = f (t, u + ϕ(t)) − f (t, ϕ(t)). (21) Its process γ satisfies γ(t; 0; u) = ψ(t; 0, u+x) − ϕ(t) for all u ∈ U −x, assuring that γ(·; 0, u) is T -periodic for all u ∈ U −x and that γ(t; 0, 0) = 0 for all t ∈ R. Then, Eq. (21) is in the particular case solved before. It can be easily shown that H(t, u + ϕ(t)) and V (t, u + ϕ(t)) are a first integral, and, respectively, an inverse Jacobi multiplier of (21). Thus, they are T -periodic in R × U 0 . The proof is done.