A Theorem for the Triple Zero bifurcation

2012 IFAC Conference on Analysis and Control of Chaotic Systems The International Federation of Automatic Control June 20-22, 2012. Cancún, México

A Theorem for the Triple Zero bifurcation Fernando Verduzco ∗ Mayra A. Mazon ∗∗ ∗

Departamento de Matematicas, Universidad de Sonora, Mexico. (e-mail: [email protected]). ∗∗ Departamento de Matematicas, Universidad de Sonora, Mexico. (e-mail: [email protected]). Abstract: Given a three-dimensional m-paremeterized vector field, which has a non-hyperbolic equilibrium point with a triple-zero eigenvalue with geometric multiplicity one, the called nondegeneracy and transversality conditions are given such that system undergoes the triple-zero bifurcation. Keywords: Triple-zero bifurcation, unfolding. 1. INTRODUCTION

librium point with a triple-zero eigenvalue with geometric multiplicity one. We are not concerned in the different bifurcations that occur in such singularity. We follow the approach given by Carrillo et al. (2010) for the TakensBogdanov bifurcation. The idea is to find, under which conditions, the family undergoes the triple-zero bifurcation, for a given unfolding. It is important to mention that, in general, this approach permit us to find, under which conditions, a nonlinear systems undergoes a bifurcation at a singularity.

The analysis of bifurcations, of codimension one and two, at equilibrium points of vector fields has been widely studied in several text-books on Dynamical Systems (see Guckenheimer-Holmes (1983), Wiggins (1990), Kuznetsov (1995)). For codimension three bifurcations, much less papers have been published, due to the existence of very complicated dynamical behavior, such as homoclinic and heteroclinic orbits. Among this codimension three bifurcations, most of them are codimension two bifurcations with some kind of degeneracy (see Dumortier et al. (1987), Dumortier et al. (1991)).

This paper is organized as follows. In Section 2 we establish the problem to solve. In Section 3 we obtain, in a different way, the classical normal form for the triple-zero singularity. We consider that this approach to calculate the normal form, can be used to the general case of a kzero singularity. In Section 4, we follow Murdock (1998) to calculate an unfolding of the singularity. In Section 5 we use the Jordan theory to simplify the Jacobian of the system at the singularity. We introduce some notation to express, in a manageable way, the nonlinear terms of the vector field. In Section 6, we follow Carrillo et al. (2010), to prove the topological equivalence between the unfolding and the simplified system, calculated at Sections 4 and 5, respectively. Finally, in Section 7 we establish the main result of the paper.

In this paper we are interested in the called triple-zero bifurcation, which consists in a triple-zero eigenvalue with geometric multiplicity one, that is, with one eigenvalue. This case has been partially studied by different authors. Medved (1984) derive, for a 3-parameterized family of three-dimensional vector fields, an unfolding and find bifurcation diagrams for the saddle-node, Hopf, TakensBogdanov and zero-Hopf bifurcations. Yu et al. (1988) and Yu et al. (1990), have proved the existence and bifurcations of a two-dimensional invariant tori around the singularity. Freire et al. (2002) study the existence of the non-degenerated codimension two bifurcations, TakensBogdanov and zero-Hopf, at the singularity, for a given unfolding. Algaba et al. (2003) analyze a family of symmetric control systems that has the circuit of Chua as a prototype. They give a partial bifurcation set on a threedimensional parameter space.

2. PROBLEM STATEMENT Consider the nonlinear system x˙ = F (x, µ), (1) where x ∈ R3 , µ ∈ Rm , with m ≥ 3, and F is sufficiently smooth. Suppose that there exists (x0 , µ0 ) ∈ R3×m such that

Dumortier, with different colleagues, have done very interesting contributions to the theory of bifurcations. In particular, for the triple-zero bifurcation, Dumortier et al. (1996) have given a characterization of such singularity, while in Dumortier et al. (2001) have approached the analysis in a different way to study global bifurcations.

H1) F (x0 , µ0 ) = 0,

In this paper we study an m-parameterized family of threedimensional vector fields which has a non-hyperbolic equi-

Our goal is to find, under which conditions, system (1) is locally topologically equivalent to the unfolding

978-3-902823-02-1/12/$20.00 © 2012 IFAC

H2) DF (x0 , µ0 ) ∼ J =

18

0 1 0 0 0 1 0 0 0

! .

10.3182/20120620-3-MX-3012.00042

CHAOS'12 June 20-22, 2012. Cancún, México

z˙1 = z2 ,


D3 = D3 + 2 D1 J + J T D1 + D2 J,

z˙2 = z3 ,

(2)

T

(4)

T 2

H 1 = 2J H1 J + (J ) H1 J.

z˙3 = ε1 + ε2 z2 + ε3 z3 + f2 (z),

If H1 = (hij ), then

where

0 0 0 0 0 2h11 0 h11 3h12

H1 =

f2 (z) = α1 z12 + α2 z1 z2 + α3 z1 z3 + α4 z22 ,

! ,

and y T H 1 y = 3h11 y2 y3 + 3h12 y32 . Then it is possible to e 23 (y). cancel the y2 y3 -terms and y32 -terms in G Observe that, if M = (mij ) = ( M 1 M 2 M 3 ), where M j is the column j of M , then M J = ( 0 M 1 M 2 ).

with α1 6= 0. 3. NORMAL FORM In this section we introduce a matrix method to find the classical normal form for the triple-zero singularity. Let us consider the nonlinear system x˙ = Jx + G2 (x) + · · · , (3) ! 010 where J = 0 0 1 , and 000  T  ! y D y G21 (y) 1 T 1  G2 (y) = G22 (y) = y D2 y , 2 G23 (y) y T D3 y represents the second order terms, then, from the normal form theory (see Guckenheimer-Holmes (1983), Wiggins (1990)) there exists a change of coordinates x = y + h2 (y), where h2 (y) is an homogeneous second order vector field, such that the original systems transforms into e 2 (y) + · · · , y˙ = Jy + G

Then, if D3 = (dij ) and D3 = (dij ), it follows, from (4), that di1 = di1 , for i = 1, 2, 3. Then we have proof the next Lemma 1. Given the nonlinear system x˙ = Jx + G2 (x) + · · · , ! ! 0 1 0 G21 (x) 0 0 1 G22 (x) , there where J = and G2 (x) = 0 0 0 G23 (x) exists a change of coordinates x = y + h2 (y), such that the original systems transforms into e 2 (y) + · · · , y˙ = Jy + G   0 e 2 (y) =  0 , with where G e 23 (y) G e 23 (y) = α1 y12 + α2 y1 y2 + α3 y1 y3 + α4 y22 , G and α1 =

where

1 ∂ 2 G23 (0). 2 ∂x21

We will call to the system e 2 (y) = G2 (y) + Jh2 (y) − Dh2 (y)Jy. G If h2 (y) = where

h21 (y) h22 (y) h23 (y)

!

e 2 (y), y˙ = Jy + G (5) the second order truncated normal form of system (3).

 y T H1 y 1 =  y T H2 y  , 2 y T H3 y 

4. UNFOLDING In this section we follow Murdock (1998) to give the classical unfolding of the singularity. Consider the truncated normal form (5) e 2 (x), x˙ = Jx + G

HiT

= Hi , for i = 1, 2, 3, then  T  y (D1 + H2 − 2H1 J) y 1 e 2 (y) =  y T (D2 + H3 − 2H2 J) y  . G 2 y T (D3 − 2H3 J) y

and consider the first order perturbation e 2 (x) + µ (p + Bx) , x˙ = Jx + G

It is known that, if M T 6= M , then y T M y ≡ 0 if and only if M T + M = 0. Therefore, it follows that

(6)

3

where µ ≈ 0, p ∈ R , and B = (bij )3×3 . Proposition 2. If α1 6= 0, then system (6) is locally topologically equivalent to e 2 (z) + S0 ε + εT S1 z, z˙ = Jz + G (7) 3 where ε ∈ R , and ! ! 0 0 0 S11 S0 = 0 0 0 , S1 = S12 , 1 0 0 S13 ! 0 0 0 with S11 = S12 = 0, and S13 = 0 1 0 . 0 0 1 Remark 1. Observe that (7) and (2) are the same system.

e 21 (y) ≡ 0 ⇔ H2 = H1 J + J T H1 − D1 , G e 22 (y) ≡ 0 ⇔ H3 = H2 J + J T H2 − D2 . G Then, e 23 (y) = 1 y T (D3 − 2H3 J) y G 2
1 T = y D3 − 2H 1 y, 2 where 19

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ε1 = µp3 ,

The perturbation (7) is called asymptotic unfolding by Murdock (1998). We proof the proposition 2 with the next lemmas. Lemma 3. The change of coordinates x = z + µ (k + Lz) , (8) where k ∈ R3 , and L = (lij )3×3 , transforms system (6) into
e 2 (z) + µ p + Bz , z˙ = Jz + G where

ε2 = µb32 , ε3 = µb33 . 5. JORDAN FORM In this section we applied the Jordan form theory to simplify system (1) which satisfy H1) − H2). Let us define A = DF (x0 , µ0 ). From H2) it follows, from the Jordan form theory, that there exists P = ( v1 v2 v3 ), such that P −1 AP = J, where Av1 = 0, Av2 = v1 , and Av3 = v2 .  T w1 Lemma 5. If P −1 =  w2T , then w3T

p = p + Jk, e + JL − LJ, B = B + kT D e 2 (z) = 1 z T Dz, e is given by lemma 1. and G 2 Proof. If we derivate (8), we obtain x˙ = (I + µL)z˙ ⇔ z˙ = (I + µL)−1 x. ˙ But, for µ ≈ 0, (I + µL)−1 ≈ I − µL + O(µ2 ). Besides,

AT w1 = w2 ,

(9)

AT w2 = w3 , AT w3 = 0.

(10) Proof. Observe that

  T w1T A w2 ⇔  w2T A  =  w3T  . 0 w3T A 

P −1 AP = J ⇔ P −1 A = JP −1

e 2 (z + µ (k + Lz)) x˙ = J (z + µ (k + Lz)) + G +µ(p + B(z + µ (k + Lz)) e 2 (z) + µ(ˆ ˆ + O(|z|2 , µ2 ), = Jz + G p + Bz)

(11)

The Taylor series of F around (x0 , µ0 ) is given by

ˆ = B + JL + k T D. e where pˆ = p + Jk and B Substituting (10-11) in (9), the result it follows. Lemma 4. If α1 6= 0, then there exist k and L such that   ! 0 0 0 0 p = 0 , and B =  0 0 0  . p3 0 b32 b33

F (x, µ) = A(x − x0 ) + Fµ (x0 , µ0 )(µ − µ0 ) 1 + D2 F (x0 , µ0 )(x − x0 , x − x0 ) 2 +Fµx (x0 , µ0 )(µ − µ0 , x − x0 ) + · · · , and, if we define the change of coordinates and parameters y = P −1 (x − x0 ), and β = µ − µ0 , (12) then, system (1) transforms into

Proof. If p = (p1 , p2 , p3 )T and k = (k1 , k2 , k3 )T , p = (p1 + k2 , p2 + k3 , p3 )T , then, we define k2 = −p1 , k3 = −p2 , and p3 = p3 . From lemma 1,   0 1 e 2 (z) = z T Dz e =  0 , G 2 e 23 (z) G   ! 0 2α1 α2 α3 e =  0 , where D e3 = α2 2α4 0 , it then, if D e α3 0 0 D3 follows that   0 e =  0 . kT D e3 kT D Then, after some calculations, and appropriate values of lij , it follows that

y˙ = P −1 F (P y + x0 , β + µ0 ) = Jy + P −1 Fµ (x0 , µ0 )β 1 + P −1 D2 F (x0 , µ0 )(P y, P y) 2 +P −1 Fµx (x0 , µ0 )(β, P y) + · · · . (13) To rewrite (13), first, let us define Definition 1. If v = (v1 . . . , vr )T ∈ Rr and M = T ( M1 , . . . , Mr ) , with Mj ∈ Rp×q , then, v · M = Σrj=1 vj Mj ∈ Rp×q . If F = (F1 , F2 , F3 )T , then F1µ (x0 , µ0 ) F2µ (x0 , µ0 ) F3µ (x0 , µ0 )

Fµ (x0 , µ0 ) =

α2 p1 + α3 p2 − b31 , k1 = 2α1 b32 = b32 − b21 + α2 k1 − 2α4 p1 ,



2

! , 3×m

 (D F1 (x0 , µ0 ))3×3 D2 F (x0 , µ0 ) =  (D2 F2 (x0 , µ0 ))3×3  , (D2 F3 (x0 , µ0 ))3×3 ! (F1µx (x0 , µ0 ))m×3 Fµx (x0 , µ0 ) = (F2µx (x0 , µ0 ))m×3 . (F3µx (x0 , µ0 ))m×3

b33 = b11 + b22 + b33 + α3 k1 . To proof proposition 2, just define ε = (ε1 , ε2 , ε3 )T , with 20

CHAOS'12 June 20-22, 2012. Cancún, México

rij = (w3 · D2 F (x0 , µ0 ))(vi , vj )

Now then,

= < w3 , D2 F (x0 , µ0 )(vi , vj ) > .

P −1 D2 F (x0 , µ0 )(P y, P y) 

but, if wiT = (wi1 , wi2 , wi3 ),

6. TOPOLOGICAL EQUIVALENCE

wiT D2 F (x0 , µ0 )(P y, P y)

Consider the nonlinear system (1) x˙ = F (x, µ), 3 where x ∈ R and µ ∈ Rm , which satisfy H1) − H2). The change of coordinate (12) transforms (1) into (14),

= Σ3j=1 wij D2 Fj (x0 , µ0 )(P y, P y)
= y T P T Σ3j=1 wij D2 Fj (x0 , µ0 ) P y
= y T P T (wi · D2 F (x0 , µ0 ))P y, then,

y˙ = Jy + R0 β + β T R1 y + f2 (y) + · · · . Then, our goal in this section is to find under which conditions, systems (14) and (7) e 2 (z) + S0 ε + εT S1 z, z˙ = Jz + G

P −1 D2 F (x0 , µ0 )(P y, P y)  = = = =

 P T (w1 · D2 F (x0 , µ0 ))P y T  P T (w2 · D2 F (x0 , µ0 ))P  y P T (w3 · D2 F (x0 , µ0 ))P

y T P T P −1 · D2 F (x0 , µ0 ) P y

y T P −1 · D2 F (x0 , µ0 ) (P, P ) y

P −1 · D2 F (x0 , µ0 ) (P, P ) (y, y).

are locally topologically equivalents. Consider the change of coordinates y = z + L0 β + β T L1 z + h2 (z), 3×m



P −1 Fµx (x0 , µ0 )(β, P y) = β T P −1 · Fµx (x0 , µ0 ) P y. We have proof the next Lemma 6. Through the change of coordinates (12), system (1) transforms into y˙ = Jy + R0 β + β T R1 y + f2 (y) + · · · , (14) 1 where f2 (y) = 2 R2 (y, y), and

where e0 = R0 + JL0 , R e1 = R1 + LT0 R2 − R e0T H + JL1 − L1 J, R fe2 (z) = f2 (z) + Jh2 (z) − Dh2 (z)Jz.

R0 = P −1 Fµ (x0 , µ0 ),

Proof. Similar to the proof of lemma 3.

−1


R1 = P · Fµx (x0 , µ0 ) P,
R2 = P −1 · D2 F (x0 , µ0 ) (P, P ). If we define R1 =

Then to proof that systems (14) and (7) are locally topologically equivalents, we have to proof that there exist L0 , L1 , h2 , and ε ∈ R3 , such that

! e0 β = S0 ε, R e 1 = ε T S1 , β R

, then, for j = 1, 2, 3,

T

R1j = (wj · Fµx (x0 , µ0 ))( v1 v2 v3 ), that is, the column k of R1j , denoted by k R1j

(17) 1 T 2 z Hz.

3×(m×3)

where L0 ∈ R , L1 ∈ R , and h2 (z) = Lemma 7. System (14), through the change of coordinates (17), transforms into e0 β + β T R e1 z + fe2 (z) + · · · , z˙ = Jz + R

In a similar way,

(R11 )m×3 (R12 )m×3 (R13 )m×3

(15)

Remark 2. The function f2 in (14) corresponds to the function G2 in lemma 1, then 1 1 (16) α1 = r11 = < w3 , D2 F (x0 , µ0 )(v1 , v1 ) > . 2 2

w1T D2 F (x0 , µ0 )(P y, P y)  = w2T D2 F (x0 , µ0 )(P y, P y)  , w3T D2 F (x0 , µ0 )(P y, P y) 

k R1j ,

e

is given by

Lemma 8. If such that

= (wj · Fµx (x0 , µ0 ))vk ,

for k = 1, 2, 3. Now then, if we define R2 =

f2 (z) = G2 (z). T w3 Fµ (x0 , µ0 ) 6= 0, then

(R21 )3×3 (R22 )3×3 (R23 )3×3

(18) (19)

e

there exist L0 and ε

e0 β = S0 ε. R

! Proof. Observe that

,

 w1T Fµ (x0 , µ0 ) R0 = P −1 Fµ (x0 , µ0 ) =  w2T Fµ (x0 , µ0 )  , w3T Fµ (x0 , µ0 )  T  L01 then, if L0 =  LT02 , we have that LT03  T  L02 + w1T Fµ (x0 , µ0 ) e0 =  LT + wT Fµ (x0 , µ0 )  . R 03 2 w3T Fµ (x0 , µ0 ) Then, if we define 

then, for j = 1, 2, 3, R2j = (wj · D2 F (x0 , µ0 ))(P, P ) = P T (wj · D2 F (x0 , µ0 ))P  T v1 =  v2T  (wj · D2 F (x0 , µ0 ))( v1 v2 v3 ), v3T that is, k R2j = P T (wj · D2 F (x0 , µ0 ))vk , for k = 1, 2, 3. In particular, if R23 = (rij ), then 21

CHAOS'12 June 20-22, 2012. Cancún, México

LT02 = −w1T Fµ (x0 , µ0 ),

Substituting (21) in (20), we obtain

LT03 = −w2T Fµ (x0 , µ0 ),

e13 = R13 + LT0 R23 − R e0T H3 R   e0T H2 J + R12 + LT0 R22 − R   e0T H1 J 2 , + R11 + LT0 R21 − R

ε1 = w3T Fµ (x0 , µ0 )β, the result it follows. Remark 3. Observe that, for the moment, L01 and ε2,3 are free.

e13 are given by therefore, the columns of R

Now we are going to find under which conditions, satisfy equation (18). Remember that e1 = R1 + LT0 R2 − R e0T H + JL1 − L1 J, R

1 1 1 e13 e0T H31 , R = R13 + LT0 R23 −R   j j 2 e13 R = Σ2j=1 R1,j+1 + LT0 Σ2j=1 R2,j+1   e0T Σ2j=1 H j −R j+1 ,   e3 = Σ3 Rj + LT Σ3 Rj R 13 j=1 1,j 0 j=1 2,j   e0T Σ3j=1 H j . −R j

and observe that ! ! 010 (L11 )m×3 (L12 )m×3 = JL1 = 0 0 1 000 (L13 )m×3 ! ! L11 L11 J L1 J = L12 J = L12 J , L13 L13 J   T L0 R21 LT0 R2 =  LT0 R22  , LT0 R23  T  e0 H R T e  e0T H  , R0 H = R e0T H R

L12 L13 0

! ,

(23)

(24)

Observe that, from (15), 1 LT0 R23

= ( L01 L02 L03 )

r11 r21 r31

! = Σ3i=1 ri1 L0i ,

then, if r11 = 2α1 6= 0 (see (16)),  −1  1 1 e0T H31 + r21 L02 + r31 L03 . e13 R13 − R R = 0 ⇔ L01 = r11 We have proof the next Lemma 9. If α1 6= 0, then there exist L01 , L1 and ε2,3 , such that e 1 = ε T S1 . βT R

then,  e11 R e1 =  R e12  R e13 R   e0T H1 + L12 − L11 J R11 + LT0 R21 − R e0T H2 + L13 − L12 J  . (20) =  R12 + LT0 R22 − R T e0T H3 R13 + L0 R23 − R − L13 J ! 0 0 , with S13 = Besides, remember that S1 = S13 ! 000 0 1 0 . Then, 001  e11 = 0,  R T e T e β R 1 = ε S1 ⇔ R = 0,  T 12 e13 = εT S13 . β R But, e11 = 0 ⇔ L12 = L11 J − (R11 + LT0 R21 − R e0T H1 ), R e12 = 0 ⇔ L13 = L12 J − (R12 + LT0 R22 − R e0T H2 ), R 

Finally, equation (19) it follows from lemma 1. 7. MAIN THEOREM Before to establish the main theorem of the paper, let us define the m-vectors d1 = Fµ (x0 , µ0 )T w3 , 2 e13 d2 = R ,

(25)

3 e13 d3 = R ,

e2 and R e3 are given by (23) and (24), respectively. where R 13 13 Theorem 10. Consider the nonlinear system x˙ = F (x, µ), 3 m where x ∈ R , µ ∈ R , with m ≥ 3, and F sufficiently smooth. Suppose that there exists (x0 , µ0 ) ∈ R3×m , such that

then   e0T H1 J L13 = − R11 + LT0 R21 − R   e0T H2 . − R12 + LT0 R22 − R

(22)

H1) F (x0 , µ0 ) = 0 0 1 0 H2) DF (x0 , µ0 ) ∼ J = 0 0 1 0 0 0 H3) α1 6= 0 (non-degeneracy)

(21)

e13 = ( β T R e1 , β T R e2 , β T R e3 ), Now then, observe that β T R 13 13 13 T and ε S13 = ( 0, ε2 , ε3 ), then,  1 e13 = 0, R T e T 2 e13 β R13 = ε S13 ⇔ ε2 = β T R ,  T e3 ε3 = β R13 .

! (non-hyperbolicity)

H4) {d1 , d2 , d3 } are linearly independents (transversality) where α1 is given by (16), and d1 , d2 , d3 are given by (25). Then, the dynamics around x = x0 , and µ ≈ µ0 , is locally topologically equivalent to the unfolding 22

CHAOS'12 June 20-22, 2012. Cancún, México

z˙1 = z2 , z˙2 = z3 ,

an unfolding. International Journal of Bifurcation and Chaos, Vol. 12, No. 12, 2799-2820. J. Guckenheimer, P.J. Holmes. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin. Y. Kuznetsov. (1995). Elements of Applied Bifurcation Theory. Springer, NY. M. Medved. (1984). On a codimension three bifurcation. Casopis pro pestovani matematiky, 109, 3-26. J. Murdock. (1998). Asymptotic Unfoldings of Dynamical Systems by Normalizing beyond the Normal Form. Journal of Differential Equations 143, 151-190. S. Wiggins. (1990). Introduction to Applied Dynamical Systems and Chaos. Springer, NY. P. Yu, and K. Huseyin. (1990). On generic dynamics related to a 3-fold zero eigenvalue. Dynamics and Stability of Systems, Vol. 5, No. 2. 113-126. P. Yu, and K. Huseyin. (1988). Bifurcations associated with a three-fold zero eigenvalue. Quart. Appl. Math., XLVIM, 193-216.

(26)

z˙3 = ε1 + ε2 z2 + ε3 z3 + where

α1 z12

+ α2 z1 z2 + α3 z1 z3 + α4 z22 ,

ε1 = dT1 (µ − µ0 ), ε2 = dT2 (µ − µ0 ), ε3 = dT3 (µ − µ0 ). Remark 4. Consider the transformation T : Rm → R3 , given by  T  d1 (µ − µ0 ) T (µ) =  dT2 (µ − µ0 )  . dT3 (µ − µ0 ) Then, the bifurcation diagram of the unfolding (26) can be seen on the original m-dimensional parameter space, as the preimage under T . In other words, transformation T permits us to find the bifurcation diagram in Rm , and the dynamical behavior in the original state variables. 8. CONCLUSION For an m-parameterized family of 3-dimensional vector fields, whose Jacobian at an equilibrium point, has a triple-zero eigenvalue with geometric multiplicity one, we have found sufficient conditions for, locally, the dynamics around the singularity and an unfolding of the triple-zero bifurcation, are topologically equivalent. 9. ACKNOWLEDGEMENTS The author M. A. Mazon was supported by CONACYT: MC-Scholarship 205202060. REFERENCES A. Algaba, M. Merino, E. Freire, E. Gamero, J. RodriguezLuis. (2003). Some results on Chua’s equation near a triple-zero linear degeneracy. International Journal of Bifurcation and Chaos, Vol. 13, No. 3, 583-608. F.A. Carrillo, F. Verduzco, J. Delgado. (2010). Analysis of the Takens-Bogdanov bifurcation on m-parameterized vector fields. International Journal of Bifurcation and Chaos, Vol. 20, No. 4, 995-1005. F. Dumortier, S. Iba˜ nez, and H. Kokubu. (2001). New aspects in the unfolding of the nilpotent singularity of codimension three. Dynamical Systems 16, 63-95. F. Dumortier, S. Iba˜ nez. (1996). Nilpotent Singularities in Generic 4-Parameter Families of 3-Dimensional Vectors Fields. Journal of Differential Equations 127, 590-647. F. Dumortier, R. Roussaurie, and J. Sotomayor. (1991). Generic 3-parameter families of planar vector fields, unfolding of saddle, focus and elliptic singularities with nilpotent linear parts. In Bifurcations of Planar Fields, Lect. Notes Math., Vol 1480, Springer-Verlag. F. Dumortier, R. Roussaurie, and J. Sotomayor. (1987). Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. Ergod. Th. and Dynam. Sys. 7, 375-413. E. Freire, E. Gamero, A.J. Rodriguez-Luis, and A. Algaba. (2002). A note on the triple-zero linear degeneracy: normal forms, dynamical and bifurcation behaviors of 23