{'fraction_non_alphanumeric': 0.04742644426945379, 'fraction_numerical': 0.0205812871358007, 'mean_word_length': 3.9669333333333334, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 15, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Normal Forms of Hopf-zero singularity

The Lie algebra generated by Hopf-zero classical normal forms is decomposed into two versal Lie subalgebras. Some dynamical properties for each subalgebra are described; one is the set of all volume-preserving conservative systems while the other is the maximal Lie algebra of nonconservative systems. This introduces a unique conservative--nonconservative decomposition for the normal form systems. There exists a Lie--subalgebra that is Lie-isomorphic to a large family of vector fields with Bogdanov--Takens singularity. This gives rise to a conclusion that the local dynamics of formal Hopf-zero singularities is well-understood by the study of Bogdanov--Takens singularities. Despite this, the normal form computation of Bogdanov-Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic non-zero condition, complete results on the simplest Hopf-zero normal forms are obtained in terms of the conservative--nonconservative decomposition. Some practical formulas are derived and the results implemented using Maple. The method has been applied on the R\"ossler and Kuramoto--Sivashinsky equations to demonstrate the applicability of our results.


Introduction
In this paper we are concerned with computing the simplest normal form of the systeṁ where h.o.t. denotes nonlinear terms (higher order terms) with respect to (x, y, z) ∈ R 3 . The system (1.1) can be transformed into the first level (classical) normal form where a ij , b ij , c ij ∈ R; see also [1,Equation 2.15]. All the existing results on the simplest normal forms of this singularity have only dealt with the cases where a 20 b 10 = 0; see [1,8,9,25,35]. The main reason for this is that terms associated with a 01 , b 10 , c 10 , and a 20 all have the same grades in the usual graded Lie algebraic structures; i.e., the gradings may not distinguish the associated four vector fields. Therefore, one needs to deal with all four terms in the computations which is a tremendously difficult job. A novelty in the results of Algaba et. al. [1] was to notice that the first two non-zero terms (associated with a 20 and b 10 ) play the key role in the computations and thus, generic conditions with respect to a 20 and b 10 are assumed. Then, they obtained the simplest normal forms and the simplest orbital normal forms. Chen et. al. [8,9] approached this family of systems using an essentially different method and provided an independent solid proof for their uniqueness. Yu and Yuan [35] made an efficient computer program to compute the simplest normal form of this family of systems. Some discussions on convergence and divergence of normal forms have been made in [13]. Indeed, we computed the numerically suggested radius of convergence associated with the second level normal forms associated with volume-preserving Hopf-zero singularity. Further, we applied the results of Iooss and Lombardi [11] on the classical normal forms of Hopf-zero singularity to obtain certain optimal truncation with exponentially small remainder. Obviously, the later works equally well for this paper. We do not address the convergence of normal forms in this paper; see [30][31][32] for related results on Bogdanov-Takens singularity. Therefore, all our claims with regards to dynamics analysis are limited to the formal flows of formal systems.
A well-noticed goal of normal form computations is for its possible applications in bifurcation and local stability analysis. We consider the cases that have not already been considered in the literature. Further, our results have other advantages such as more simple computations, dynamically meaningful decomposition of vector fields, revealing some possible symmetries, possible application in developing computer ODE-solvers, classification and introduction of important families of vector fields, descriptive study of the Lie subalgebras, a comparison between dynamics of different singularities, and possible applications for more complicated singularities.
A motivation of this paper is to use the term associated with a 01 as the main player in calculations; this is to complete the existing results on this problem with a quadratic non-zero term. (Throughout this paper the only assumption is a 01 = 0.) The associated grading in our approach distinguishes this vector field (grade 1) from the other three (grade 2). This is achieved through a sl 2 -representation for the classical normal forms. This greatly simplifies the computations; we obtain complete results on the simplest normal form of these systems without any extra generic conditions. Our approach can be applied on many well known models such as Röseler and Kuramoto-Sivashinsky equations.
Dynamically meaningful decomposition of a system is a substantial asset in applying many techniques in the study of nonlinear dynamics. A powerful tool for such kind of decompositions is sl 2 -representation of vector fields. This has been mainly applied on nilpotent singularities. For example, this decomposes a two dimensional Bogdanov-Takens normal form into sum of an Eulerian and a Hamiltonian vector field. This kind of decompositions is greatly appreciated in the global analysis of nonlinear dynamical systems such as Melnikov theory, perturbation theory for normally hyperbolic invariant manifolds, and Kolmogorov-Arnold-Moser (KAM) theory; see [34,Chapter 33]. Furthermore, the dynamics analysis of each decomposed vector field helps in a better understanding of the whole vector field as if each component competes and interacts with the others in order to determine the whole dynamics. For instance, this has been used in the study of reaction-diffusion equations, where reaction and diffusion terms do the interactions and competitions to describe the dynamics.
There are at least two more advantages for such decompositions; there are stated by [6,29]. Different splitting procedures of differential equations can be used in designing geometric integrals. Decompositions might enhance the efficiency of ODE-solvers by knowing an integral for a piece of the vector field. This can also have qualitative and quantitative purposes; see [6]. Barrio and Palacián propose an algorithm working with Lie transformations of ODEs that have been split into conservative and dissipative components. Their algorithm is stated as more efficient than the standard method, since they work with scalar expressions of the conservative part rather than its associated vector field terms.
The computational burden has been the main obstacles of most classification problems in the normal form literature. A systematic approach is required for such computations; a Liegraded structure is an important instrument where grade-homogenous parts of vector fields are simplified inductively. In this process a basis for the space of grade-homogenous vector fields are applied. The mostly used choice for the basis has been the monomial vector fields oblivious of their dynamics. This usually involves large matrices and it is hard to find the patterns of computations. The conservative and nonconservative polynomial vector fields play this role in this paper instead of the monomials. Then, computations yield more solid patterns compared to when monomial vector fields are used. Indeed both transformation generators and normal forms are presented through conservative and nonconservative grade-homogeneous vector fields. This is a new feature of this paper that distinguishes our results from the existing results on the simplest normal forms of this singularity. This is accomplished via a sl 2 -representation for the classical normal form vector fields. This may be surprising to see the theory of sl 2representation is applied to a non-nilpotent system. Another novelty of our results is to use sl 2 -style for the second level normal forms; sl 2 -styles have only been used for the first level normal forms in the existing literature.
Any Lie (sub)algebra structure may have interesting dynamics interpretations and normal form theory provides a powerful tool for such descriptions. For our first instance, recall that a Lie subalgebra generates a group of transformations and the group establishes an equivalence relation. Hence, it gives us a classification within the Lie algebra through infinite level normal forms. These usually introduce important families of vector fields. Here, we denote L for the Lie algebra generated by the first level Hopf-zero normal forms. Two transversal Lie subalgebras for L are presented. These represent the quasi-Eulerian vector fields and volume-preserving vector fields with a first integral. The second and more important example is described as follows. There exists a Lie subalgebra from L that is Lie-isomorphic to a Lie subalgebra L b from the Lie algebra generated by all two dimensional vector fields with Bogdanov-Takens singularity; see Theorem 2.5. This gives rise to the fact that local dynamics of any planar reduced (by ignoring the phase coordinate) system from Hopf-zero singularity can be embedded into the flow of a Bogdanov-Takens singularity; see Theorem 2.7. Therefore, the local planar flow associated with Hopf-zero is well-understood by studying that of Bogdanov-Takens; see our further detailed discussion following Remark 2.6. In other words, the local reduced system of Hopf-zero holds less complexity than Bogdanov-Takens. The Lie isomorphism further provides an explanation for why the sl 2 -representation works fine for non-nilpotent singularities and also suggests that our techniques may be applicable to some other non-nilpotent singularities.
The rest of this paper is organized as follows. The conservative-nonconservative decomposition for normal forms are introduced in Section 2. This is achieved by a sl 2 -representation for L and presenting two transversal Lie subalgebras. Some properties associated with each family are described. The second level normal form is computed in Section 3. In Section 4 we obtain the simplest normal forms. The procedure is divided into three cases and accordingly their simplest normal forms are computed in three subsections. Some practical formulas are derived and applied on Rössler and Kuramoto-Sivashinsky equations in Section 5.
Thereby, terminologies of "system", "vector field" and "differential operator" are exchangeably used. For any vector field w, we define Now L is a Lie algebra by the Lie bracket ad v (w) := [v, w] = vw − wv for any v, w ∈ L . Any sl 2 is a Lie algebra and is represented by a triad {M, N, H}. In this paper they are introduced by It The idea is to construct certain orbits from which the vector fields from L are decomposed in terms of the orbit elements. This is done by the following definition. This provides a tool to use a similar approach to the method applied by Baider and Sanders [3][4][5]. These give rise to the following theorem.
Theorem 2.2. The formulas for F l k , E l k and Θ l k in terms of (x, y, z)-coordinates are given by A straightforward calculation proves the following lemma.
Lemma 2.3. The structure constants for the Lie algebra L is governed by The algebra of first integrals for any vector field v = a −1 Hence, any v ∈ T has only phase components in cylindrical coordinates. The set of all formal first integrals for v ∈ T is the algebra generated by x and y 2 + z 2 ; see [13, The following theorem indicates that the space of all volume-preserving and conservative normal forms is given by F ⊕ T; see [24] for relevant results on three dimensional volumepreserving vector fields and their normal forms. Furthermore, it recalls that vector fields from E are nonconservative.
Theorem 2.4. The following holds.
1. Any differential system governed by Equation (1.2) is associated with a vector field v ∈ L and vice versa.
2. The vector spaces F , E , and T are transversal Lie subalgebras in L , i.e., 3. The Lie subalgebra T is a Lie ideal for L . Thus, F ⊕ T is also a Lie subalgebra for L .
4. The algebra of first integrals for any nonzero element from T is < x, y 2 + z 2 > .

Let
6. There is no nonzero first integral for any nonzero vector field from E . Furthermore, nonzero vector fields from E are not volume preserving.
7. The space F ⊕ T is the maximal vector space of volume-preserving vector fields. The algebra of first integrals for any v ∈ F ⊕ T is nontrivial.
8. The flow generated by any v ∈ T is static in the amplitude and x-coordinates. In other words, it can only be dynamic in the phase coordinate.
9. For any v ∈ F ⊕ E , the generated flow is static in the phase coordinate.
Proof. The space L is defined such that claim 1 holds. The proofs for 2 and 3 follow Lemma 2.3. The claims 4, 5 and 6 are proved by [ , respectively. Since L = E ⊕ F ⊕ T, the proof of 7 is straightforward. Claim 8 is true because T is generated by all vector fields from L that have (only) nonzero phase components.
Since F ⊕ E is transversal to the Lie algebra T, the flow generated by v ∈ F ⊕ E is static in phase coordinate.
Theorem 2.5. The space F ⊕ E is a Lie algebra and there exists a Lie-isomorphism h to a proper Lie subalgebra L b of the Lie algebra generated by all two dimensional Bogdanov-Takens singularities.
Proof. Denote L BT for the Lie algebra generated by all vector fields of the form The negative sign is chosen such that it matches with the notation of [5]. Define a map Remark 2.6. The claim 5 from Theorem 2.4 implies that f l k (x, y, z) := x l+1 (y 2 + z 2 ) k−l+1 is associated with F l k in L and from [5] Here the sl 2 -orbits given by (2.1) are terminated when they lead to non-permissible vector fields (i.e., l is increased to k + 1), while the sl 2 -orbits given by [5, Equations 3.6a and 3.6b] are terminated by zero (i.e., A −2 k = 0). Now we conclude that the local dynamics of a Hopf-zero normal form is well-understood by the study of Bogdanov-Takens singularities. The basis of our claim is as follows. There is a common practice to ignore the phase component of the Hopf-zero normal form and obtain a planar reduced system. Then, study the dynamics of the planar reduced system and extract the full three dimensional dynamics from the planar reduced system; see [2,18,[20][21][22]. Given this, we prove that for any planar reduced system obtained from a Hopf-zero normal form, there exists a Bogdanov-Takens singularity from L b such that the flow of the planar reduced system is embedded into the flow of Bogdanov-Takens singularity; see Theorem 2.7. It is interesting to note that the dynamics of Bogdanov-Takens singularities are expected to be more rich than that of the planar reduced systems obtained from Hopf-zero. This is because of 8 two reasons. First, the embedding between the two flows assigns the square of the amplitude variable from the reduced systems into a state variable from Bogdanov-Takens systems that is not necessarily nonnegative. This excludes certain dynamical complexities. Second (and more important than the first), a complement space to L b is expected to absorb most of the dynamics of Bogdanov-Takens singularity. This is because all nonzero nonlinear terms from L b appearing in a Bogdanov-Takens singularity are simplified in the sl 2 -classical normal forms; see Remark 3.2. This contributes to the possible exclusion of many complex dynamical behaviors.
Theorem 2.7. For any Hopf-zero classical normal form, there exists a Bogdanov-Takens singularity such that the flow generated by the planar reduced system (associated with Hopf-zero) is embedded into the flow generated by the Bogdanov-Takens system.
Proof. Consider the vector fields from L in cylindrical coordinates. Now for any v ∈ L , let v :=ṽ +v, whereṽ ∈ T andv ∈ F ⊕ E .
Define w := h(v) ∈ L b andv as the two dimensional vector field obtained fromv. Denote nonnegative real numbers by R + . The changes of variableš is a homeomorphism transformingv into w. Hence,ȟ embeds the flow ofv into the flow of w. This completes the proof.
Notation 2.8. Throughout this paper we use Pochhammer k-symbol notation, that is, for any natural number k and real number b. Further, for integer numbers m, n, N, we denote when there exists an integer k such that m − n = kN.
where v k ∈ L k and L k denotes homogenous vector fields with grade k for any k. Let Next we inductively define There exists a complement space C n,N such that im L n,N ⊕ C n,N = L n . The vector field is called an N-th (infinite) level normal form when w n ∈ C n,N (w n ∈ C n,n ) for any n. Then, there exist invertible transformations such that v can be transformed into the n-th and infinite level normal form; see [15].

The second level normal forms
In this section we compute the second level normal form of the system (1.1). Any such system is transformed into Equation 1.2 and then by the case 1 of Theorem 2.4, it can be represented by where Θ and θ are denoted for the new and old phase variables, respectively. The changes of variables ϕ omits Θ 0 0 from the vector field v (2) . Once the simplest normal form computation is exhausted, the map ϕ −1 adds Θ 0 0 back into the system; also see [14,Theorem 4.1]. This technique has been used in perturbation theory, see [26,Lemma 5.3.6]. This was first brought to our attention by James Murdock. This is new in normal form theory and substantially reduces the computational burden. After eliminating Θ 0 0 , we may change a 0 to any arbitrary numberã 0 via the time rescaling t :=ã 0 a 0 τ where t and τ denote the old and new variables associated with time, respectively. Therefore, without the loss of generality we may assume Remark 3.2. All nonlinear Hamiltonian and Eulerian terms that appear in the first level normal form of Bogdanov-Takens singularity in [5] are of the form A −1 k and B 0 k while these terms do not belong to L b ; see Theorem 2.5. Therefore, the computations in hypernormalization steps in this paper are essentially independent from [5] despite similarity of the procedures and the existence of a Lie-isomorphism between L b and F ⊕ E .
For hypernormalization we need to use a normal form style. A style is a rule on how to choose complement spaces in the normal form computation. We use the sl 2 -style for the second level normal forms. The sl 2 -style in [5] states that the only nonlinear terms from ker ad A −1 0 can stay in the classical normal forms. Those are A −1 k and B 0 k for any k; see [5]. Lemma 3.1 proves that the terms F k k , E k k , and Θ k k (see Equations (2.1)-(2.3)) may stay in our second level normal form system. These are described by ker ad F 1 0 , if we would define F 1 0 . Indeed, we embed L into a bigger algebra (i.e., as a proper Lie subalgebra), where the bigger algebra contains F 1 0 . As far as algebra is concerned, this is simply performed via a simple generalization of Lemma 2.3. However, this extension may not have a justification in nonlinear dynamics; i.e., assigning any term in an ODE system or a near-identity transformation to those extra generated vector fields (beyond L ) is not permissible. Indeed, from the bigger algebra, F 1 0 is only used in definitions of G given by Equations (4.6), (4.20) and (4.32). The map G is merely an instrument that substantially facilitates the computation. The original ideas behind definition of G and our gradings come from [5].
Once the second level normal form is calculated, the sl 2 -style is extended to a formal basis style. Formal basis style uses an order on, the second level, normal form terms to distinguish between alternative terms for elimination. Here, we give priority of elimination to conservative terms over nonconservative terms of the same grade in style I and vice versa in style II; see [15,16,27,28] for further information on formal basis style. Let be a basis for the vector space L N and δ denote for a grading function. Two formal basis styles are defined through the following orderings on B N and are used in this paper.
• Style I: E l k ≺ F m n ≺ Θ i j for any l, k, m, n, i, j. Furthermore, F l k ≺ F m n , E l k ≺ E m n and Θ l k ≺ Θ m n if k < n. This gives a priority for elimination of E l k -terms rather than F m n -terms.
• Style II: F m n ≺ E l k ≺ Θ i j for any l, k, m, n, i, j. Besides, F l k ≺ F m n , E l k ≺ E m n and Θ l k ≺ Θ m n if k < n. The priority of elimination is with F m n -terms rather than E l k -terms.

The simplest normal forms
Throughout this paper we assume there exist a l = 0 and b k = 0 for some l and k. Define Then, in order to obtain complete results for the simplest normal forms we divide the problem into the following three cases: In this section we compute the simplest normal forms for the three cases (4.2) in the following three subsections.

Case i: r < s.
Assume v (2) and define δ(F l k ) = δ(E l k ) = r(k − l) + k and δ(Θ l k ) = r(k − l) + r + k + 1. plays an important role in further normalization of v (2) . For any arbitrary α ∈ F, α > 0, through changes of variables t := αsign(a r ) a r 1 r+1 τ, x := αsign(a r ) a r 1 r+1 X, y := Y, and z := Z, we can change a r into α sign(a r ). Thereby, without the loss of generality we assume that a r in Equation (4.4) is a non-algebraic number. When it comes to practical normal form computation using a computer, this assumption is not valid; computers do not recognize irrational numbers. However, this does not violate the practical computations; see [13, Remak 3.2] for our reason in details.
Besides, Equations (4.16) represent an infinite level normal form.
Proof. For s ≥ r 2 we change the grading function on Θ-terms by This grading puts the term Θ p 1 p 1 with a lower grading than F r 2 r 2 and E s s . Then, [13,Equation 3.7] implies that F k+1 r eliminates Θ 2k(r+1)+r+p 1 2k(r+1)+r+p 1 for any k ∈ N in the r 2 -th normal form level. Now assume that s < r 2 and the grading function follows Equation (4.3). By Theorem 4.2 there exist state solutions E m n , F m n , for (m, n) = j(r + 1) + s − 1, 2k + rj + s , such that 2 2k−j 2k(r + 1) + s + 2 (j + 1)(r + 1) + s and g l follows Equation (4.14). Since a r is changed to be a non-algebraic number, for any k ∈ N either E 2k(r+1)+r+s 2k(r+1)+r+s or F 2k(r+1)+r+s 2k(r+1)+r+s can be eliminated from the (s + 1)-th level of normalization.

Case ii: s < r.
For this case we assume The grading function is defined by δ(F l k ) = δ(E l k ) = s(k − l) + k and δ(Θ l k ) = s(k − l) + k + s + 1.
where in Then, for h 0 = 0 and < 0. By an induction argument we may conclude that for any odd number k the number h k is negative and h k is positive for any even number k. Therefore, h k = 0 for any k and by Equations (4.24-4.26), the proof is complete. We only use style II in the following theorem. Assume that the (s + 1)-th level coefficients a i and b j are nonzero for some i, j ≥ 1. Define s 1 := s, s 2 := min j | j > s , r := min i | a i = 0, i ≥ 1 . Theorem 4.6. There exist invertible transformations sending (1.1) into the (r + s + 2)-th level normal form where α k = 0 for k ≡ s+1 2s, β k = 0 for k ≡ s+1 s, γ k = 0 for k ≡ s+1 s, and if s 2 < r, we have β k = 0 for k = s 2 + s 2 + s, when s 2 = r, α k = 0 for k = s 2 + s 2 + s, Furthermore, Equations (4.27) is indeed the infinite level normal form.
Proof. For s 2 < r, by Lemma 4.5 there exists a B s 2 s+s 2 such that E s 2 +s 2 +s s 2 +s 2 +s .
So, in both styles we can remove E s 2 +s 2 +s s 2 +s 2 +s from the system. For s 2 = r, by Lemma 4.5 there exist A s 2 s+s 2 and B s 2 s+s 2 such that Since both coefficients are nonzero, the proof is complete.

Case iii: r = s.
Let 28) and assume that as bs is a non-algebraic number. This section is similar to the case III of Baider and Sanders [4] which is the most difficult case of the three. We essentially use the idea of Wang et al. [19,33]; they assume a certain ratio be non-algebraic. This prevents that the ratio be a root for any polynomial that appears in the computations and the results readily follows. When the corresponding fraction is an algebraic number, both problems (the case in this subsection and the third case of [4]) remain unsolved. Denote where in • style I, β k = 0 for k ≡ s+1 −1 and k ≡ s+1 2s, and for k ≡ s+1 −1 we have γ k = 0.
Proof. Define G := ad(F 1 0 ) • ad(X s ) (4.32) and Hence, and w −1 0 = u 0 0 = 0. The sequences u i m and w i m for i = 0, −1 and m > 0 follow the recurrence relations On the other hand [X s , E 0 k ] = 2b s k+1 u 0 k+1 (k + 1)(s + 1) + 1 F ks+k+s ks+k+s + w 0 k+1 (k + 1)(s + 1)E ks+k+s ks+k+s , Using the recurrence relations (4.34-4.37), one can compute u −1 k+2 , w −1 k+2 , u 0 k+1 , and w 0 k+1 in terms of polynomials of as bs with rational coefficients. Since as bs is non-algebraic, all these four numbers are not zero and the proof is complete by the rules established in styles I and II.
Remark 4.8. The condition as bs being a non-algebraic was an essential assumption in the proof of Theorem 4.7. However, we usually truncate the normal form system up to certain degree, say N. Then, we may only need that as bs be distanced from the zeros of the polynomials generated for computing u −1 k+2 , w −1 k+2 , u 0 l+1 , and w 0 l+1 for k = 0, 1, . . . , ⌊ N −3 s+1 ⌋ and l = 0, 1, . . . , ⌊ N s+1 ⌋ − 1, for an instance of this see Example 5.4.

Examples
In this section we provide the formulas for the first few coefficients of the simplest normal forms in terms of the coefficients of the original system (before classical normal forms). These are very useful for practical applications. Then, they are applied on the Rössler and Kuramoto-Sivashinsky equations to demonstrate the applicability of our results. The following proposition presents the formulas for the first few coefficients of the classical normal form in terms of conservative-nonconservative decomposition introduced in this paper.
Proposition 5.1. Consider a differential system where c 0,2,0 + c 2,0,0 = 0. Denote its cubic-truncated classical normal form by Then, Proof. The formulas are derived by explicit calculation. They are implemented by treating the coefficients as symbols in Maple and verified by the program.
We are interested to derive formulas for the coefficients of the simplest normal forms in terms of the system (5.1). Given the formulas (5.2-5.4), we may present the formulas in terms of the coefficients of classical normal forms. Instead, we define the numbers a 0 , a 1 , a 2 , a 3 , b 1 , b 2 and c 1 in terms of the coefficients of classical normal forms by This system has Hopf-zero singularity for two sets of parameters. One is for parameter values a = c, b = 1, 0 < a 2 < 2, (5.7) while the system associated with the other set of parameter values has a simple dynamics; see [1]. Hence, we denote the vector field associated with (5.6) by R a . Therefore, Proposition 5.1 implies that the classical normal form of Equations (5.6) is given by Assume that the solutions are traveling waves h(x, t) := u(x + µt). This is a common technique in transforming a partial differential equation into a system of ordinary differential equations. This gives rise toẋ = y, for some parameter µ. For µ = 0, the system has a Hopf-zero singularity at the origin; see Chang [7]. Our formulas for classical normal form of up to degree six is given by Since the coefficients of F and E-terms are not changed in the second level normal form, s = 1 and a 1 b 1 = 2. This falls within the case iii (see Subsection 4.3), while the ratio a 1 b 1 is a rational number. Yet the number a 1 b 1 = 2 is distanced from the zeros of the generated polynomials (see Remark 4.8 and the last paragraph in the proof of Theorem 4.7) in the computation of u −1 k+2 , w −1 k+2 , u 0 k+1 , and w 0 k+1 for k = 0, 1, 2. Hence, the infinite level normal form truncated up to degree six is governed by where style I is used. The equivalent infinite level normal form system in cylindrical coordinates isẋ = ρ 2 − 2x 2 + 187 15 x 4 + 754127923063 747079200 x 6 , ρ = xρ − 374 15 x 3 ρ − 2262383769189 747079200 x 5 ρ, (5.10) θ = 1.