Meromorphic and formal first integrals for the Lorenz system

Kaiyin Huang; Shaoyun Shi; Wenlei Li

doi:10.1080/14029251.2018.1440745

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Volume 25, Issue 1, February 2018, Pages 106 - 121

Meromorphic and formal first integrals for the Lorenz system

Authors

Kaiyin Huang

School of Mathematics, Jilin University, Changchun 130012, P. R. China,keiyinhuang@gmail.com

Shaoyun Shi

School of Mathematics, Jilin University, Changchun 130012, P. R. China

State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, P. R. China,shisy@jlu.edu.cn

Wenlei Li^*

School of Mathematics, Jilin University, Changchun 130012, P. R. China

Beijing Computational Science Research Center, ZPark II, No. 10 Dongbeiwang, West Road, Haidian District, Beijing 100094, China,lwlei@jlu.edu.cn

^*Corresponding author.

Corresponding Author

Wenlei Li

Received 17 September 2017, Accepted 13 October 2017, Available Online 6 January 2021.

DOI: 10.1080/14029251.2018.1440745 How to use a DOI?
Keywords: Meromorphic first integrals; formal first integrals; Lorenz system
Abstract: The Lorenz system
x˙=σ(y−x), y˙=rx−y−xz, z˙=−βz+xy,
is completely integrable with two functional independent first integrals when σ = 0 and β, r arbitrary. In this paper, we study the integrability of the Lorenz system when σ, β, r take the remaining values. For the case of σβ ≠ 0, we consider the non-existence of meromorphic first integrals for the Lorenz system, and show that it is not completely integrable with meromorphic first integrals, and furthermore, if 2(σ+1)2+4σ(r−1)/β is not an odd number, then it also dose not admit any meromorphic first integrals and is not integrable in the sense of Bogoyavlensky. For the case of σ ≠ 0, β = 0, we study the existence of formal first integrals and present a necessary condition of the Lorenz system processing a time-dependent formal first integral in the form of Φ(x, y, z)exp(λt).
Copyright: © 2018 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Consider the Lorenz system

x˙=−σx+σy, y˙=rx−y−xz, z˙=−βz+xy, (1.1)

where σ, r and β are parameters and the dot denotes derivative with respect to the time t. This system was first derived in [13] to describe atmospheric convection and was the first chaotic example showing the existence of the strange attractor. It is also well known for a variety of problems in lasers, dynamos, thermosyphons, brushless DC motors, electric circuits, chemical reactions and forward osmosis.

The Lorenz system (1.1) has been widely investigated from the dynamical point of view, for more details please see the monograph [25]. Another important topic related to the Lorenz system (1.1) is the integrability. Kus [12] and Gupta [9] proposed a direct method to find some nonautonomous first integrals of the form P(x,y,z)eαt with P(x, y, z) ∈ ℂ[x, y, z] and α ∈ ℝ. Later, by means of the weight homogeneous polynomials and the Darboux integrability theory, Zhang in his paper [29] characterized all the exponential factors and the Darboux first integrals of (1.1). Llibre and Valls in [15] studied the formal and analytic first integrals of the Lorenz system (1.1). Ballesteros et al. [4] applied the Poisson-Lie approach to the integrability of Lorenz system and obtained new integrable deformations for Lorenz system. For more details, see [10, 14, 27] and references quoted therein.

The existence of first integrals for differential equations plays a fundamental role in the qualitative theory of differential equations. In particular, for a given 3-dimensional differential system, if it admits a first integral, we can reduce the dimension of the considered system by one. The nonexistence of first integrals can allow us to expect that the considered system is complex and admits a chaotic behavior. Therefore, it is an important issue to recognize the values of the parameters of given differential systems where they admit first integrals. However, there is no general way to deal with this problem. Many scholars have developed a lot of methods to study the existence of first integrals of the differential equations such as Ziglin analysis, Morales-Ramis theory, the Painlevé analysis, Darboux theory of integrability and the Carleman embedding procedure. See [1, 6, 17, 24] and the references therein.

Let U be an open set in ℂ³. A non-constant function Φ(x, y, z) ∈ C¹(U, ℂ) is called a first integral of system (1.1) if it stays constant along all solution curves (x(t), y(t), z(t)) of system (1.1), i.e.,

(−σx+σy)∂Φ∂x+(rx−y−xz)∂Φ∂y+(−βz+xy)∂Φ∂z≡0.

Similarly, we can also define a non-constant differentiable function Ψ(x, y, z, t) as a time-dependent first integral of system (1.1) if Ψ(x, y, z, t) satisfies

(−σx+σy)∂Ψ∂x+(rx−y−xz)∂Ψ∂y+(−βz+xy)∂Ψ∂z+∂Ψ∂t≡0.

If the first integral Φ(x, y, z) or Ψ(x, y, z, t) is a formal power series in (x, y, z) or a meromorphic function with respect to (x, y, z), then it is called a formal first integral or meromorphic first integral of (1.1), respectively.

The aim of this paper is to make a complete investigation to the integrability and nonintegrability of the Lorenz system (1.1), which is well known to show different dynamical properties according to different values of parameters. We will mainly divide our study into three cases: σ = 0; σ β ≠ 0; and σ ≠ 0, β = 0.

The case when σ = 0 is trivial, since system (1.1) can be reduced to a linear system. Moreover, as pointed out by Llibre and Valls [15], system (1.1) with σ = 0 can also be viewed as a completely integrable system with two functionally independent first integrals

Φ1=x and Φ2=F1(x,y,z)exp(−2arctanF2(x,y,z)F3(x)),

where

F1=x(r2x3−(1+β)rx2y+bxy2+x3y2+β(β−1)rxz −2rx3z−β(β−1)yz+(1−β)x2yz+bxz2+x3z2),F2=(β−1)(rx−y)+(β+1)xz−2x2y(β+1)((r−z)x−y)F3,F3=4(β+x2)β+1−1.

Our first result deals with the Lorenz system (1.1) with σβ ≠ 0. In this case, we investigate the non-integrability of (1.1). Unlike Hamiltonian systems where the definition of the integrability is well-defined in the sense of Liouville, there is no universal definition of integrability for general dynamical systems. In our work, we focus on two kinds of notions of the integrability, complete integrability and B-integrability(see section 2 for the definitions and comments on two notions). Let Γ be the flow generated by the particular solution (x, y, z) = (0, 0, e ^−βt) of system (1.1). The first main result of the paper is the following.

Theorem 1.1.

Assume that σβ ≠ 0. Then system (1.1) is not completely integrable with meromorphic first integrals in a neighborhood of Γ. Moreover, if

2(σ+1)2+4σ(r−1)|β|

is not an odd number, then system (1.1) does not possess any meromorphic first integral and is not B-integrable in the meromorphic category.

We note that Theorem 1.1 characterizes first integrals for system (1.1) with σβ ≠ 0 in the category of meromorphic functions, and its proof is based on the differential Galois theory. This method has its roots in the work of Morales-Ruiz, Ramis, Simó, Baider, Churchill, Rod and Singer [3, 7, 22–24]. Roughly speaking, their results show that if Hamiltonian system with n degrees of freedom admits n meromorphic first integrals which are in involution and independent, then the identity component of the differential Galois group of the normal variational equations along a non-equilibrium solution must be commutative. Many scholars applied Morales-Ramis theory to large numbers of practical models, see for instance [8, 21, 26]. For general dynamical systems, the authors in [18, 19] presented some results which reveal the relation between the meromorphic first integrals and the differential Galois group of normal variational equations, which will be used to prove Theorem 1.1.

Next we shall study system (1.1) with σ ≠ 0, β = 0.

Theorem 1.2.

Assume that σ ≠ 0, β = 0, m≔(σ−1)2+4σr≠0 , and (σ + 1) / m ∉ ℚ∩ (−1, 1). Then following statements hold.

(a)
System (1.1) has one and only one formal first integral in the sense of functional independence.
(b)
If (1.1) has one time-dependent formal first integral of the form Φ(x, y, z) exp (−λt), then
λ=m2(α1−α2)−(σ+1)2(α1+α2). (1.2)
(c)
If (1.1) has one time-dependent formal first integral of the form Φ(x, y, z) exp (−λt), and (σ + 1) / m ∉ ℚ, then there exists a positive integer k ∈ ℕ, such that λ = −k(σ + 1).

The outline of this paper is as follows. In Section 2, we present some preliminary results that will be used later. The proof of Theorem 1.1, 1.2 are given in Sections 3 and 4, respectively.

2. Preliminaries

Consider the following n-dimensional system

dxdt=F(x), x∈ℳ, t∈ℂ, (2.1)

where F(x) is an n-dimensional vector-valued analytic function, and ℳ is an n-dimensional complex analytic manifold.

Definition 2.1.

We say that the system (2.1) is completely integrable if it admits n − 1 functionally independent first integrals Φ₁, ⋯, Φ_n₋₁.

Definition 2.2.

We say that the system (2.1) is is B-integrable if it admits k functionally independent first integrals Φ₁, ⋯, Φ_k and (n − k) vector fields w₁ = F, ⋯, w_n _{− k} such that

[wi,wj]=0, and wj[Φi]=0, for 1≤i≤k, 1≤j≤n−k.

Assume system (2.1) is completely integrable. Then the orbits of this system are contained in the curves

{x∣Φ1(x)=c1,⋯,Φn−1(x)=cn−1, where c1,⋯,cn−1∈ℂ}.

Hence, its general solutions can be obtained analytically in an ”explicit” way. Furthermore, the topological structure of a completely integrable system is also simple. Llibre et al. [16] proved that a completely integrable system is orbitally equivalent to a linear differential system.

The notion of B-integrability was introduced by Bogoyavlensky [5]. It can be shown that if a system (2.1) is B -integrable, then it is integrable by quadrature. By definition, we can easily see that B-integrability is a weaker definition of complete integrability. Actually, B-integrability is also a natural generalization of the Liouvillian integrability from Hamiltonian systems to general dynamical systems, see [5] for details.

Let ψ(t) be a non-equilibrium solution of system (2.1). The variational equations along the phase curve Γ of ψ(t) read

dξdt=A(t)ξ, A(t)=∂F∂x|x=ψ(t), ξ∈TΓℳ, (2.2)

where T_Γℳ is the vector bundle of Tℳ restricted on Γ. Further, denoting by N = T_Γℳ / Γ the normal bundle, and using the nature projection π: T_Γℳ → N, the variational equations can be reduced to the normal variational equations

η˙=π*(T(F)(π−1η)), η∈N. (2.3)

Now we proceed to consider the differential Galois group of (2.3). Roughly speaking, the differential Galois group of (2.3) is a matrix group G ⊂ G L(n − 1, ℂ) acting on the fundamental solutions of (2.3) such that it dose not change polynomial and differential relations between them, for more details see [24]. We also recall the identity component G⁰ of G is the unique connected component containing the identity, which is a normal subgroup of G with finite index.

The main ingredients for the proofs of theorem 1.1 are the following theorems, which characterise the relation between the existence of meromorphic first integrals of (2.1) and the property of the identity component of the differential Galois group associated with (2.3), and give the necessary conditions for the complete integrability and B-integrability of (2.1), respectively.

Theorem 2.1.

[2] Assume that system (2.1) is B-integrable in the meromorphic category in a neighbourhood of a phase curve Γ. Then the identity component of the differential Galois group of the normal variational equations (2.3) along Γ is Abelian.

Theorem 2.2.

[18, 19] Assume that system (2.1) has m(1 ≤ m < n) functionally independent meromorphic first integrals in a neighborhood of Γ. Then the Lie algebra 𝒢 of the differential Galois group G of equations (2.2) has m meromorphic invariants, and the identity component G⁰ of G has at most (n − m − 1)(n − 1) generators, i.e.,

G0={(e𝔗1t1⋅e𝔗2t2⋯e𝔗ktk)s∣(t1,⋯,tk)∈𝒱˜⊂ℂk,s∈ℕ},

where {𝔍₁, ⋯, 𝔍_k} is a basis of 𝒢 with k ≤ (n − m − 1)(n − 1), 𝒱˜ is a neighborhood of the original element in ℂ^l. Especially,

(1)
If m = n − 1, i.e., system (2.1) is completely integrable, then 𝒢 = {0}, G⁰ = {1}, where 1 denote the identity element of G.
(2)
If m = n − 2, then 𝒢, G⁰ have at most n − 1 generators.
(3)
If n = 3 and m = 1, then 𝒢, G⁰ are solvable.

In general, for a n-th (n ≥ 2) order linear differential equations, it is very difficult to compute the corresponding differential Galois group in most cases. However, for n = 2, 3, Kovacic [11] and Singer [28] have proposed an algorithm to get the Galois group completely. In what follows, we list some results of second order differential equation by Kovacic [11].

Let us consider the following second order linear differential equation

d2ξdx2+a(x)dξdx+b(x)ξ=0, a(x),b(x)∈ℂ(x), (2.4)

where ℂ (x) stands for the field consisting of all rational functions on the complex plane ℂ. Making a well-known change of variable

ξ=χexp(−12∫x0xa(s)ds),

we can eliminate the term involving dξ / dx and obtain the reduced form of (2.4)

d2χdx2=r(x)χ, r(x)=a24+12dadx−b. (2.5)

The differential Galois group G of system (2.5), belonging to SL(2, ℂ), can be classified with the following four cases [11].

Theorem 2.3.

One and only one of the following four cases can occur.

Case 1. G is conjugate to a triangular group. In this case, (2.5) has a solution of the form e∫ω with ω ∈ ℂ(x).

Case 2. G is not of case 1, but is conjugate to a subgroup of

D={(c00c−1)∣c∈ℂ,c≠0}∪{(0c−c−10)∣c∈ℂ,c≠0}.

In this case, (2.5) has a solution of the form e∫ω with ω algebraic over ℂ(x) of degree 2 and case 1 does not hold.

Case 3. G is not of case 1 or case 2, but is a finite group. In this case, all solutions of (2.5) are algebraic over ℂ(x).

Case 4. G = SL(2, ℂ). Then (2.5) is not integrable in Liouville sense.

Recall that the order of the poles of r(x) = p(x) / q(x), where polynomials p(x) and q(x) are relatively prime, is the multiplicity of the zero of q(x), and the order of r at ∞ is deg q(x) − deg p(x). Kovacic [11] presented a complete algorithm to analysis which cases the differential Galois group G of system (2.5) falls into, see Appendix A for the Kovacic’s algorithm.

The next result [11] provides some necessary conditions for cases 1-3.

Theorem 2.4.

The following conditions are necessary for the respective types of Theorem 2.3 to hold.

Case 1. Every pole of r must have even order or else have order 1. The order of r at ∞ must be even or else be great than two.

Case 2. r must have at least one pole such that either has odd order greater than 2 or else has order two.

Case 3. The order of a pole of r cannot exceed 2 and the order of r at ∞ must be at least two. If the partial fraction expansion of r is

r=∑iαi(x−βi)2+∑iγjx−di

then

1+4αi∈ℚ, ∑jγj=0, ∑iαi+∑iγjdi∈ℚ.

It should be pointed out that the identity component G⁰ in case 1-3 are solvable, but in case 4 is not, see Lemma 2.2 in [24]. Form the proof of Theorem 2.4 by Kovacic, we also see that the necessary condition for case 3 is based on the assumption that G is a finite group and admits an algebraic solution, which has nothing to do with the assumption that case 1,2 do not hold. Hence, if the necessary condition for case 3 evaluates to false, then G is not a finite group.

3. The case of σβ ≠ 0

Suppose σ β ≠ 0, then (x, y, z) = (0, 0, e^−βt) is a non-equilibrium solution of system (1.1). Let Γ be the flow generated by this solution. Taking (x, y, z) = (ξ, η, ζ + e ^−βt) in (1.1) and neglecting quadratic terms of (ξ, η, ζ), we obtain the variational equations along Γ

(ξ˙η˙ζ˙)=(−σσ0r−e−βt−1000−β)(ξηζ). (3.1)

Then, the equations for variables (ξ, η) form a closed subsystem

(ξ˙η˙)=(−σσr−e−βt−1)(ξη), (3.2)

which is the so-called normal variational equations along Γ. A straightforward computation yields a second-order equation

ξ¨+(σ+1)ξ˙−σ(r−1−e−βt)ξ=0, (3.3)

which is equivalent to system (3.2). In order to transform (3.3) into an equation with rational coefficients, we apply a variable change τ = e^−βt and obtain

ξ″+β−σ−1βτξ′−σ(r−τ−1)β2τ2ξ=0, (3.4)

where the prime denotes the derivative with respect to τ. Further, under the transform ξ(τ)= χ(τ)τ1+σ−β2β , system (3.4) can be rewritten as

χ″=r(τ)χ, r(τ)=(σ+1)2−β2+4σ(r−1)−4στ4β2τ2. (3.5)

The following lemma plays a key role in our first result.

Lemma 3.1.

Suppose σβ ≠ 0. Then the following statements hold.

(1)
The differential Galois group G of (3.5) is not finite.
(2)
The identity component G⁰ of G is solvable if and only if
2(σ+1)2+4σ(r−1)|β|
is an odd number.

Proof. (1) Obviously, r(τ) has two singular points 0 and ∞ with order 2 and 1, respectively. In view of Theorem 2.4, the necessary conditions for case 1,3 cannot hold. Therefore, the differential Galois group G of (3.5) is not finite.

(2) Note that in case 1-3 the identity component G⁰ of G are solvable, but in case 4 is not. Based on the discussion above, we see that G⁰ is solvable if and only if it belongs to case 2 in Theorem 2.3. In what follows, we apply the Kovacic’s algorithm to (3.5) step by step to get the necessary and sufficient condition such that case 2 holds. We only consider the case β > 0, the case β < 0 can be dealt similarly.

Step 1 Some simple calculations lead to

E0={2,2±2(σ+1)2+4σ(r−1)β}∩ℤ, E∞={1}.

Step 2 If 2(σ+1)2+4σ(r−1)/β is not an odd number, then d≜12(e∞−e0) is not a non-negative integer for any e₀ ∈ E₀ and e_∞ ∈ E_∞, which means case 2 cannot happen. If 2(σ+1)2+4σ(r−1)/β is an odd number, then there exists only one option

e0=2−2(σ+1)2+4σ(r−1)β, e∞=1,

such that d=12(e∞−e0)∈ℕ∪{0} . Hence, we obtain θ=1−2d2τ and progress to the next step.

Step 3 Let P(τ)=∑k=0dakτk , where

ad=1, ak−1=−f(k,d)g(k,d)ak, k=d,d−1,…,1,f(k,d)=k(k−1)(k−2)+3(1−2d)2k(k−1)+d(2d−1)k, g(k,d)=4σβ2(k−1−d),

then polynomial P(τ) satisfies

P‴+3θP″+(3θ2+3θ′−4r)P′+(θ″+3θθ′+θ3−4rθ−2r′)P=0,

i.e.,

P‴+3(1−2d)2τP″+(d(2d−1)τ2+4σβ2τ)P′−4dσβ2τ2P=0.

It follows that case 2 holds.

Summarizing the above discussion, when β > 0, we see that the differential Galois group of (3.5) belongs to case 2 if and only if 2(σ+1)2+4σ(r−1)/β is not an odd number. The proof is completed.

Remark 3.1.

One can also get the proof of Lemma 3.1 by observing that (3.3) can be transformed into the Whittaker equation for which their differential Galois group is studied, see the reference [24] for details.

Proof of Theorem 1.1.

Let us observe that the differential Galois group of (3.2) can be viewed as a normal subgroup of the differential Galois group of (3.5), see Theorem 2.5 in [24]. So, by Lemma 3.1, the differential Galois group of (3.2) is also not finite. In addition, the identity component G⁰ is a normal subgroup of G with finite index, so G⁰ is the trivial subgroup {1} if and only if G is a finite group [24]. Therefore, the identity component G⁰ of (3.2) is not {1} and the Lorenz system (1.1) is not completely integrable with meromorphic first integrals by Theorem 2.2.

Similarly, if

2(σ+1)2+4σ(r−1)|β|

is not an odd number, according to Lemma 3.1, the identity component of differential Galois group of (3.5) as well as (3.2) is not solvable. Note that an Abelian group is a solvable group. Therefore, the identity component of differential Galois group of (3.2) is also not Abelian. In view of Theorem 2.1 and 2.2, we can trivially complete Theorem 1.1.

Remark 3.2.

Let β = 8/3, σ = 10 and r = 28, then it is not difficult to calculate that

2(σ+1)2+4σ(r−1)|β|=312018 (3.6)

is not an odd number, by Theorem 1.1, we conclude that system (1.1) admits no meromorphic first integrals. On the other hand, it is well known that the Lorenz system (1.1) has a strange attractor [25] under the above conditions. This fact can be seen as a new evidence of the connection between the chaos and the non-integrability.

Remark 3.3.

If σβ ≠ 0 and 2(σ+1)2+4σ(r−1)/β is an odd number, then system (1.1) has at most one meromorphic first integral in sense of functional independence. Specifically, let β = 2σ = 1, r = 0, one can find a meromorphic first integral

Φ(x)=2(x2−z)2y2−1+2x2z,

which means system (1.1) has one and only one meromorphic first integral Φ(x) in sense of functional independence.

4. The case of σ ≠ 0, β = 0

Assume σ ≠ 0 and β = 0. We make the change of variables

(xyz)=(1σm0σ+λ1σσ+λ2m0001)(x1x2x3),

and transform system (1.1) to

{x˙1=λ1x1+ax1x3+bx2x3,x˙2=λ2x2+cx1x3+dx2x3,x˙3=ex12+fx1x2+gx22, (4.1)

where

λ1=−(σ+1)−m2,λ2=−(σ+1)+m2,m=(σ−1)2+4σr,a=σm,b=σ2m2,c=−1,d=−σm,e=σ+λ1σ,f=σ−1m,g=(σ+λ2)σm2.

Lemma 4.1.

Assume that β = 0, m ≠ 0 and (σ + 1) / m ∉ ℚ∩ (−1, 1). Then system (1.1) has at most one formal first integral.

Proof. For the singularity (0, 0, 0), system (1.1) has the eigenvalues λ₁, λ₂, λ₃ = 0. By Theorem 1 in [30], we know the number of functionally independent formal first integrals of system (1.1) is no more than the number of linearly independent elements of the set

𝒢={(k1,k2,k3)∈(ℤ+)3:k1λ1+k2λ2+k3λ3=0,k1+k2+k3≥1}.

Clearly, as m ≠ 0 and (σ + 1) / m ∉ ℚ∩ (−1, 1), the elements in 𝒢 have the form (k₁, k₂, k₃) = (0, 0, m) with m ∈ ℤ ⁺. It follows that system (1.1) has at most one formal first integral in the sense of functionally independence.

Now let us introduce two linear partial differential operators

ℒ=λ1x1∂∂x1+λ2x2∂∂x2,ℒλ=λ1x1∂∂x1+λ2x2∂∂x2−λ,ℋ=−(ax1x3+bx2x3)∂∂x1−(cx1x3+dx2x3)∂∂x2−(ex12+fx1x2+gx22)∂∂x3,

then one can regard ℒ and ℒ_λ as linear maps from H_k _{+ 1} into H_k _{+ 1}, and ℋ as a linear map from H_k into H_k _{+ 1}, where H_k ⊂ ℂ [x₁, x₂, x₃] denote the linear space formed by the homogeneous polynomials of degree k.

Lemma 4.2.

Assume that β = 0, m ≠ 0 and (σ + 1) / m ∉ ℚ∩ (−1, 1). Then Img ℋ ⊆ Img ℒ.

Proof. Let

𝒮={P∣P=∑|α|=k+1Ck+1,αx1α1x2α2x3α3, α1+α2>0},

where α=(α1,α2,α3)∈(ℤ+)3,|α|=α1+α2+α3,Ck+1,α∈ℂ,l∈ℕ/{0} . Then 𝒮 is a subset of H_k _{+ 1}, and Img ℋ ⊆ 𝒮. On the other hand, according to the assumption of as m ≠ 0 and (σ + 1) / m ∉ ℚ∩ (−1, 1), it is easy to check that

λ1α1+λ2α2≠0

for any α₁, α₂ ∈ ℤ ⁺ with α₁ + α₂ > 0. Now for any element of 𝒮

P=∑|α|=k+1Ck+1,αx1α1x2α2x3α3,

set

P˜=∑|α|=k+1C˜k+1,αx1α1x2α2x3α3,

where C˜k+1,α=0 if α₁ + α₂ = 0 and C˜k+1,α=Ck+1,α/(λ1α1+λ2α2) if α₁ + α₂ > 0, then P˜∈Hk+1 and ℒ[P˜]=P , which means 𝒮 ⊆ Img ℒ. Therefore, Img ℋ ⊆ 𝒮 ⊆ Img ℒ.

Proof of Theorem 1.2. (a)

Note that if (1.1) admits a formal first integral Φ˜(x,y,z) , then we can rewrite Φ˜=Φ(x1,x2,x3) in the variables (x₁, x₂, x₃) and Φ must be a formal first integral of (4.1) and vice versa. Hence, we need only to study the existence and uniqueness of formal first integrals for system (4.1).

Let Φ = Φ_l + Φ_l _{+ 1} + ⋯ with

Φk=∑|α|=kCk,αx1α1x2α2x3α3, k=l,l+1,⋯,

and Φ_l ≢ 0.

By the definition of first integrals, Φ is a first integral of (4.1) if and only if

(λ1x1+ax1x3+bx2x3)∂Φ∂x1+(λ2x1+cx1x3+dx2x3)∂Φ∂x2+(ex12+fx1x2+gx22)∂Φ∂x3=0,

i.e.,

ℒ[Φ]=ℋ[Φ],

or equivalently

ℒ[Φl]=0, (4.2)

ℒ[Φj+1]=ℋ[Φj], j=l,l+1,⋯. (4.3)

The next thing to do is to show that the equations (4.2)–(4.3) are formally solvable. In fact, (4.2) has a solution Φl*=x3l . By Lemma 4.2, ℋ[Φl*]∈Imgℋ⊆Imgℒ , so we can get Φl+1*∈Hl+1 such that ℒ[Φl+1*]=ℋ[Φl*] . By induction, we could get Φj* , j=l,l+1,⋯ , such that (4.3) hold. Let Φ*=Φl*+Φl+1*⋯ , then Φ* is a formal first integral of (4.1). Moreover, by Lemma 4.1, the proof is completed.

(b) Assume that (1.1) has a time-dependent formal first integral of the form Φ˜(x,y,z)e−λt , then (4.1) has a corresponding formal time-dependent formal first integral of the form Φ(x1,x2,x3)e−λt , i.e,

ℒλ[Φ]=ℋ[Φ]. (4.4)

Expanding Φ(x₁, x₂, x₃) as

Φ=Φl+Φl+1+⋯,

with Φk=∑|α|=kCk,αx1α1x2α2x3α3,k=l,l+1,⋯ , and Φ_l ≢ 0.

Comparing the lowest order terms of (4.4) yields

ℒλ[Φl]=0

or

∑|α|=lCl,α(λ1α1+λ2α2−λ)x1α1x2α2x3α3=0.

Hence, λ = λ₁α₁ + λ₂α₂ has to be fulfilled for any nonzero coefficient C_l_,_α, namely,

λ=(σ−1)2+4σr2(α1−α2)−(σ+1)2(α1+α2). (4.5)

This relation implies (1.2).

(c) Repeating the same arguments, we get (4.5) and claim that (α1,α2)∈(ℤ+)2 satisfying λ = λ₁α₁ + λ₂α₂ is unique. In deed, suppose there exists (α˜1,α˜2)∈(ℤ+)2 such that (α˜1,α˜2)≠(α1,α2) and λ=λ1α˜1+λ2α˜2 . Then, we have

λ1(α˜1−α1)+λ2(α˜2−α2)=0,

or

σ+1(σ−1)2+4σr=α˜2−α2−α˜1+α1α˜2−α2+α˜1−α1∈ℚ,

which contradicts assumptions. Without loss of generality, we set Φl=x1α1x2α2x3l−α1−α1 .

Next, identifying the homogenous component of degree l + 1 of (4.4) yields

ℒλ[Φl+1]=ℋ[Φl]. (4.6)

With some calculations, we obtain

ℋ[Φl]=−(aα1+dα2)x1α1x2α2x3l−α1−α1+1 −bα1x1α1−1x2α2+1x3l−α1−α1+1 −cα2x1α1+1x2α2−1x3l−α1−α1+1 −eα3x1α1+2x2α2x3l−α1−α1−1 −fα3x1α1+1x2α2+1x3l−α1−α1−1 −gα3x1α1x2α2+2x3l−α1−α1−1.

Using the same argument as in the proof of statement (a), the left side of (4.6) has no term x1α1x2α2x3l−α1−α1+1 and so

aα1+dα2=σm(α1−α2)=0. (4.7)

Taken together with (4.5) and (4.7), we have α₁ = α₂ ≔ α and

−λσ+1=α∈ℕ.

Remark 4.1.

Consider the case of σ=13 , β = 0, r = 2, it is easy to verify the conditions of Theorem 1.2. In this case, we can set λ=−43 . Then, the Lorenz system may have a formal integrals of motion of the form Φ(x,y,z)exp(43t) . In fact, system (1.1) admits the time-dependent first integral

I=(−x2+13y2+23xy+x2z−34x4)e43t.

Remark 4.2.

The statement (a) in Theorem 1.2 can also be derived from Theorem 1 in [20]. However, the criterion we present is more elementary and crucial to the proof of the statement (b, c) in Theorem 1.2. Moreover, we can adopt the same procedure as in the proof of Theorem 1.2 to other nonlinear differential systems which have a fixed point with some single multiply zero eigenvalues and other ℕ -independent nonzero eigenvalues. Then we can obtain similar results about the existence of formal first integrals of them.

Remark 4.3.

In statement (b), we show that if −λ /(σ + 1) ∈ ℕ, then there is no term x1αx2αx3l+1−2α in ℋ[Φ_l]. Furthermore, some tedious calculations lead to the coefficient of x1α1x2α2x3l+2−α1−α1 in ℋ[Φ_l _{+ 1}]

bcα−λ1+λ2+bcαλ1−λ2=0,

the coefficient of x1α1x2α2x3l+3−α1−α1 in ℋ[Φ_l _{+ 2}]

bcα(α+1)(−λ1+λ2)2(a(α−1)+d(α+1))+bcα(α+1)(λ1−λ2)2(a(α+1)+d(α−1))=0,

the coefficient of x1α1x2α2x3l+4−α1−α1 in ℋ[Φ_l _{+ 3}]

bcα(α+1)(−λ1+λ2)3(a(α−1)+d(α+1))2+bcα(α+1)(λ1−λ2)3(a(α+1)+d(α−1))2=0.

This leads us to conjecture that there is no term x1α1x2α2x3k−α1−α1 in ℋ[Φ_k], k = l, l + 1, ⋯, which means (4.6) are formal solved and (4.1) has a time-dependent formal first integral.

Acknowledgments

This work is supported by NSFC grant (No. 11771177, 11301210), China Automobile Industry Innovation and Development Joint Fund (No. U1664257), Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 2017TD-20), NSF grant (No. 20140520053JH) and ESF grant (No. JJKH20170776KJ) of Jilin, China.

Appendix A. Kovacic’s Algorithm

Let r be the complex rational function that defines the second order linear differential equation ξ‴ = rξ. Let 𝒫 be the set containing ∞ and the poles of r.

A1. Kovacic’s Algorithm of Case 1

Step 1. For each c ∈ 𝒫 we define [r]c, αc± as follows:

(a)
If c is a pole of order 1, then [r]c=0, αc+=αc−=1 .
(b)
If c is a pole of order 2, then [r]c=0, αc±=12±121+4b , where b is coefficients of 1/(x − c)² in the partial fraction expansion for r.
(c)
If c is a pole of order 2 v ≥ 4, then [r]c=a(x−c)v+⋯+d(x−c)2 of negative order part of the Laurent series expansion of r at c, αc±=12(±ba+v) , where b is the coefficient of 1/(x−c)^v ^{+ 1} in r minus the coefficient of 1/(x−c)^v ^{+ 1} in [r]c2 .
(d)
If the order of r at ∞ is >2, then [r]∞=0, α∞+=0, α∞−=1 .
(e)
If the order of r at ∞ is 2, then [r]∞=0, α∞±=12±121+4b , where b is coefficients of 1/x² in the Laurent series expansion of r at ∞.
(f)
If ∞ is a pole of order −2v ≤ 0, then [r]∞=axv+⋯+d of positive order part of the Laurent series expansion of r at ∞, α∞±=12(±ba−v) , where b is the coefficient of x^v^{− 1} in r minus the coefficient of x^v⁻¹ in [r]∞2 .

Step 2. Let d=∑c∈𝒫t(c)αcs(c) , where s(c) ∈{+, −} for any c ∈ 𝒫, t(∞) = +1 and t(c) = −1 for any c ∈ 𝒫 \ {∞}. If d is a non−negative integer, then let w=∑c∈𝒫s(c)[rc+αcs(c)x−c)+s(∞)[∞] , otherwise, the family is discarded. If no families remain under consideration, case 1 of Theorem 5. cannot happen.

Step 3. For each family retained from step 2, we search for a monic polynomial P of degree d such that the equation P″ + 2 w P′ + (w′ + w²−r) P = 0 holds. If such a polynomial exists, then ξ=Pe∫w is a solution of ξ″ = rξ. If no such polynomial is found for any family retained from Step 2, case 1 of Theorem 2.3 cannot happen.

A2. Kovacic’s Algorithm of Case 2

Step 1. For each c ∈ 𝒫 we define E_c as follows:

(a)
If c is a pole of order 1, then E_c = {4}.
(b)
If c is a pole of order 2, then Ec={2+k1+4b,k=0,±2}∩ℤ , where b is coefficients of 1/(x − c)² in the partial fraction expansion for r.
(c)
If c is a pole of order v > 2, then E_c = {v}.
(d)
If the order of r at ∞ is > 2, then E_∞ = {0, 2, 4}.
(e)
If the order of r at ∞ is 2, then E∞={2+k1+4b,k=0,±2}∩ℤ , where b is coefficients of 1/x² in the Laurent series expansion of r at ∞.
(f)
If ∞ is a pole of order v < 2, then E_∞ = {v}.

Step 2. Let d=12∑c∈∈𝒫t(c)ec , where e_c ∈ E_c for any c ∈ 𝒫, t(∞) = + 1 and t(c) = −1 for any c ∈ 𝒫 \ {∞}. If d is a non-negative integer, then let θ=12∑c∈𝒫ecx−c , otherwise, the family is discarded. If no families remain under consideration, case 2 of Theorem 5. cannot happen.

Step 3. For each family retained from step 2, we search for a monic polynomial P of degree d such that the equation P‴ + 3θP″ + (3θ² + 3θ′ −4r)P′ + (θ″ + 3θθ′ + θ³−4rθ−2r′) P = 0 holds. If such a polynomial exists, let ϕ=θ+P′P and let w be a solution of the equation w2+ϕw+(12ϕ′+12ϕ2−r)=0 , then ξ=e∫w is a solution of ξ″ = rξ. If no such polynomial is found for any family retained from Step 2, case 2 of Theorem 2.3 cannot happen.

A3. Kovacic’s Algorithm of Case 3

Step 1. For each c ∈ 𝒫 we define E_c as follows:

(a)
If c is a pole of order 1, then E_c = {12}.
(b)
If c is a pole of order 2, then Ec={6+12kn1+4b,k=0,±1,±2,⋯,±n2}∩ℤ , where b is coefficients of 1/(x − c)² in the partial fraction expansion for r.
(c)
E∞={6+12kn1+4b,k=0,±1,±2,⋯,±n2}∩ℤ , where b is coefficients of 1/x² in the Laurent series expansion of r at ∞.

Step 2. Let d=n12∑c∈𝒫t(c)ec , where e_c ∈ E_c for any c ∈ 𝒫, t(∞) = + 1 and t(c) = − 1 for any c ∈ 𝒫 \ {∞}. If d is a non-negative integer, then let θ=n12∑c∈𝒫ecx−c,S=∏c∈𝒫\{∞}(x−c) , otherwise, the family is discarded. If no families remain under consideration, case 3 of Theorem 5. cannot happen.

Step 3. For each family retained from step 2, we search for a monic polynomial P of degree d such that the recursive equations

Pn=P,Pi−1=−SPi+((n−i)S′−Sθ)Pi−(n−i)(i+1)S2rPi+1;(i=n,n−1,⋯,0),

with P _{− 1} ≡ 0 holds. If such a polynomial exists, let w be a solution of the equation ∑i=0nSiPi(n−i)!wi=0 , then ξ=e∫w is a solution of ξ″ = rξ. If no such polynomial is found for any family retained from Step 2, case 3 of Theorem 2.3 cannot happen.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 25 - 1
Pages: 106 - 121
Publication Date: 2021/01/06
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TY  - JOUR
AU  - Kaiyin Huang
AU  - Shaoyun Shi
AU  - Wenlei Li
PY  - 2021
DA  - 2021/01/06
TI  - Meromorphic and formal first integrals for the Lorenz system
JO  - Journal of Nonlinear Mathematical Physics
SP  - 106
EP  - 121
VL  - 25
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1440745
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ID  - Huang2021
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