Qual. Theory Dyn. Syst. https://doi.org/10.1007/s12346-018-0273-4

Qualitative Theory of Dynamical Systems

Normal Forms of Planar Polynomial Differential Systems Abderrahmane Turqui1

· Dahira Dali1

Received: 19 April 2017 / Accepted: 5 February 2018 © Springer International Publishing AG, part of Springer Nature 2018

Abstract Using the invariant theory, we develop an algorithmic method, which is based on the construction of a matrix of linear transformation, to compute normal forms of planar polynomial differential systems. We illustrate our method in the case where the planar polynomial differential system is cubic. Keywords Polynomial differential system · Invariant · Linear transformation · Normal form Mathematics Subject Classification 34C20 · 14L36 · 34G14 · 15A72

1 Motivations The invariant theory [9,11,12,18] is a powerful tool in the study of the qualitative theory of differential equations. It allows us to characterize geometric properties of a given polynomial differential system by means of algebraic and semi-algebraic relations between its coefficients. In the case of the planar quadratic and cubic differential systems, many difficult problems are solved, such as those related to the number and nature of singular points, topological and geometric classification (see e.g. [14,17,19– 21]).

B

Abderrahmane Turqui [email protected] Dahira Dali [email protected]

1

Faculty of Mathematics, University of Sciences and Technology Houari Boumediene, El Alia, BP 32, 16111 Bab Ezzouar, Algiers, Algeria

A. Turqui, D. Dali

The invariant theory has revolutionized the qualitative study of polynomial differential systems in the early 1960s, through Sibirskii, when he assimilated their coefficients to tensor components [21]. However, the computation of the invariants is complicated because one has to manipulate polynomials of several indeterminates of higher degrees and to solve algebraic systems when the algebra of the invariants is of finite type. Numerous works devoted to solving algebraic systems [5,10,13,16] and many works are devoted to the calculation and decomposition of algebraic invariants [2,6,7,22]. The algebraic invariants are particularly interesting for classifying normal forms, trajectories of polynomial differential systems, etc. Sibirskii gave normal forms of homogeneous planar quadratic systems with respect to the general linear group with the help of Aronhold identities [21] . Date in his paper [8] gave another classification of the same systems, using a cubic form and later [1] gave a new classification with a smaller number of normal forms. In [3,4,15] the authors constructed normal forms of planar polynomial differential systems with invariant coefficients with respect to the linear group G L(2, R). In this paper, we consider a polynomial differential systems of finite degrees with coefficients in a field of characteristic zero. These systems are defined on finite dimensional phase spaces and allowed the action of a linear group. The main result is the construction of a matrix of linear transformation, that transforms any coefficient of a polynomial differential system into an invariant coefficient with respect to the general linear group. Using this matrix, we will develop an algorithmic method to construct normal forms of given polynomial differential systems. In the last section, we give an application in the case of planar cubic differential systems.

2 Preliminaries We consider the polynomial differential systems of degree at most k with coefficients in a field k of characteristic zero: n k  n   dx j = aj + ··· aαj 1 α2 ···αr x α1 x α2 · · · x αr , 1 ≤ j ≤ n, dt r =1 α1 =1

(1)

αr =1

j

where for j = 1, n and for r = 1, k, a j ∈ T01 and aα1 α2 ...αr ∈ Tr1 , where for r = 0, k, Tr1 denotes the space of tensors 1 time contravariant and r times covariant symmetric with respect to the lower subscripts. Tr1 corresponds to the homogeneous part of degree r of the polynomials of the right hand side of system (1). Writing these differential using tensors is due to Sibirskii [21]. Using Einstein’s notation n systems υi δ i the complete polynomial differential system (1) can be written as υi δ i = i=1  dx j = aj + aαj 1 α2 ...αi x α1 x α2 · · · x αi , j, α1 , · · · , αk ∈ {1, 2, . . . , n} . dt k

i=1

(2)

Normal Forms of Planar Polynomial Differential Systems

Let C(n, k, k) be the set of all coefficients on the right hand side of system (2), and x = (x 1 , x 2 , . . . , x n ) the vector of the unknown variables of (2). C(n, k, k) can be identified as the direct sum T01 ⊕ T11 ⊕ · · · ⊕ Tr1 , 0 ≤ r ≤ k. Let G be a linear group of transformations of the phase space kn . We denote by S(A) the differential system (2) where the set A of its corresponding coefficients is a subset of C(n, k, k). That is, if P(C(n, k, k)) is the power set of C(n, k, k), then the set of all differential systems of the form (2) can be written as S (C(n, k, k)) = {S(A), A ∈ P (C(n, k, k)) , A = ∅} . A polynomial function C(a, x) : C (n, k, k) × kn → k is a covariant with respect to the group G, or G-covariant of S (C(n, k, k)) (or C (n, k, k)) if there exists a character  of the linear group G, such that ∀g ∈ G, ∀a ∈ C(n, k, k), C(ρ(g)a, ρ(g)x) = (g)C(a, x), where ρ is a representation of the considered group. If  ≡ 1, the covariant is said to be absolute, otherwise it is said to be relative. If the C(a, x) is constant with respect to x, then it is said to be a G-invariant. We will simply say G-covaraint in such a case. In the case of the general linear group G L(n, k) it is known [21] that the character  (Q) has the form  (Q) = |Q|−w where w is an integer, which is called the weight of the covariant C(a, x). A G-covariant C(a, x) is said to be reducible if it can be expressed as a polynomial function of G-covariants of lower degree. If C(a, x) is reducible, we write C(a, x) ≡ 0 (modulo G). A finite family B of G-covariants of S(C(n, k, k)) is called a system of generators if any G-covariant of S(C(n, k, k)) is reducible to zero modulo B. A system B of generators is said to be minimal if none of them is generated by the others. The action of the general linear group G L(n, k) on kn : (Q, x) → Qx, induces the representation ρ:

G L(n, k) Q

→ →

G L(C(n, k, k)) ρ(Q),

defined by = ρ(Q)a j j ρ(Q)aα1 α2 ...αr =

j

Qi ai , j β β β Q i Pα11 Pα22 . . . Pαrr aβi 1 β2 ...βr ,

(3)

where j, α1 α2 . . . αr = 1, n, r = 1, k, and Q is a matrix of G L(n, k) and P its inverse. The G L(n, k)-covariants of S(C(n, k, k)) are called centro-affine covariants of S(C(n, k, k)). The formula (3) is called the formula of the centro-affine transformations. One can check with the help of the definition of G L (2, R)-invariant that the following polynomial expressions in coefficients of C(2, 2, R),

A. Turqui, D. Dali

  C1 = tr a ij

i, j=1,2



=

α=1,2

C2 = 2 det(a ij )i, j=1,2 =

aαα ,



p q

ar as ε pq εr s ,

r,s, p,q=1,2

are centro-affine invariants of S(C(2, 1, R)), and C3 =



α β α,β=1,2 aαβ x ,



C4 =

r,s=1,2 a

r xsε , rs

are centro-affine covariants of S(C(2, 2, R)) where ε pq = q − p, εr s = s − r. Let E be a k-vector space of dimension n, and let p and q be two non negative integers. A contraction over the tensor space E ⊗ p ⊗ E ∗⊗q is the map: ϕ : E ⊗ p ⊗ E ∗⊗q → E ⊗ p−1 ⊗ E ∗⊗q−1 , defined by

n    i 1 ...i p i ...i αi m+1 ...i p ξ j11 ... jm−1 . ϕ ξ j1 ... jq = l−1α jl+1 ... jq α=1

If p = q then a sequence of p contractions is called a complete contraction. A covariant alternation over the tensor space E ⊗ p ⊗ E ∗⊗q , where p ≥ n and q ≥ n, is the map: φ : E ⊗ p ⊗ E ∗⊗q → E ⊗ p ⊗ E ∗⊗q−n , defined by n  n n     i ...i i ...i φ ξ j11 ... jpq = ... ξ j11 ...αp k1 =1 k2 =1

kn =1

k1 ... αk2 ...αkn ... jq

εαk1 αk2 ...αkn ,

and a contravariant alternation over the tensor space E ⊗ p ⊗ E ∗⊗q where p ≥ n and q ≥ n, is the map: ψ : E ⊗ p ⊗ E ∗⊗q → E ⊗ p−n ⊗ E ∗⊗q , defined by n  n n     i 1 ...αk ... αk2 ...αkn ...i p i 1 ...i p ψ ξ j1 ... jq = ... ξ j1 ... jq 1 εαk1 αk2 ...αkn , k1 =1 k2 =1

kn =1

where the tensor εαk1 αk2 ...αkn (εαk1 αk2 ...αkn ) with αk1 , αk2 . . . , αkn = 1, 2, . . . , n, is an n-vector, the valence of which coincides with the dimension of the space and the coordinates of which are equal to ⎧ if (αk1 , . . . , αkn ) is an even permutation, ⎨1 εαk1 αk2 ...αkn = εαk1 αk2 ...αkn = −1 if (αk1 , . . . , αkn ) is an odd permutation, ⎩ 0 otherwise.

Normal Forms of Planar Polynomial Differential Systems

Consider the tensor product: (T01 )⊗d0 ⊗ (T11 )⊗d1 ⊗ · · · ⊗ (Ts1 )⊗dr ⊗ k⊗δ , 1 ≤ r ≤ k.

(4)

A tensor T is said to be a tensor of type (d0 , d1 , . . ., dr , δ) associated with S(C(n, k, k)) if T ∈ (T01 )⊗d0 ⊗ (T11 )⊗d1 ⊗ · · · ⊗ (Ts1 )⊗dr ⊗ k⊗δ , 0 ≤ r ≤ k, d0 , d1 , . . ., dr , δ ∈ N. We will then write T (d0 , d1 , . . ., dr , δ). If the coefficients of the tensor T are in a subset A of C(n, k, k), then the tensor T (d0 , d1 , . . ., dr , δ) is said to be a tensor associated with S(A). Two fundamental theorems about centro-affine covariants of S(C(n, k, k)) are known. Theorem 2.1 (Gurevich [11]) The expressions obtained with the help of successive alternations and complete contraction over the tensor products (4) form a system of generators of centro-affine covariants of S(C(n, k, k)). We denote by A(n, k, k) the k−algebra of the centro-affine covariants of S(C(n, k, k)). A centro-affine covariant of S(C(n, k, k)) is said to be of type (d0 , d1 , . . ., dr , δ) if it j is homogeneous of degree di (i = 0, 1, . . . , r ) with respect to coordinates of aα1 α2 ···αi n and of degree δ with respect to coordinates of x ∈ k . It is clear that a centro-affine covariant C of the system of generators of centroaffine covariants of S(C(n, k, k)) obtained with the help of successive alternations and complete contraction from a tensor T (d0 , d1 , . . ., dr , δ) associated with S(C(n, k, k)) is a centro-affine covariant of type (d0 , d1 , . . ., dr , δ). Theorem 2.2 ([12]) The k-algebra A(n, k, k) is of finite type. This theorem is called the Hilbert basis theorem. In the rest of this paper, let F = { f 1 , . . . , f s } be a minimal system of generators of centro-affine covariants of S(C(n, k, k)).

3 Monoid of Centro-Affine Covariants Let a ji , i = 1, n, be the coefficients of the tensor a j (once contravariant and 0 time j j covariant), and aαli1 ···αir , l = 1, n, and r = 1, k, the coefficients of the tensor aα1 ···αr , (once contravariant and r time covariants) where ji , jl , αi1 , . . . ,αir ∈ {1, · · · , n} . For p0 = ( p10 , . . . , p 0j0 ) ∈ N j0 , we use (a j ) p0 to denote the product 0

p 0j

1

p1

(t10 ) p1 · · ·(t 0j0 ) (t11 ) p1 · · ·(t 1j1 )

0

j1

2

j

; for p1 = ( p11 , . . . , p 1j1 ) ∈ N j1 we use (aα1 ) p1 to denote the product j

; and for p2 = ( p12 , . . . , p 2j2 ) ∈ N j2 , we use (aα1 α2 ) p2 to denote the

product (t12 ) p1 · · ·(t 2j2 )

p 2j 2

, etc. For pr = ( p1r , . . . , prjr ) ∈ N jr , where 1 ≤ r ≤ k, we r r j use (aα1 ···αr ) pr to denote the product (t1r ) p1 · · ·(t rjr ) p jr , and for α = (δ1 , . . . , δn ) ∈ Nn , δ δn we use x α to denote the product (x 1 ) 1 · · ·(x n ) .

A. Turqui, D. Dali

A monomial associated with S (C(n, k, k)) is a finite product of the form (a j )d0 (aαj 1 )d1 (aαj 1 α2 )d2 · · ·(aαj 1 ···αr )dr (x)α , 1 ≤ r ≤ k. In general, a monomial is not a centro-affine covariant of S (C(n, k, k)). 0 0 j One writes (a j )d0 to denote the product (a 1 )d1 · · · (a n )dn , (aα1 )d1 to denote the 1 1 1 j d product (a11 )d11 (a21 )d12 · · · (ann ) nn …and (aα1 ···αr )dr to denote the product r

r

r

1 1 n (a1...1 )d1...1 (a1...12 )d1...12 · · · (an...n )dn...n , 1 , . . . , d 1 ), . . . , (d r , . . . , d r where (d10 , . . . , dn0 ), (d11 nn n...n ), 1 ≤ r ≤ k, and 1...1 (δ1 , . . . , δn ) are respectively the partitions of the non negative integers d0 , d1 , d2 , j . . . , dr and δ, and jr is the number of coefficients of the tensor aα1 ···αr , (1 ≤ r ≤ k). The monomial

(a j )d0 (aαj 1 )d1 (aαj 1 α2 )d2 · · ·(aαj 1 ···αr )dr (x)α , 1 ≤ r ≤ k, will be called a monomial of type (d0 , d1 , d2 , . . . , dr , δ). For example, among the monomials a ij11 a ij22 : (a11 )2 , a11 a21 , a11 a22 , (a21 )2 , a21 a12 , a21 a22 , (a12 )2 , a12 a22 , (a22 )2 , monomials of type (0, 2, 0) are (a11 )2 , a11 a22 , a21 a12 and (a22 )2 . Let us order the tensors coefficients a j , aαj 1 , aαj 1 α2 , . . . , aαj 1 ···αr , (1 ≤ r ≤ k), of C(n, k, k) and the components x 1 , . . . , x n of the contravariant vector x in the following manner: a j ≺ ai 1 ...is ≺ x i for all i, j, ∈ {1, . . . , n}; a j ≺ a if j < for all i, j,

∈ {1, . . . , n}; and a j1 ... js ≺ a

1 ... s if s1 < s2 or (s1 = s2 and the first non null com1 2 ponent of the vector ( j, j1 , . . . , js1 )−( , 1 , . . . , s2 ) = ( j − , j1 − 1 , . . . , js1 − s2 ) is negative). For example, for the planar cubic differential system S(C(2, 3, k)), we have j

1 1 2 2 ≺ a112 ≺ a122 ≺ a211 ≺ x 1 ≺ x 2. a 1 ≺ a 2 ≺ a11 ≺ a21 ≺ a12 ≺ a22 ≺ a11

(5)

The set of all monomials will be denoted by M, while the set of all monomials of type (d0 , d1 , d2 , . . . , dr , δ) will be denoted by M(d0 ,d1 ,d2 ,...,dr ,δ) . Since the number of partitions of the non negative integers d0 , d1 , d2 , . . . , dr , (1 ≤ r ≤ k), and δ are finite, M(d0 ,d1 ,d2 ,...,dr ,δ) is a finite set and hence can be written as m 1 , m 2 , . . . , m n 0 , where n 0 is the number of elements of M(d0 ,d1 ,d2 ,...,dr ,δ) . For example,    2 2 . M(0,2,0) = a11 , a11 a22 , a21 a12 , a22

Normal Forms of Planar Polynomial Differential Systems

If we define (a j ) p0 (aαj 1 ) p1 (aαj 1 α2 ) p2 · · ·(aαj 1 ···αr ) pr (x)δ × (a j )q0 (aαj 1 )q1 (aαj 1 α2 )q2 · · ·(aαj 1 ···αr )qr (x)μ = (a j )

p0 +q0

(aαj 1 )

p1 +q1

(aαj 1 α2 )

p2 +q2

· · ·(aαj 1 ···αr )

pr +qr

(x)δ+μ

for i = 1, . . . , r and pi , qi ∈ N jr (1 ≤ r ≤ k), then M is a monoid with the identity 1. The total ordering defined by (5) can be extended to a total lexicographic ordering for the set M. Recall that a monomial order for a monoid is a binary relation that is (i) total, (ii) compatible with the product, and (iii) well ordered (so that any nonempty subset of the monoid has a smallest element). By treating the tensors coefficients as alphabets, the total ordering defined by (5) can be extended to a total lexicographic ordering for the set M in the usual manner (see e.g. [16]): (a j ) p0 (aαj 1 ) p1 (aαj 1 α2 ) p2 · · ·(aαj 1 ···αr ) pr (x)δ ≺ (a j )q0 (aαj 1 )q1 (aαj 1 α2 )q2 · · ·(aαj 1 ···αr )qr (x)μ , if, and only if, the first non zero component of the vector ( p0 − q0 , p1 − q1 , p2 − q2 , . . . , pr − qr , α − μ) is positive. This ordering is a monomial order, since it suffices and is easy to check that if m 1 and m 2 are two monomials such that m 1 ≺ m 2 then for any monomial m, one has mm 1 ≺ mm 2 . Let us consider some examples of centro-affine invariants of S(C(2, 2, k)):  = aαα is a sum of a11 and a22 . Since a11 ≺ a22 , we may write T r a ij   i, j=1,2 T r a ij as a sum of terms ordered in an increasing manner: i, j=1,2

  T r a ij

i, j=1,2

= a11 + a22 .

β

α , K = a α x β and L = a p x q ε : We may do the same for I = aβα aα , J = a β aαβ pq αβ

 2  2 I = a11 + 2a21 a12 + a22 , 1 2 1 2 J = a 1 a11 + a 1 a21 + a 2 a12 + a 2 a22 , 1 1 1 2 2 1 2 2 K = a11 x + a12 x + a21 x + a22 x , L = a1 x 2 − a2 x 1

In general, let ≺ be a monomial order for the monoid M, the lexicographic order for example. Then any centro-affine covariant C of S (C(n, k, k)) can be written as C = α1 m 1 + α2 m 2 + · · · + αn 0 m n 0 , where for i = 1, . . . , n 0 , αi ∈ k and m 1 ≺ m 2 ≺ · · · ≺ m n 0 . Such a sum is called a sum arranged in increasing order.

A. Turqui, D. Dali

4 Matrix of Transformation We first consider for example a subset S(A) where

 A = a ∈ C(2, 1, R) : a j = 0 = aβα , j, α, β = 1, 2  a 1 a11 + a 2 a21 a 1 a12 + a 2 a22 . We can easily check that |Q 1 | and the matrix Q 1 = a1 a2 is a centro-affine covariant of S(C(2, 1, R)) and if we assume that |Q 1 | = 0 then we of S(C(2, 1, R)). can remark that ρ(Q 1 )a 1 is not a centro-affine   1 2 covariant a a1 − a 2 a11 a 1 a22 − a 2 a21 . Then |Q 2 | Now let’s consider the matrix Q 2 = a 2 a22 a21 is not a centro-affine invariant of S(C(2, 1, R)) and if we assume that |Q 2 | = 0 then ρ(Q 2 )a 1 is a centro-affine invariant of S(C(2, 1, R)) but ρ(Q 2 )a 2 is not. In this section, for a given differential system S(A) of S(C(n, k, k)), we search a centro-affine transformation Q such that for all a ∈ A, ρ(Q)a and det Q are centroaffine invariants of S(C(n, k, k)). Let us consider the square matrix Q defined by 

j

j

Q i j = Ti j , j, i j = 1, 2, . . . , n,

(6)

j

where for j = 1, . . . , n and 0 ≤ r ≤ k, Ti j (d0 ( j) , . . . , dr ( j)), (i j = 1, 2, . . . , n) is a tensor once covariant and 0 time contravariant associated with S(C(n, k, k)). If j the tensors Ti j (d0 ( j) , . . . , dr ( j)), j = 1, . . . , n are made up from coefficients of a subset A of S(C(n, k, k)) then the matrix (6) is said to be associated with S(A) and denoted by Q A . For example, let us consider the systems S(A), S(B) and S(C) where 

1 1 1 2 2 2 = a12 = a22 = a11 = a12 = 0, a22 =1 , A = a ∈ C(2, 2, R) : a 1 = −a 2 = 1, a11  a2 2 2 1 1 i = a12 = 22 = 1, a11 = a12 = 0, aαβγ B = a ∈ C(2, 3, R) : a 1 = a 2 = a11 3  = 0, j, α, β, γ = 1, 2 , 

C = a ∈ C(2, 1, R) : a 1 = a 2 = 1, a11 = 0, a21 = 1 = a12 = 1, a22 = 0 . If we consider the once covariant and 0 time contravariant tensor J j (0, 0, 1, 0) = aαα j and Ti (1, 0, 0, 0) = a α εiα where α, i, j = 1, 2 then  QA =

T1 J1

T2 J2



 =

a2 α aα1

−a 1 α aα2



 =

−1 0

 −1 , 1

Normal Forms of Planar Polynomial Differential Systems

is the matrix associated with S(A), and  QB =

−a 1 α aα2

a2 α aα1



 =

1 1

 −1 , 3

is the matrix associated with S(B). Note that one cannot define a matrix associated with the system S(C) from the tensor Ti (1, 0, 0, 0) since C does not contain the tensor j aαβ , j, α, β = 1, 2. But if we consider Vi (1, 0, 0, 0) = a α εiα , i, α = 1, 2 and W j (1, 1, 0, 0) = β a α aα εβ j , j, α, β = 1, 2, then  QC =

V1 V2 W1 W2



 =

−a 1 a2 β β α α a aα εβ1 a aα εβ2



 =

 1 −1 , −1 1

is the matrix associated with S(C). However if we consider the system S(D) where

 j j D = a ∈ C(2, 3, R) : a j = aαβ = 0 and aαβγ = 0, j, α, β, γ = 1, 2 we may remark that we cannot construct tensors once covariant and 0 time contravariant associated with S(D). The question then arises as to whether a matrix Q A associated with S(A) can be determined. Let T (d0 , d1 , . . ., dr , δ), 1 ≤ r ≤ k, be a tensor associated with S(C(n, k, k)). Proposition 4.1 If d2 + 2d3 + · · · + (r − 1)dr − d0 − δ ≡ λ [n] where λ is a non negative integer such that 1 ≤ λ ≤ n − 1, then we can construct from any tensor T (d0 , d1 , . . ., dr , δ), 1 ≤ r ≤ k, at least one tensor λ-times covariant and 0 time contravariant denoted by Ti1 ...iλ (d0 , d1 , . . ., dr , δ), i 1 , . . . , i λ = 1, . . . , n. Proof Let T (d0 , d1 , . . ., dr , δ) be a tensor associated with S(C(n, k, k)) such that d2 + 2d3 + · · · + (r − 1)dr − d0 − δ ≡ λ [n] . Then T (d0 , d1 , . . ., dr , δ) is p times contravariant and q times covariant where p − q = mn + λ, m ∈ Z. Hence, after p operations of contraction and m operations of alternation we obtain from the tensor T (d0 , d1 , . . ., dr , δ) at least one tensor λ−times covariant and 0 time contravariant.   Hence, If d2 + 2d3 + · · · + (r − 1)dr − d0 − δ ≡ 1 [n] then we can construct from any tensor T (d0 , d1 , . . ., dr , δ), 1 ≤ r ≤ k, at least one tensor once covariant and 0 time contravariant denoted by Ti (d0 , d1 , . . ., dr , δ), i = 1, . . . , n. For example, we take systems in C(2, 2, R) and consider two tensors as follows: β

− − − − − a−β a−− a−− a−− x −, T (1, 1, 5, 1) = a α a− a−α − − − − − − pq J (1, 1, 5, 1) = a − a− a−− a − p− aq− a−− a−− x ε ,

A. Turqui, D. Dali

where by ‘−’ and ‘− ’ we denote free indexes. We observe that both tensors have the same type (d0 , d1 , d2 , δ) = (1, 1, 5, 1). In both cases the same d2 −d0 −δ = 3 ≡ 1[2]. T is a tensor 6 times contravariant and 9 times covariant, whereas J is a tensor 8 times contravariant and 9 times covariant but since λ = 1 we can construct tensors one time covariant and 0 time contravariant from T (1, 1, 5, 1) and J (1, 1, 5, 1) with the help of operations of alternations and complete contraction. For example, μ

δ λ ξ aδβ aλi aμω aξων x ν ε pq Ti (1, 1, 5, 1) = a α a βp aqα

and

β γ

ξ

λ aλω aξων x ν ε pq , Ji (1, 1, 5, 1) = a α ai aαβ a δpγ aqδ

where α, β, δ, γ , δ, λ, μ, ν, ξ, ω, p, q, i = 1, 2 obtained respectively from T (1, 1, 5, 1) and J (1, 1, 5, 1). Now let’s consider n tensors T j (d0 ( j) , . . . , dr ( j)) associated with S(C(n, k, k)) such that d2 ( j) + 2d3 ( j) + · · · + (r − 1)dr ( j) − d0 ( j) ≡ 1 [n] , j = 1, n, 0 ≤r ≤ k,

(7)

If for j = 1, 2, . . . , n and for r = 0, . . . , k the tensor T j (d0 ( j) , . . . , dr ( j)) is associated with a given system S(A) then the system (7) is said to be the system associated with S (A). For example, the system (7) associated with S (A) where 

A = a ∈ C(2, 1, R) : a j = 0 = aβα , j, α, β = 1, 2 is given by



− d0 (1) ≡ 1 [2] − d0 (2) ≡ 1 [2]

and the system (7) associated with S (A) where

 γ α A = a ∈ C(2, 3, R) : a β = 0; aαi = 0; aβγ l = 0; i, α, β, γ , l = 1, 2 is given by the system ⎧ ⎨ d2 (1) + 2d3 (1) − d0 (1) ≡ 1 [2] ⎩

d2 (2) + 2d3 (2) − d0 (2) ≡ 1 [2]

In view of Proposition 4.1 to construct a matrix Q A associated with a given differj ential system S(A) it suffices to find n tensors Ti j (d0 ( j) , . . . , dr ( j)), j = 1, n; i j = 1, . . . , n and r ∈ {1, 2, . . . , k} associated with the system S(A) satisfying the system (7) associated with the system S(A). j For example, if A contains a tensor aα1 α2 ...αr0 = 0, j, α1 , α2 , . . . , αr0 = 1, n such that r0 ≡ 0 [n] then any family of n tensors T j (d0 ( j) , . . . , dr ( j)), j = 1, n;

Normal Forms of Planar Polynomial Differential Systems

r ∈ {1, 2, .., k} associated with S(A) such that dr0 ( j) ≡ −1 [n] and dl ( j) ≡ 0 [n] if l = r0 , satisfies ( 7). j If A contain at least a tensor aα1 α2 ...αr0 = 0, j, α1 , α2 , . . . , αr0 = 1, n such that r0 ≡ 2 [n] then any family of n tensors T j (d0 ( j) , . . . , dr ( j)), j = 1, n; r ∈ {1, 2, .., k} associated with S(A) such that dr0 ( j) ≡ −1 [n] and dl ( j) ≡ 0 [n] if l = r0 , satisfies ( 7) etc. Lemma 4.2 Let S(A) be a subset of S(C(n, k, k)). If Q A is a matrix associated with S(A) then the determinant |Q A | is a centro-affine invariant of S(C(n, k, k)). Proof If Q A is defined by (6) then |Q A | = Ti11 Ti22 . . . Tinn εi1 i2 ...in , hence, in view of Gurevich theorem |Q A | is a centro-affine invavariant of S(C(n, k, k)). The Lemma is proved.   Theorem 4.3 If Q A is an invertible matrix (6) associated with a subset S(A) of j S(C(n, k, k)) then for j, α1 , α2 , . . . , αr = 1, n and r = 0, k; |Q A |r ρ(Q A )aα1 α2 ...αr is a centro-affine invariant of S(C(n, k, k)) where ρ is the representation of the general linear group G L(n, k). Proof First we prove the theorem in the case where n = 2, let A be a subset of C(2, k, k) and Q A be a matrix associated with S(A) defined as follows  QA =

S1 J1

S2 J2

 ,

where Si (i = 1, 2) and J j ( j = 1, 2) are tensors once covariant and 0 time contravariant associated with S(A) of type (d0 ( j), . . . , dl ( j)), 1 ≤ l ≤ k. Assume that |Q A | = 0 then  1  1 R1 T Q −1 , = A |Q A | T 2 R 2 i j where Q −1 A is the inverse of Q A and T (i = 1, 2) and R ( j = 1, 2) are defined by



T j = Jα ε jα ( j = 1, 2) R j = Sβ εβ j ( j = 1, 2)

hence, by formula (3) we have ρ (Q A ) a 1 = (Q A )i1 a i = Si a i ; ρ (Q A ) a 2 = (Q A )i2 a i = Ji a i ;

A. Turqui, D. Dali

and 1 |Q A |r ρ (Q A ) a11..1 = (Q A )i1 aβi 1 ...βh δ1 δ2 ...δl T β1 T β2 . . . T βh R δ1 R δ2 . . . R δl   22..2    h times l times

= Si aβi 1 ...βh δ1 δ2 ...δl T β1 T β2 . . . T βh R δ1 R δ2 . . . R δl ; 2 |Q A |r ρ (Q A ) a11..1 = (Q A )i2 aβi 1 ...βh δ1 δ2 ...δl T β1 T β2 . . . T βh R δ1 R δ2 . . . R δl   22..2    h times l times

= Ji aβi 1 ...βh δ1 δ2 ...δl T β1 T β2 . . . T βh R δ1 R δ2 . . . R δl ; where r =l +h and 1 ≤ r ≤ k. The theorem is proved in the case when n = 2. Now j let Q A = Ti j be a matrix associated with S(A) where n  2 and for j = 1, n 1≤ j≤n 1≤i j ≤n

j

, Ti j ( ji = 1, 2, . . . , n) is a tensor once covariant and 0 time contravariant associated with S (A) of type (d0 ( j), . . . , dl ( j)) (1 ≤ l ≤ k) . Assume that |Q A | = 0 then Q −1 A =



1 j P i |Q A | i

 1≤ j≤n 1≤i j ≤n

where Q −1 A is the inverse of Q A and for i, ji = 1, 2, . . . , n : T i+1 . . .Tαnn εα1 α2 ...αi−1 ji αi+1 ...αn , α1 , α2 . . ., αn = 1, n. Pi i = Tα11 Tα22 . . .Tαi−1 i−1 αi+1 j

Then for j = 1, . . . , n we have j

ρ (Q A ) a j = Ti j a i j ; |Q A |r ρ (Q A ) a11..1 22..2 ...nn . . . n = Ti j aαj1 ...αs β1 ...βs ...l1 ...ls P1α1 P1α2 . . . 1 2 n          j i

j

s1 times s2 times

sn times

αs

β

βs

l

P1 1 P2 1 . . .P2 2 . . .Pnl1 . . .Pnsn ; where j, i j ; α1 , . . . , αs1 ; β1 . . . βs2 ; . . . ; l1 , . . . , lsn = 1, n , r = 1, 2, . . . , k and r =   s1 + s2 + · · · + sn .This completes the proof. For example, let’s consider the subset S(A) of the set S(C(2, 3, R)) of complete planar cubic differential systems where

 γ α A = a β = 0; aαi = 0; aβγ l = 0; i, α, β, γ , l = 1, 2

Normal Forms of Planar Polynomial Differential Systems

and the tensors Ti11 (0, 0, 1, 0) and Ti22 (1, 0, 0, 1) associated with S(A) defined by ⎧ 1 α ⎨ Ti1 (0, 0, 1, 0) = aαi 1

(i 1 , α = 1, 2)

⎩ T 2 (1, 0, 0, 1) = a β a γ (i , β, γ = 1, 2) βγ i 2 2 i2  QA =

α aα1 γ β a aβγ 1

α aα2 γ β a aβγ 2 β

 is the matrix associated with S(A). Note that the deter-

γ

minant |Q A | = a α aβp aαγ q ε pq is a centro-affine covariant and since |Q A | = 0 we can see that for any coefficients a in A ρ(Q A )a is a centro-affine. For example α 1 α 2 α β a + aα2 a = aαβ a . ρ(Q A )a 1 = (Q A )i1 a i = Q 11 a 1 + Q 12 a 2 = aα1

Now, let Q A be an invertible matrix associated with a given subset S(A) of S(C(n, k, k)) then this matrix transforms the given system S(A) into a system S(B) where

B = bαj 1 α2 ...αr : for j = 1, n ; r = 1, k; bαj 1 α2 ...αr  = ρ(Q A )aαj 1 α2 ...αr and aαj 1 α2 ...αr ∈ A In view of Theorem 4.3, for j, α1 , α2 , . . . , αr = 1, n and r = 0, k, |Q A |r ρ(Q A ) j aα1 α2 ...αr is a centro-affine invariant of S(C(n, k, k)). To decompose the determinant |Q A | and the centro-affine invariants |Q A |r ρ(Q A ) j aα1 α2 ...αr , j, α1 , α2 , . . . , αr = 1, n and r = 0, k in F the idea is to compute f 1 (b), . . . , f s (b) where b ∈ B. Since f 1 , . . . , f s are still invariant under any centro-affine transformation, they are invariant under the transformation Q A . We are then led to the following algebraic system ⎧ ⎨ P1 (b) = 0 ... m = 1, . . . , s ⎩ Pm (b) = 0,

(8)

w

where for j = 1, . . . , m, P j (b) = f i j (b) −  i j f i j ,  = |Q A | , f i j ∈ F and wi j is its weight. Solving this system with the help of Gröbner bases, we obtains for j, α1 , α2 . . . , αr = 1, n and r = 0, k:  = C( f i1 , . . . , f im ) and bαj 1 α2 ...αr =

Cαj α

1 2 ...αr

( fi1 , . . . , fim )

(C( f i1 , . . . , f im ))r j

,

(9)

where for j = 1, . . . , n and r = 1, . . . , k; C( f i1 , . . . , f im ) and Cα1 α2 ...αr ( f i1 , . . . , f im ) are polynomial functions of f i1 , . . . , f im .

A. Turqui, D. Dali j

Remark 4.4 In view of Gurevich theorem the determinant |Q A | and |Q A |r bα1 α2 ...αr for j, α1 , α2 , . . . , αr = 1, n and r = 0, k are elements of the system of generators of the centro-affine covariants of S(C(n, k, k)). Then if F is not known, it suffices to reduce j the set {|Q A |r bα1 α2 ...αr : j, α1 , α2 , . . . , αr = 1, n; r = 0, k} modulo G L(n, k). This remark is important because the description of the algebra of the centro-affine covariants of given differential systems is difficult. The covariants are polynomials in several variables (more then 12 for the planar polynomial differential systems).

5 Normal Forms of Polynomial Differential Systems Let us consider the set S(C(n, k, k)) where C(n, k, k) has a tensor (aαj 1 α2 ...αr ) j,α1 ,α2 ,...,αr 0

0 =1,n

= 0, 1 ≤ r0 ≤ k,

where r0 ≡ 0 [n]. In this section, our goal is to develop a constructive method to determine normal forms with respect to the centro-affine group G L(n, k) and characterize each one of them by algebraic relations in terms of centro-affine invariants of the given systems S(C(n, k, k)). An autonomous differential equation of the first order F(t, x, ddtx ) = 0 is said to be invariant under a linear group G if for all g ∈ G, the transformation y = g(x) leads to the differential equation F(t, y, dy dt ) = 0. We can see, using the formula (3), that for any subset A of C(n, r, k), r = 1, . . . , n; the differential system S(A) is invariant under the linear group G L(n, k). This motivates the following definition: Two subsets S(A) and S(B) of S(C(n, r, k)) are said to be G L(n, k)-equivalent or centro-affine equivalent and we write B = ρ(Q)A if, and only if, there exists Q in G L(n, k) such that for j, α1 , α2 . . . , αr = 1, n and r = 0, k, bαj 1 α2 ...αr = ρ(Q)aαj 1 α2 ...αr , j

j

where aα1 α2 ...αr ∈ A and bα1 α2 ...αr ∈ B. For example, let’s consider the following planar quadratic differential systems  dx dt dy dt

= x − y + x2 − xy = x 2 + y2,

 dx and

dt dy dt

= 23 x − 21 y − 21 x y + 23 x 2 = 4x 2 − 23 x y + 23 x + 21 y 2 − 21 y,

that is, the subsets S (A) and S (B) of S(C(2, 2, R)) where

1 2 A = a ∈ C(2, 2, R) : a 1 = a 2 = a12 = a22 = a22 = a12 = 0,  1 1 2 2 a11 = −a21 = a11 = −a12 = a11 = a22 =1 ,

Normal Forms of Planar Polynomial Differential Systems

 3 1 1 2 B = a ∈ C(2, 2, R) : a 1 = a 2 = a22 = 0, a12 = a11 = a11 = −a12 = , 2 1 1 2 2 a22 = a21 = a12 = −a22 = − , a11 =4 . 2  S (A) and S (B) are centro-affine equivalent since ρ(Q)A = B, where Q = Let R be a binary relation on S(C(n, k, k)) defined by

1 1

 0 . 2

S(A)RS(B) ⇐⇒ ∃Q ∈ G L(n, k), B = ρ(Q)A. R is an equivalence relation on S(C(n, k, k)) since ∀A, B, C ∈ C(n, k, k) we can easily check that (i) (ii) (iii)

ρ(In )A = Awhere In is the unit matrix, ρ(Q)A = B ⇐⇒ ρ(Q −1 )B = A, ρ(Q)A = B and ρ(Q −1 )B = C then ρ(Q Q −1 )A = C.

The quotient set by R is denoted by S/R. Let S(A) and S(B) be two subsets of S(C(n, k, k)) and, Q A and Q B their associated matrices, respectively. Lemma 5.1 If |Q A | |Q B | = 0 then S(A)RS(B) if, and only if, ρ(Q A )A = ρ(Q B )B. Proof Let S(A) and S(B) be two subsets of S(C(n, k, k)) and, Q A and Q B their associated matrices such that |Q A | |Q B | = 0. If S(A)RS(B) then S(ρ(Q A )A)RS(ρ(Q B ) B); indeed it is clear that S(A)RS(ρ(Q A )A) and S(B)RS(ρ(Q B )B). Hence, there exists Q 0 ∈ S L(n, k) such that ρ(Q A )ρ(Q 0 )A = ρ(Q B )B. In view of Theorem 4.3 we have ρ(Q A )A = ρ(Q B )B. Conversely, If ρ(Q A )A = ρ(Q B )B i.e. for all j, α1 , α2 . . . , αr = 1, n and r = 0, k, ρ(Q A )aαj 1 α2 ...αr = ρ(Q B )bαj 1 α2 ...αr , where aα1 α2 ...αr ∈ A and bα1 α2 ...αr ∈ B, then B = ρ(Q −1 B Q A )A or S(A)RS(B). The lemma is proved.   j

j

Corollary 5.2 Let S(A) and S(B) be two subsets of S(C(n, k, k)), and C be the centro-affine invariant given by (9) such that, for all a ∈ A and b ∈ B, C( f 1 (a), . . . , f s (a)).C( f 1 (b), . . . , f s (b)) = 0. Then S(A)RS(B) if, and only if, for all a ∈ A and for all b ∈ B, (C( f 1 (b), . . . , f s (b) ))r Cαj α

1 2 ...αr

= (C( f 1 (a), . . . , f s (a)) Cα r

( f 1 (a), . . . , f s (a))

j 1 α2 ...αr

( f 1 (b), . . . , f s (b)),

A. Turqui, D. Dali

where for j, α1 , α2 , . . . , αr = 1, n; r = 0, k, Cαj α ...αr is the centro-affine invariant 1 2 of S(C(n, k, k)) given by (9). Hence, the following set  N=

( f1 , . . . , fs )

Cαj α

1 2 ...αr

(C( f 1 , . . . , f s ))r

 : j, α1 , α2 . . . , αr = 1, n; r = 0, k; C( f 1 , . . . , f s ) = 0

is a normal form of S/R. It is clear that S/R is finite. We shall now show step by step how normal forms of given differential systems S(C(n, k, k)) can be determined. To obtain the first normal form we start with A1 = C(n, k, k) Q A1 , compute the determinant 1. Set A1 = C(n, k, k). Compute an arbitrary  matrix   Q A  and decompose it in F. Then  Q A  = C1 ( f 1 , . . . , f s ) where C1 is a 1 1 polynomial function of f 1 , . . . , f s . 2. Assume C1 ( f 1 , . . . , f s ) = 0 and compute ρ(Q A1 )A1 . 3. Decompose each element of ρ(Q A1 )A1 in F by solving the algebraic system (8) or reduce ρ(Q A1 )A1 modulo G L(n, k) if F is not known and then for j, α1 , α2 , . . . , αr = 1, n and r = 0, k we obtain ρ(Q A1 )aαj

1 α2 ...αr

=

Cαj α

1 2 ...αr

( fi1 , . . . , fim )

(C1 ( f 1 , . . . , f s ))r

,

then the set  N1 =

Cαj α

1 2 ...αr

( fi1 , . . . , fim )

(C1 ( f 1 , . . . , f s ))r

: j, α1 , α2 . . . , αr = 1, n;

r = 0, k where C1 ( f 1 , . . . , f s ) = 0



is the first normal form. A second normal form is constructed in the same manner when taking A2 = {a ∈ A1 , C1 ( f 1 , . . . , f s ) = 0}. The procedure stops when A1 ∪ A2 ∪ · · · ∪ Al = C(n, k, k) and then S/R = {N1 , N2 , . . . , Nl } . Now, we are able to give an algorithm to construct a normal form of a given polynomial differential system. Algorithm • Step 1 Enter the coefficients C(n, k, k), a minimal generating family F = { f 1 , f 2 , . . . , f s } of centro-affine covariants of S(C(n, k, k)), A1 = C(n, k, k), S/R = ∅ and l = 1. • Step 2 Order M by a monomial order. • Step 3 Deduce from F a Gröbner basis B.

Normal Forms of Planar Polynomial Differential Systems

  • Step 4 Compute an arbitrary Q Al associated with S (Al ). Compute  Q Al  and ρ(Q Al )Al .   • Step 5 Decompose  Q Al  and the elements of ρ(Q Al )Al in F, with the help of Gr öbner basis B. Then Nl = ρ(Q Al )Al and S/R = S/R∪{Nl }. • Step 6 While  A1 ∪ A2 ∪ · · · ∪ Al = C(n, k, k) do l = l + 1, Al+1 = a ∈ Al ,  Q Al  = 0 and go to step 4 else return S/R.

6 Application Note that the knowledge of a minimal system of generators F of centro-affine covariants of a given polynomial differential system S(C(n, k, k)) simplifies the problem of decomposing the coefficients of the normal forms. However, when a minimal system of generators is not known we can reduce the family of new coefficients. We shall now show how to construct normal forms of given polynomial differential systems S(C(n, k, k)) in the case where a minimal system of generators F is known and in the case where a minimal system of generators F is not known. Let’s consider as example, the set of the complete planar cubic differential systems S(C(2, 3, R)). A minimal system of generators of centro-affine covariants of the set of the complete quadratic differential systems S(C(2, 2, R)) is known. Theorem 6.1 ([2,21,22]) The family E = {I1 , . . . , I36 , K 1 , . . . , K 33 } where I1 I2 I3 I4 I5 I6 I7 I8 I9

= aαα , β = aβα aα , γ β = a αp aαq aβγ ε pq , β γ = a αp aβq aαγ ε pq , β γ = a αp aγ q aαβ ε pq , β γ δ pq = a αp aγ aαq aβδ ε , β γ δ pq r s α = a pr aαq aβs aγ δ ε ε , β γ δ = a αpr aαq aδs aβγ ε pq εr s , β γ δ pq r s = a αpr aβq aγ s aαδ ε ε ,

I19 I20 I21 I22 I23 I24 I25 I26 I27

β

= aβα aαγ a γ , β = aγα aαβ a γ , p = aαβ a α a β a q ε pq , α aβ aγ aδ , = aαβ γδ α aβ aγ aδ , = aβγ αδ β γ = aγα aδ aαβ a δ , α a β a γ a δ ε pq , = aαp γ q βδ α a β a γ a δ ε pq , = aαp γ q βδ p α β γ q = aα aβγ a a a ε pq ,

I10 I11 I12 I13 I14 I15 I16 I17 I18

I28 I29 I30 I31 I32 I33 I34 I35 I36

β γ

μ

δ a ε pq , = a αp aδ aμ aαq βγ β γ δ μ = a αp aqr aβs aαγ aδμ ε pq εr s , γ β δ a μ ε pq εr s , = a αp aqr aβs aαδ γμ γ β μ δ α = a p aqr aγ s aαβ aδμ ε pq εr s , β γ δ μ ν = a αp ar aαq aβs aγ δ aμν ε pq εr s , β γ δ μ ν = a αpr aqk aαs aδl aβγ aμν ε pq εr s εkl , γ β μ δ a a ν a τ ε pq εr s , = a αp ar aδ aαq βs γ τ μν α β = a aαβ , p = a α a q aα ε pq , β

(10)

γ

= aβα aαγ aδμ a δ a μ , β γ = aγα aαβ aδμ a δ a μ , β γ = a αp aαq aβδ aγδ μ a μ ε pq , β γ = a αp aαq aβμ aγδ δ a μ ε pq , β γ = a αp aβq aαμ aγδ δ a μ ε pq , α aβ aγ aδ aμaν , = aβν αγ δμ α a β a γ a δ a μ a ν ε pq , = aμp αq βν γ δ β γ δ μ = a αp aν aαq aβμ aγ δ a ν ε pq , γ β δ a μ a ν ε pq εr s , = a αpr aνq aαs aβγ δμ

(11)

A. Turqui, D. Dali α xβ, K 1 = aαβ p K 2 = aα x α x q ε pq , β K 3 = aβα aαγ x γ , β K 4 = aγα aαβ x γ , p α β q K 5 = aαβ a x x ε pq , α aβ x γ x δ , K 6 = aαβ γδ α aβ x γ x δ , K 7 = aβγ αδ β γ K 8 = aγα aδ aαβ x δ , α a β a γ x δ ε pq , K 9 = aαp γ q βδ α a β a γ x δ ε pq , K 10 = aαp δq βγ

K 21 K 22 K 23 K 24 K 25 K 26 K 33

K 11 K 12 K 13 K 14 K 15 K 16 K 17 K 18 K 19 K 20

α xβ xγ xqε , = aα aβγ pq β γ δ μ α = aβ aαγ aδμ x x , β γ = aγα aαβ aδμ x δ x μ , β γ = a αp aαq aβδ aγδ μ x μ ε pq , γ β = a αp aαq aβμ aγδ δ x μ ε pq , β γ = a αp aβq aαμ aγδ δ x μ ε pq , α aβ aγ x δ x μ x ν , = aβν αγ δμ β γ α = aμp aαq aβν aγδ δ x μ x ν ε pq , β γ δ μ = a αp aν aαq aβμ aγ δ x ν ε pq , β γ δ μ = a αpr aνq aαs aβγ aδμ x ν ε pq εr s ,

= a p x q ε pq , p = aα a α x q ε pq , q = a p aαβ x α x β ε pq , q α β γ = a p aα aβγ x x ε pq , p = a α a β aαβ x q ε pq , β γ = a α aαδ aβγ x δ , β γ μ = a α a pα aqβ aγδ ν aδμ x ν ε pq ,

p

K 27 K 28 K 29 K 30 K 31 K 32

β

(12)

γ

= a α aαγ aβδ x δ , p γ = a α a β aγ aαβ x q ε pq , β γ δ x μ, = a α aδ aαμ aβδ γ β δ x μ, = a α aγ aαμ aβδ β γ δ α = a aαγ aβδ aμν x μ x ν γ δ μ = a α a β aαβ aμν aγ δ x ν ,

(13)

form a minimal system of generators of centro-affine covariants of systems S(C(2, 2, R)). Hence, the centro-affine invariants of the linear and the quadratic parts of the considered set S(C(2, 3, R)) can be decomposed in E = {I1 , . . . , I36 , K 1 , . . . , K 33 } and for the set of the centro-affine covariants of the cubic parts it suffices to reduce it modulo G L(2, R). Let us summarize step by step how a normal form of S(C(2, 3, R)) can be constructed. 1. A1 = C(2, 3, R), an arbitrary matrix Q A1 of the form (6) can be  Q A1 =

a α ε1α β a α aα εβ1

a α ε2α β a α aα εβ2



 =

a2 − a α aα2

− a1 a α aα1



  ,  Q A1  = a α a q aαp ε pq .

  We see immediately that  Q A1  = −I18 . 2. Assume I18 = 0 and compute ρ(Q A1 )A1 :  ρ(Q A1 )A1 =

1 , b1 , b1 , b1 , b1 , b1 , b1 , b1 , b11 , b21 , b11 12 22 111 112 122 222 2 , b2 , b2 , b2 , b2 , b2 , b2 b2 , b12 , b22 , b11 12 22 111 112 122 222



Normal Forms of Planar Polynomial Differential Systems

where b2 = a p aα a α ε pq , q

b1 = a p a q ε pq , b11 =

1 α δ β λ I18 a a aλ aδ εαβ ,

b21 =

1 α p q I18 a a aα ε pq ,

b12 =

1 α β p q λ I18 a a aα aλ aβ ε pq ,

b22 =

1 α β p q I18 a a aα aβ ε pq ,

1 = b22

1 α β p q 2 a a a aαβ ε pq , I18

2 = b22

1 μ α β p q 2 a a a aμ aαβ ε pq , I18

1 = b21

1 α β p λ q 2 a a a aα aλβ ε pq , I18

2 = b21

1 μ α β λ p q 2 a a a aα aμ aλβ ε pq , I18

1 = b11

1 α β p q δ λ 2 a a a aδλ aβ aα ε pq , I18

2 = b11

1 μ α β q p δ λ 2 a a a aδλ aμ aβ aα ε pq , I18

1 = b111

1 α β δ p γ λ μ q 3 a a a a aα aβ aδ aγ λμ ε pq , I18

2 = b111

1 α β δ ν γ λ μ p q 3 a a a a aα aβ aδ aν aγ λμ ε pq , I18

1 = b211

1 α β δ p λ γ q 3 a a a a aβ aα aγ λδ ε pq , I18

2 = b211

1 α β δ ν γ λ p q 3 a a a a aα aβ aν aγ λδ ε pq , I18

1 = b221

1 α β δ p γ q 3 a a a a aα aβδγ ε pq , I18

2 = b221

1 α β δ ν γ p q 3 a a a a aα aν aγβδ ε pq , I18

1 = b222

1 α β δ p q 3 a a a a aαβδ ε pq , I18

2 = b222

1 α β δ ν p q 3 a a a a aν aαβδ ε pq . I18

3. Decompose each element of the linear and quadratic parts of ρ(Q A1 )A1 in F = E by solving the algebraic system (8), that is the system  w ⎧ ⎨ Ii1 (b) −  Q A1  i1 Ii1 = 0 ...  w ⎩ Iim (b) −  Q A1  im Iim = 0

(14)

where for 1 ≤ m ≤ 36, Iim ∈ E, Iim = 0 and wi1 , . . . , wim are the weights of Ii1 , . . . , Iim where, w1 = 0, w7 = 2, w13 = 2, w19 = 0, w25 = 1, w31 = 1, w2 = 0, w8 = 2, w14 = 2, w20 = 0, w26 = 1, w32 = 1, w3 = 1, w9 = 2, w15 = 3, w21 = 1, w27 = 1, w33 = 0, w4 = 1, w10 = 1, w16 = 2, w22 = 0, w28 = 0, w34 = 1, w5 = 1, w11 = 2, w17 = 0, w23 = 0, w29 = 0, w35 = 1, w6 = 1, w12 = 2, w18 = 1, w24 = 0, w30 = 1, w36 = 2, and the new coefficients b of ρ(Q A1 )A1 are the unknown variables. Note that ⎧ 1 ⎪ ⎨b = 0 b21 = − 1 ⎪ ⎩ 2 b2 = 0.

A. Turqui, D. Dali

Then the algebraic system (14) can be written as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

1 b22

=

I21 2 I18

− b2

= I18

− b11  1 2 b1 + 2b12 1 + b2 b12 22 1 + b2 b11 12 1 1 2 b1 b22 − b22

= I1

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 + b2 − b2 b1 ⎪ − b11 b12 ⎪ 12 1 22 ⎪ ⎪ ⎩ 1 1 2 2 1 b1 b11 − b11 + b1 b21

= I2 = − II17 18 = = = =

(15)

I20 I18 I27 2 I18 I19 I18 I24 I18 .

Solving this system, we obtain ⎧ ⎪ b2 ⎪ ⎪ ⎪ ⎪ ⎪ b11 ⎪ ⎪ ⎪ ⎪ ⎪ b12 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ b22 ⎪ ⎪ ⎪ ⎪ b2 ⎪ ⎪ ⎨ 22 1 b12 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ b12 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ b11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ b11 ⎪ ⎪ ⎪ ⎪ ⎩

= − I18 = − I1   = 21 I2 − I12 =

I21 2 I18

=− = = = =

1 2 I18

(I27 + I21 I1 )

1 2 (I27 + I21 I1 − I17 I18 ) I18   1 −2I1 I27 + I2 I21 + 2I18 I19 − 3I12 I21 + 2I1 I17 I18 2 I18   1 I20 I18 + 2I1 I27 − I2 I21 − 2I18 I19 + 3I12 I21 − 2I1 I17 I18 2 I18 1 3 2 2 2 (I2 I27 − 2I24 I18 − 7I1 I21 − 5I1 I27 + 5I1 I17 I18 I18

+ 3I1 I2 I21 − 2I1 I20 I18 − I2 I17 I18 + 4I1 I18 I19 ).

Theorem 6.2 The elements of the set L = {I, J, K , L , M, N , O, P} where γ

μ q

γ

μ p q

γ

p q

I = a α a β a δ a p aα aβλ aδ aγ λμ ε pq , M = a α a β a δ a ν aα aβλ aδ aν aγ λμ ε pq , γ q

J = a α a β a δ a p aβλ aα aγ λδ ε pq , γ q

K = a α a β a δ a p aα aβδγ ε pq ,

L = a α a β a δ a p aαβδ ε pq , q

N = a α a β a δ a ν aα aβλ aν aγ λδ ε pq , γ

O = a α a β a δ a ν aα aν aγβδ ε pq , p q

P = a α a β a δ a ν aν aαβδ ε pq , p q

are centro-affine invariants of S(C(2, 3, R)) which are “polynomially independent” in the sense that none of the elements I, J, K , L , M, N , O, P can be written as a polynomial function of the others.

Normal Forms of Planar Polynomial Differential Systems

Indeed, in view of Gurevich theorem I, J, K , L , M, N , O, P are centro-affine invariants of the system of generators of A(2, 3, R) of the centro-affine invariants of the differential systems S(C(2, 3, R)). Recall that the k-algebra A(n, k, k) of the centro-affine covariants of S(C(n, k, k)) is multigraded: A(n, k, k) = ⊕d0 ,d1 ,...,dr ∈N A(d0 ,d1 ,...,dr ) , where A(d0 ,d1 ,...,dr ) is the vector subspace of the invariants of type (d0 , d1 , . . . , dr ), (1 ≤ r ≤ k) . The elements of the subfamily L1 = {I, J, K , L , M, O} are polynomially independent. Indeed, if one invariant of L1 can be generated by the others then its type can be decomposed as linear combination of their types (see e.g. [7]), which is impossible since I ∈ A(4,3,0,1) , M ∈ A(4,4,0,1) , J ∈ A(4,2,0,1) , N ∈ A(4,3,0,1) , K ∈ A(4,1,0,1) , O ∈ A(3,2,0,1) , L ∈ A(4,0,0,1) , P ∈ A(4,1,0,1) . We show in the same manner that the G L (2, R)—invariants J, L , M, N , O, P are polynomially independent. Now it suffices to show that I cannot be generated from 2 is a term of I but is N , and K cannot be generated from P. Note that (a 1 )4 (a12 )3 a222 2 2 1 4 not a term of N and (a ) a1 a112 is a term of K but is not a term of P. This proves that the elements of the set L are polynomially independent. Corollary 6.3 The elements of the set L ∪ E are centro-affine invariants of S(C(2, 3, R)) and they are polynomially independent. Proof Using the algorithm 2 in page 1918 of [7] we see that none of elements of L can be generated by elements of L ∪ E. Obviously, the elements of E cannot be generated by elements of L since E ⊂ ⊕d0 ,d1 ,d2 ∈N A(d0 ,d1 ,d2 ,0) and all centro-affine covariants   of L are of types (d0 , d1 , d2 , d3 ) with d3 = 0. Then one gets a first normal form of the complete planar cubic differential systems when I18 = 0: dy 1 1 = − I1 y 1 − y 2 + 2 (I20 I18 + 2I1 I27 − I2 I21 − 2I18 I19 + 3I12 I21 dt I18 1 I21 − 2I1 I17 I18 )(y 1 )2 + 2 (I27 + I21 I1 − I17 I18 ) y 1 y 2 + 2 (y 2 )2 I18 I18 I J K L + 3 (y 1 )3 + 3 3 (y 1 )2 y 2 + 3 3 (y 2 )2 y 1 + 3 (y 2 )3 , I18 I18 I18 I18 and  1 1 dy 2 = −I18 + I2 − I12 y 1 + 2 (I2 I27 − 2I24 I18 − 7I13 I21 − 5I12 I27 dt 2 I18 + 5I12 I17 I18 3I1 I2 I21 − 2I1 I20 I18 − I2 I17 I18 4I1 I18 I19 )(y 1 )2

A. Turqui, D. Dali

1 (−2I1 I27 + I2 I21 + 2I18 I19 − 3I12 I21 + 2I1 I17 I18 )y 1 y 2 2 I18 1 M N − 2 (I27 + I21 I1 )(y 2 )2 + 3 (y 1 )3 + 3 3 (y 1 )2 y 2 I18 I18 I18 O 2 2 1 P 2 3 + 3 3 (y ) y + 3 (y ) . I18 I18 +

Hence, by Corollary 5.2 the following theorem holds. Theorem 6.4 Two planar cubic differential systems S(A) and S(B) such that I18 (a).I18 (b) = 0 for all a ∈ A and b ∈ B are centro-affine equivalent if, and only if, I1 (a) = I1 (b), I17 (a) = I17 (b), J (a) = J (b), O(a) = O (b) ,

I2 (a) = I2 (b), I18 (a) = I18 (b), I21 (a) = I21 (b), I27 (a) = I27 (b), I19 (a) = I19 (b), I24 (a) = I20 (b), I20 (a) = I20 (b), I (a) = I (b), K (a) = K (b) , L(a) = L (b) , M(a) = M (b) , N (a) = N (b) , P(a) = P (b) ,

where a ∈ A and b ∈ B.

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