POLYNOMIAL FIRST INTEGRALS FOR THE CHEN AND LÜ SYSTEMS

where a, b, c ∈ R are parameters is known as the Chen system [Chen & Ueta, 1999]. It exhibits chaotic phenomena which resembles some familiar features from both the Lorenz and the Rössler attractors, for suitable choices of the parameters. Despite of its similar structure to the Lorenz system, it is not topologically equivalent. This is why Lü and Chen investigated the real differential system


Introduction and statement of the main results
The following real differential system ẋ = a(y − x), where a, b, c ∈ R are parameters is known as the Chen system [Chen & Ueta, 1999].It exhibits chaotic phenomena which resembles some familiar features from both the Lorenz and the Rössler attractors, for suitable choices of the parameters.Despite of its similar structure to the Lorenz system, it is not topologically equivalent.This is why Lü and Chen investigated the real differential system ẋ = a(y − x), where a, b, c ∈ R are parameters, which now is usually called the Lü system [Lü & Chen, 2002].The Lü system connects the Lorenz system and the Chen system and represents a transition from one to the other.For more details see [Lü & Chen, 2002].Moreover, recently Lü and Zhang [Lü & Zhang, 2007] and Lü [Lü, 2009] characterize the invariant algebraic surfaces of the Chen and of the Lü systems, respectively.Furthermore, Lü in [Lü, 2007] characterize the Darboux first integrals of the Chen system.These recent years the dynamics of the Chen system has been analyzed from many different points of view.See for instance [Bashkirtseva et al., 2010, Hou et al., 2008, Mahmoud et al., 2007, Yao & Liu, 2010].
The vector field associated to (1) is and the vector field associated to ( 2) is Let U be an open subset of R 3 such that R 3 \ U has zero Lebesgue measure.We say that a real function ) is constant for all values of the solution (x(t), y(t), z(t)) of X where it is defined, i.e., if XH = 0.
Two functions f 1 (x, y, z) and f 2 (x, y, z) are independent if their gradients are linearly independent vectors for all (x, y, z) ∈ R 3 except perhaps for a set of zero Lebesgue measure.
If the vector field X has two independent first integrals H 1 and H 2 , we say that it is completely integrable.In this case the orbits of X are contained in the curves When a = 0 the Chen and Lü differential systems are completely integrable as it is shown in the following result.Its proof is tedious but it follows easily from direct computations.
Theorem 1.1.When a = 0 the Chen and Lü systems are integrable with the first integrals H 1 = x and H 2 .
(a) For the Chen system where A = c(x 2 + bz) − x 2 z and B = x 2 − bc.
(b) For the Lü system In view of Theorem 1.1 from now on we consider only the case a ̸ = 0 in both the Chen and the Lü systems.
Our main result for the Chen and Lü systems are the following two theorems.
Theorem 1.2.The Chen differential system with a ̸ = 0 has a polynomial first integral if and only if b = c = 0.In this case any polynomial first integral is a polynomial in the variable y 2 + z 2 + 2az.Theorem 1.2 provides a counterexample to statement (b) of Theorem 1 of [Lü, 2007], where the author claims that the Chen system has no Darboux polynomial first integrals and of course the polynomial first integral y 2 + z 2 + 2az is a Darboux polynomial for the Chen system.
Theorem 1.3.The Lü system with a ̸ = 0 has a polynomial first integral if and only if b = c = 0.In this case any polynomial first integral is a polynomial in the variable y 2 + z 2 .
The proof of Theorem 1.2 is given in section 2, but we only provide an sketch of the proof of Theorem 1.3 in section 3. We note that in the proof of Theorems 1.2 we use similar arguments to the ones used in the papers [Lü & Zhang, 2007] and [Lü, 2009].

Proof of Theorem 1.2
In this section we prove Theorem 1.2.For simplifying the computations we introduce the following weight change of variables (3) with µ ∈ R \ {0}.Then the Chen system (1) becomes where the prime denotes the derivative of the variables with respect to T .
A polynomial G(X, Y, Z) is said to be weight homogeneous of degree m ∈ N with respect to the Here N is the set of positive integers.
Let g(x, y, z) be a polynomial first integral of system (1).Set where G i is the weight homogeneous part with weight degree-m−i of G, and m is the weight degree of G with the weight exponent s = (1, 2, 2).
From the definition of a polynomial first integral we have where we still use x, y, z instead of X, Y, Z.
Equating in (5) the terms with µ i for i = 0, 1, . . ., m + 2, we get for j = 2, 3, . . ., m + 2, where G j = 0 for j > m and L is the linear partial differential operator of the form The characteristic equations associated to the linear partial differential operator are This system has the general solution where d 1 and d 2 are constants of integration.According to this, we make the change of variables Its inverse transformation is In the following, for simplicity, we only consider the case Under the changes of variables ( 8) and ( 9), the first equation of (6) becomes the following ordinary differential equation (for fixed u, v): where Ḡ0 is G 0 written in the variables u, v and w.In what follows we always use the notation θ to denote θ(x, y, z) written in the variables u, v, w.
The last equation has the following general solution Ḡ0 = T0 (u, v), where T0 is an arbitrary smooth function in u and v. So, Since G 0 is a weight homogeneous polynomial and the weight degrees of u and v in the variables x, y, z are 2 and 4, respectively, G 0 should have weight degree either m = 4n, or m = 4n − 2 for some convenient n ∈ N.So G 0 has the form with the weight degree 4n, or with the weight degree 4n − 2.
Case 1: G 0 has the form (11). Substituting G 0 into the second equation of ( 6), we can prove that Using the transformations ( 8) and ( 9) and working in a similar way to solve Ḡ0 we get the following ordinary differential equation (for fixed u and v): it is easy to deduce that the integration of the previous equation with respect to w is where T1 (u, v) is an arbitrary smooth function in u and v. Since Ḡ1 is a weight homogeneous polynomial of weight degree 4n − 3, we must have T1 (u, v) = 0, and where a 0 = a n+1 = 0.It is easy to prove that since a ̸ = 0, conditions ( 12) are equivalent to one of the following conditions, either b = 2a = −c, n = 1/2 and there exists i 0 ∈ {1, . . ., n − 1} such that Since n is a natural number the two first cases are not possible, and therefore only the third case is possible.In this last case it is obvious that the Chen system has no polynomial first integral of the given form.
These equations imply that i = 0 and a (4) i = 0 for i = 1, . . ., n.Thus we have that G 5 = 0 and By recursive calculations we can prove that for i ≥ 0 and 2i ≤ m.Hence the polynomial first integral is The generator of the polynomial first integral is y 2 + z 2 + 2az.This completes the proof of Theorem 1.2.

Proof of Theorem 1.3
In this section we prove Theorem 1.3.We introduce the weight change of variables in (3) and system (2) becomes where again the prime denotes the derivative with respect to T .
Let g(x, y, z) be a polynomial first integral of system (2) and set G(X, Y, Z) as in (4).We have where again we still use x, y, z instead of X, Y, Z.
Proceeding as in the proof of Theorem 1.2 we get that G 0 has either the form (10) with weight degree 4m, or the form (11) with weight degree 4m − 2.
Proceeding analogously to Case 1 in the proof of Theorem 1.2 we get that the Lü system has no polynomial first integrals if G 0 is of the form (11). Again, proceeding as in Case 2 of the proof of Theorem 1.2 we get that G 1 = 0 and again from ( 13) and since a ̸ = 0, the unique possibility is b = c = 0, a 0 ̸ = 0 and a i = 0 for i = 1, . . ., n. Therefore G 0 = a 0 (y 2 + z 2 ) n , and i (x 2 − 2az) 2i−1 (y 2 + z 2 ) n−i .
Then from (15) with j = 3, and proceeding as in the proof of Theorem 1.2 we get that a (2) i = 0 for i = 1, . . ., n, and consequently G 2 = G 3 = 0.By recursive calculations, and proceeding as in Case 2 in the proof of Theorem 1.2 we can prove that G i = 0 for i = 1, . . ., n. Hence the generator of the polynomial first integrals is y 2 + z 2 .This completes the proof of Theorem 1.3.