Non-existence Criteria for Laurent Polynomial First Integrals

Shaoyun Shi^{∗}and Yuecai Han School of Mathematics, Jilin University, Changchun 130012, P. R. China

Abstract In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial first integrals for a general nonlinear systems of ordinary differential equations ẋ = f(x), x ∈ ℝ^{n} with f(0) = 0. We show that if the eigenvalues of the Jacobi matrix of the vector field f(x) are ℤ-independent, then the system has no nontrivial Laurent polynomial integrals.

Keywords. First integrals, integrability, Laurent polynomials, non-integrability, partial integrabiltiy.

AMS(MOS). Mathematics Subject Classification. 58F07, 34A20.

1 Introduction

A system of differential equations

ẋ = f(x)  (1)

is called completely integrable, if it has a sufficiently rich set of first integrals such that its solutions can be expressed by these integrals. According to the famous Liouville-Arnold theorem, a Hamiltonian system with n degrees of freedom is integrable if it possesses n independent integrals of motion in involution. Here, a single-valued function ϕ(x) is called a first integral of system (1) if it is constant along any solution curve of system (1). If ϕ(x) is differentiable, then this definition can be written as the condition

⟨\frac{dϕ}{dx}, f(x)⟩ = 0.

Obviously, if no such a nontrivial integral exists, then system (1) is called nonintegrable. So finding some simple test for the existence or non-existence of nontrivial first integrals(in given function spaces, such as those of polynomials, rational, or analytic functions) is an important problem in considering integrability and nonintegrability, see Costin[2], Kozlov[6] and Kruskal and Clarkson [7].

∗Supported by NSFC grant TY10126013.

The theory of Ziglin[17] has been proved to be one of the most successful approach for proving non-integrability of the n degrees of freedom Hamiltonian systems. This theory has been shown to be useful for many systems, such as the motion of rigid body around a fixed point[17], homogeneous potentials[16], and a reduced Yang-Mills potentials[5], etc. The technique consists of linearizing a system around particular solutions(forming linear manifolds). The linearized equation is then studied. If its monodromy group is too large(i.e., the branches of solutions have too many values) then the linearized equation has no meromorphic first integrals. It is then shown that this implies that the original system has no meromorphic first integrals.

One of the pioneering authors in considering the non-integrability problems for non-Hamiltonian systems is Yoshida. Using a singularity analysis type method, he was able to derive necessary conditions for algebraic integrability for similarityinvariant system[15]. In [3], Furta suggested a simple and easily verifiable criterion of non-existence of nontrivial analytic integrals for general analytic autonomous systems. He proved that if the eigenvalues of the Jacobi matrix of the vector field f(x) at some fixed point are ℕ-independent, then the system ẋ = f(x) has no nontrivial integral analytic in a neighborhood of this fixed point. Based on this key criterion, he also made some further study on non-integrability of general semiquasihomogeneous systems. Some related results can be found in [4, 9, 10, 11, 12, 13].

However, there are still many systems encountered in physics which do not fall in the set of completely integrable or completely non-integrable systems. Indeed, if a system admits a certain number of first integrals less than the number required for the complete integration, then non-integrability cannot be proved in general. Such systems will be called partially integrable[4]. Some works have been done in this direction, see [1, 8, 14].

The function space we are interested in this paper is the Laurent polynomial ring C[x_{1}^{±}, ⋯, x_{n}^{±}]. A Laurent polynomial P(x) in the n variables x = (x_{1}, x_{2}, ⋯, x_{n}) is given by

P (x) = ∑_{(k_{1},···,k_{n})∈A} P_{k_{1}···k_{n}}x_{1}^{k_{1}} ⋯ x_{n}^{k_{n}},

where P_{k_{1}···k_{n}} ∈ ℂ and A, the support of P (x), is a finite subset of integer group ℤ^{n}. We will give a simple criterion for non-existence of Laurent polynomial first integrals for general nonlinear analytic systems.

The outline of this paper is as follows. We will first give a simple criterion of non-existence of Laurent polynomial first integral for general analytic systems of differential equations in Section 2. Then in Section 3, we consider the partial integrability for general nonlinear systems. Some examples are presented in Section 4 to illustrate our results.

2 A criterion for non-integrability

Consider an analytic system of differential equations

ẋ = f(x), x = (x_{1}, ⋯, x_{n}) ∈ ℂ^{n}  (2)

in a neighbourhood of the trivial stationary solution x = 0. Let A denote the Jacobi matrix of the vector field f(x) at x = 0. System (2) can be rewritten as

ẋ = Ax + f̃(x), x = (x_{1}, ⋯, x_{n}) ∈ ℂ^{n}  (3)

near some neighborhood of the origin x = 0, where f̃(x) = o(x).

Theorem 2.1. If the eigenvalues λ_{1}, ⋯, λ_{n} of A are ℤ-independent, i.e., they do not satisfy any resonant equality of the following type

∑_{j=1}^{n} k_{j}λ_{j} = 0, k_{j} ∈ ℤ, ∑_{j=1}^{n} |k_{j}| > 0,  (4)

then system (3) does not have any nontrivial Laurent polynomial integral.

Proof. Since after a nonsingular linear transformation, A can be changed to a Jordan canonical form, for simplicity, we can assume A is a Jordan canonical form, i.e.,

A = ( \begin{matrix}J_{1} \\ & J_{2} \\ & & . & . & . \\ & & & & & Jm̃ \end{matrix} ) , J_{r} = ( \begin{matrix}λ_{r} & 1 \\ & ... & ... \\ & & ... & 1 \\ & & & λ_{r} \end{matrix} ) ,

where J_{r}(r = 1, ⋯, m̃) is a Jordan block with degree equal to i_{r}, i_{1} + ⋯ + im̃ = n. Suppose system (3) has a Laurent polynomial integral

P (x) = ∑_{(k_{1},···,k_{n})∈A} P_{k_{1}···k_{n}}x_{1}^{k_{1}} ⋯ x_{n}^{k_{n}},

where A is the support of P (x), then P (x) has to satisfy the following partial differential equation

⟨\frac{dP}{dx} (x), Ax + f̃(x)⟩ ≡ 0,  (5)

where ⟨·, ·⟩ denotes the standard scalar product in ℂ^{n}.

Note that P (x) can be rewritten as

P (x) = P_{l}(x) + P_{l+1}(x) + ⋯ + P_{p}(x), l ≤ p, l, p ∈ ℤ,  (6)

where P_{k}(x) are homogeneous Laurent polynomials of degree k in x and P_{l}(x) ≢ 0, P_{p}(x) ≢ 0.

Substitute (6) into (5) and equate all the terms in (5) of the same order with respect to x with zero and consider the first nonzero term in (6), we get

⟨\frac{dP_{l}}{dx} (x), Ax⟩ ≡ 0.

This means that P_{l}(x) is an integral of the linear system

ẋ = Ax.  (7)

Make the following transformation of variables

x = Cy,

where

C = ( \begin{matrix}C_{1} \\ & C_{2} \\ & & . & . & . \\ & & & & & Cm̃ \end{matrix} ) , C_{r} = ( \begin{matrix}1 \\ & ϵ \\ & & . & . & . \\ & & & & & ϵ^{i_{r}−1} \end{matrix} ) ,

ϵ > 0 is a constant.

Under the transformation, system (7) can be rewritten as

ẏ = (B + ϵB̃)y,  (8)

where

B = ( \begin{matrix}B_{1} \\ & B_{2} \\ & & . & . & . \\ & & & & & Bm̃ \end{matrix} ) , B_{r} = ( \begin{matrix}λ_{r} \\ & λ_{r} \\ & & . & . & . \\ & & & & & λ_{r} \end{matrix} ) ,

B̃ = ( B̃_{1} ; B̃_{2} ; ... ; B̃m̃ ) , B̃_{r} = ( \begin{matrix}0 & 1 \\ & ... & ... \\ & & ... & 1 \\ & & & 0 \end{matrix} ) .

So Q(y, ϵ) = P_{l}(Cy) is an integral of linear system (8), i.e.,

⟨\frac{dQ}{dy}(ϵ, y), (B + ϵB̃)y⟩ ≡ 0.  (9)

Since

P_{l}(x) = ∑_{k_{1}+···+k_{n}=l} P_{k_{1}···k_{n}}x_{1}^{k_{1}}x_{2}^{k_{2}} ⋯ x_{n}^{k_{n}},

Q(y, ϵ) has the form

Q(y, ϵ) = P_{l}(Cy) = ϵ^{L}Q_{L}(y) + ϵ^{L+1}Q_{L+1}(y) + ⋯ + ϵ^{M}Q_{M}(y),  (10)

where L, M ∈ ℤ are certain integers, L ≤ M, Q_{i}(v) are homogeneous form of degree l and Q_{L}(y) ≢ 0, Q_{M}(y) ≢ 0.

By (9) and (10), Q_{L}(y) has to satisfy the following equation

⟨\frac{dQ_{L}}{dy} (y), By⟩ ≡ 0.  (11)

Since

Q_{L}(y) = ∑_{k_{1}+···+k_{n}=l} Q_{k_{1}···k_{n}}^{L}y_{1}^{k_{1}} ⋯ y_{n}^{k_{n}} ≢ 0,

by (11)

∑_{k_{1}+···+k_{n}=l}[(k_{1} + ⋯ + k_{i_{1}})λ_{1} + (k_{i_{1}+1} + ⋯ + k_{i_{1}+i_{2}})λ_{2} + ⋯

+ (k_{i_{1}+···+i_{m̃−1}+1} + ⋯ + k_{n})λm̃]Q_{k_{1}···k_{n}}^{L}y_{1}^{k_{1}} ⋯ y_{n}^{k_{n}} ≡ 0.

Thus, a resonant condition of (4) type has to be fulfilled for some nonzero coefficient Q_{k_{1}···k_{n}}^{L}, which contradicts the conditions of Theorem 2.1. The proof is now complete.

3 A criterion for partial integrability

By the proof of Theorem 2.1, we can see that if system (3) has a nontrivial Laurent polynomial integral P (x), then linear system (7) must have a nontrivial homogeneous Laurent polynomial integral P_{l}(x). So in general at least one resonant relationship of type (4) must be satisfied, and the set

G = {k = {k_{1}, ⋯, k_{n}} ∈ ℤ^{n} : ∑_{j=1}^{n} k_{j}λ_{j} = 0}

is a nonempty subgroup of ℤ^{n}.

Lemma 1. Let system (3) have s(s < n) nontrivial Laurent polynomial integrals P^{1}(x), ⋯, P^{s}(x). If any nontrivial homogeneous Laurent polynomial integral Q_{q}(x) of linear system (7) is a smooth function of P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x), i.e., Q_{q}(x) = H(P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x)), then any nontrivial Laurent polynomial integral Q(x) of system (3) is a smooth function of P^{1}(x), ⋯, P^{s}(x).

Proof: Under the transformation x = εy, system (3) can be rewritten as

ẏ = Ay + εf̃(y, ε).  (12)

We can also rewrite the integrals P^{i}(x) and Q(x) of system (3) as follows

P̃^{i}(y, ε) = P^{i}(εy) = ε^{l_{i}}(P_{l_{i}}^{i}(y) + εP_{l_{i}+1}^{i}(y) + ⋯ + ε^{p_{i}−l_{i}}P_{p_{i}}^{i}(y)),  (13)
Q̃(y, ε) = Q(εy) = ε^{q}(Q_{q}(y) + εQ_{q+1}(y) + ⋯ + ε^{p−q}Q_{p}(y)),  (14)

where P_{l_{i}+j}^{i}(x) and Q_{q+j} are homogeneous Laurent polynomials. Therefore P̃^{1}(y, ε), ⋯, P̃^{s}(y, ε) and Q̃(y, ε) are integrals of system (12).

Since

Q_{q}(x) = H(P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x)),  (15)

under the transformation x = εy, we have

ε^{q}Q_{q}(y) = Q_{q}(εy) = H(P_{l_{1}}^{1}(εy), ⋯, P_{l_{s}}^{s}(εy)) = H(ε^{l_{1}}P_{l_{1}}^{1}(y), ⋯, ε^{l_{s}}P_{l_{s}}^{s}(y)). (16)

By (15) and (16) we obtain

H(ε^{l_{1}}P_{l_{1}}^{1}(y), ⋯, ε^{l_{s}}P_{l_{s}}^{s}(y)) = ε^{q}H(P_{l_{1}}^{1}(y), ⋯, P_{l_{s}}^{s}(y)).  (17)

Let H^{(0)} = H. Then the function

Q̃^{(1)}(y, ε) = Q̃(y, ε) − H^{(0)}(P̃^{1}(y, ε), ⋯, P̃^{s}(y, ε))

is an integral of system (12), since Q̃(y, ε) and P̃^{1}(y, ε), ⋯, P̃^{s}(y, ε) are all integrals of system (12).

By (13), (14), (15) and (17), it is not difficult to see that the function Q̃^{(1)}(y, ε) is at least of q + 1 order with respect to ε and can be rewritten as

Q̃^{(1)}(y, ε) = ε^{q_{1}}(Q_{q_{1}}^{(1)}(y) + ∑_{j=1}^{∞} ε^{j}Q_{q_{1}+j}^{(1)}(y)),

where q_{1} ≥ q + 1 is an integer, Q_{q_{1}+j}^{(1)}(y) is a homogeneous form of degree q_{1} + j.

Now Q_{q_{1}}^{(1)}(y) is also an integral of linear system (7). According to the assumptions of the lemma, Q_{q_{1}}^{(1)} = H^{(1)}(P_{l_{1}}^{1}, ⋯, P_{l_{s}}^{s}). So the function

Q̃^{(2)}(y, ε) = Q̃^{(1)}(y, ε) − H^{(1)}(P̃^{1}(y, ε), ⋯, P̃^{s}(y, ε))

is also an integral of system (12) which is at least of q_{1} + 1 degree with respect to ε. By repeating infinitely this process, we obtain that

Q̃(y, ε) = ∑_{j=0}^{∞} H^{(j)}(P̃^{1}(y, ε), ⋯, P̃^{s}(y, ε)),

which is equivalent to the fact that

Q(x) = ∑_{j=0}^{∞} H^{(j)}(P^{1}(x), ⋯, P^{s}(x)) = F(P^{1}(x), ⋯, P^{s}(x)),

for a certain smooth function F.

Lemma 2. Assume A is diagonalizable and rank G = s. If linear system (7) has s homogeneous Laurent polynomial integrals P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x) which are functionally independent, then any other nontrivial homogeneous Laurent polynomial integrals Q_{q}(x) of system (7) is a function of P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x).

Proof. For simplicity, we assume A has already a diagonal form diag(λ_{1}, ⋯, λ_{n}). Let τ_{1} = (τ_{11}, ⋯, τ_{1n}), ⋯, τ_{s} = (τ_{s1}, ⋯, τ_{sn}) are generating elements of group G.

It is clearly that ω^{1}(x) = x_{1}^{τ_{11}} ⋯ x_{n}^{τ_{1n}}, ⋯, ω^{s}(x) = x_{1}^{τ_{s1}} ⋯ x_{n}^{τ_{sn}} are s rational integrals of linear system (7). Furthermore, ω^{1}(x), ⋯, ω^{s}(x) are functionally independent.

In fact, since τ_{1} = (τ_{11}, ⋯, τ_{1n}), ⋯, τ_{s} = (τ_{s1}, ⋯, τ_{sn}) are generating elements of group G,

( \begin{matrix}τ_{11} & τ_{21} & ⋯ & τ_{s1} \\ τ_{12} & τ_{22} & ⋯ & τ_{s2} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ τ_{1n} & τ_{2n} & ⋯ & τ_{sn} \end{matrix} )

is full-ranked. Therefore, it must have a subdeterminant of degree s which is nonzero, without loss of generality, we can assume

| \begin{matrix}τ_{11} & τ_{21} & ⋯ & τ_{s1} \\ τ_{12} & τ_{22} & ⋯ & τ_{s2} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ τ_{1s} & τ_{2s} & ⋯ & τ_{ss} \end{matrix} | ≠ 0.

Then

| \begin{matrix}\frac{∂ω^{1}}{∂x_{1}} & \frac{∂ω^{1}}{∂x_{2}} & ⋯ & \frac{∂ω^{1}}{∂x_{s}} \\ \frac{∂ω^{2}}{∂x_{1}} & \frac{∂ω^{2}}{∂x_{2}} & ⋯ & \frac{∂ω^{2}}{∂x_{s}} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ \frac{∂ω^{s}}{∂x_{1}} & \frac{∂ω^{s}}{∂x_{2}} & ⋯ & \frac{∂ω^{s}}{∂x_{s}} \end{matrix} | = n ; ∏ ; i=1 ω^{i}(x)x_{i}^{−1} | \begin{matrix}τ_{11} & τ_{12} & ⋯ & τ_{1s} \\ τ_{21} & τ_{22} & ⋯ & τ_{2s} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ τ_{s1} & τ_{s2} & ⋯ & τ_{ss} \end{matrix} | ≠ 0.

And hence

( \begin{matrix}\frac{∂ω^{1}}{∂x_{1}} & \frac{∂ω^{1}}{∂x_{2}} & ⋯ & \frac{∂ω^{1}}{∂x_{n}} \\ \frac{∂ω^{2}}{∂x_{1}} & \frac{∂ω^{2}}{∂x_{2}} & ⋯ & \frac{∂ω^{2}}{∂x_{n}} \\ ⋯ & ⋯ & ⋯ & ⋯ \\ \frac{∂ω^{s}}{∂x_{1}} & \frac{∂ω^{s}}{∂x_{2}} & ⋯ & \frac{∂ω^{s}}{∂x_{n}} \end{matrix} )

is full-ranked, i.e., ω^{1}(x), ⋯, ω^{s}(x) are functionally independent. Now assume

Q_{q}(x) = ∑_{k_{1}+···+k_{n}=q} Q_{k_{1}···k_{n}}x_{1}^{k_{1}} ⋯ x_{n}^{k_{n}}

is a nontrivial homogeneous Laurent polynomial integral of linear system (7). Then

⟨\frac{dQ_{q}}{dx} (x), Ax⟩ = ∑_{k_{1}+···+k_{n}=q}(k_{1}λ_{1} + ⋯ + k_{n}λ_{n})Q_{k_{1}···k_{n}}x_{1}^{k_{1}} ⋯ x_{n}^{k_{n}} ≡ 0.

Hence for every nontrivial monomial x_{1}^{k_{1}} ⋯ x_{n}^{k_{n}} of above expansion, geometric point k = (k_{1}, ⋯, k_{n}) ∈ G. Therefore there exist a_{1}, ⋯, a_{s} ∈ ℤ such that

k = a_{1}τ_{1} + ⋯ + a_{s}τ_{s},

or equivalently

k_{i} = a_{1}τ_{1i} + ⋯ + a_{s}τ_{si}, i = 1, 2, ⋯, n,

and we have

x_{1}^{k_{1}} ⋯ x_{n}^{k_{n}} = (x_{1}^{τ_{11}} ⋯ x_{n}^{τ_{1n}})^{a_{1}} ⋯ (x_{1}^{τ_{s1}} ⋯ x_{n}^{τ_{sn}})^{a_{s}}
= (ω^{1}(x))^{a_{1}} ⋯ (ω^{s}(x))^{a_{s}}.

So there exist a Laurent polynomial G such that

Q_{q}(x) = G(ω^{1}(x), ⋯, ω^{s}(x)).  (18)

Similarly, there also exist Laurent polynomials F_{i} such that

P_{l_{i}}^{i}(x) = F_{i}(ω^{1}(x), ⋯, ω^{s}(x)), i = 1, 2, ⋯, s.  (19)

Since P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x) are functionally independent, the matrix

\frac{∂(P_{l_{1}}^{1}, ⋯, P_{l_{s}}^{s})}{∂(x_{1}, ⋯, x_{n})} = \frac{∂(F_{1}, ⋯, F_{s})}{∂(ω^{1}, ⋯, ω^{s})} · \frac{∂(ω^{1}, ⋯, ω^{s})}{∂(x_{1}, ⋯, x_{n})}

is full-ranked. On the other hand, ω^{1}(x), ⋯, ω^{s}(x) are functionally independent, thus the matrix

\frac{∂(ω^{1}, ⋯, ω^{s})}{∂(x_{1}, ⋯, x_{n})}

is also full-ranked. Therefore the matrix

\frac{∂(F_{1}, ⋯, F_{s})}{∂(ω^{1}, ⋯, ω^{s})}

is full-ranked(nondegenerated). By Inverse Function Theorem and (19)

ω^{i}(x) = G_{i}(P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x)), i = 1, 2, ⋯, s,  (20)

where G_{i} is a smooth function. Combining (18) and (20) we have

Q_{q}(x) = G(ω^{1}(x), ⋯, ω^{s}(x))
= G(G_{1}(P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x)), ⋯ G_{m}(P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x)))
= F(P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x)).

The proof is complete.

The following theorem is the direct result of Lemma 1 and Lemma 2.

Theorem 3.1. Assume system (3) has s(s < n) nontrivial Laurent polynomial integrals P^{1}(x), ⋯, P^{s}(x) and matrix A is diagonalizable. If P_{l_{1}}^{1}(x), ⋯, P_{l_{s}}^{s}(x) are functionally independent and rank G = s, then any other nontrivial Laurent polynomial integral Q(x) of system (3) must be a function of P^{1}(x), ⋯, P^{s}(x).

4 Examples

Example 1. Consider the following system

ẋ_{j} = α_{j}x_{j} + β_{j}x_{j+1} + ∑_{k=1}^{n} a_{jk}x_{j}x_{k}, j = 1, 2, ⋯, n,  (21)

where α_{j}, β_{j} and a_{jk} are real constants, β_{n} = 0. Some ecological systems, such as Lotka-Volterra systems, can be reduced to such forms by a linear transformation near an equilibrium.

According to Theorem 2.1, we know that if α_{1}, ⋯, α_{n} are ℤ-independent, i.e., they do not satisfy any resonant equality

∑_{i=1}^{n} k_{i}α_{i} = 0, k_{i} ∈ ℤ, ∑_{i=1}^{n} |k_{i}| ≥ 1,

then system (21) does not have any Laurent polynomial first integral.

Example 2. We give another artificial example illustrating Theorem 3.1. Consider following system

ẋ_{1} = x_{1},
ẋ_{2} = x_{2},
ẋ_{3} = x_{3},  (22)
ẋ_{4} = αx_{4} + f(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}),
ẋ_{5} = βx_{5} + g(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}),

where x = (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) ∈ ℂ^{5}, f(x) = o(x), g(x) = o(x).

System (22) has two Laurent polynomial integrals P_{1}(x) = x_{1}x_{2}^{−1} and P_{2}(x) = x_{1}x_{3}^{−1} which are functionally independent. According to Theorem 3.1, we can conclude that any other nontrivial Laurent polynomial integral of system (22) is a smooth function of P_{1}(x) and P_{2}(x) if the rank of the group

G = {k = (k_{1}, k_{2}, k_{3}, k_{4}, k_{5}) ∈ ℤ^{5} : k_{1} + k_{2} + k_{3} + k_{4}α + k_{5}β = 0}

is equal to 2. This condition is equivalent to that for any k̃_{1}, k̃_{2}, k̃_{3} ∈ ℤ,

k̃_{1} + k̃_{2}α + k̃_{3}β ≠ 0,

since (1, −1, 0, 0, 0) and (1, 0, −1, 0, 0) are two generating elements of G.

Acknowledgments

The authors would like to thank the anonymous referees for very helpful comments.

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