Orbital Hypernormal Forms

: In this paper, we analyze the problem of determining orbital hypernormal forms—that is, the simplest analytical expression that can be obtained for a given autonomous system around an isolated equilibrium point through time-reparametrizations and transformations in the state variables. We show that the computation of orbital hypernormal forms can be carried out degree by degree using quasi-homogeneous expansions of the vector ﬁeld of the system by means of reduced time-reparametrizations and near-identity transformations, achieving an important reduction in the computational effort. Moreover, although the orbital hypernormal form procedure is essentially nonlinear in nature, our results show that orbital hypernormal forms are characterized by means of linear operators. Some applications are considered: the case of planar vector ﬁelds, with emphasis on a case of the Takens–Bogdanov singularity.


Introduction
The theory of normal forms is a basic tool for the study of several problems in differential equations: bifurcations, analysis of stability, the center problem, the reversibility problem, the integrability problem, etc. The basic concepts and different approaches for this theory can be found, for instance, in [1][2][3][4][5][6][7]. The classical theory of normal forms, established by Poincaré, is applied to systems with nonzero linear parts and uses near-identity transformations to eliminate nonessential terms in the local dynamical behavior of the system. To be precise, let us consider a smooth autonomous system 9 x " Fpxq, with x P R n , having an isolated equilibrium point at the origin (i.e., Fp0q " 0). Usually, the normal form technique is used to simplify the vector field F degree by degree, through Taylor expansions. To achieve the quoted simplification, one makes successive near-identity transformations of the form x " y`P k pyq, where P k is a k-degree polynomial vector field satisfying the homological equation. In this way, the k-degree term of the Taylor expansion of F is simplified by eliminating the part belonging to the range of the homological operator. An important fact is that the homological operator depends on the linearization matrix DFp0q, and then the structure of the normal form is determined by this matrix.
The classical normal form does not provide, in general, the simplest form, and further reductions are possible, leading to simpler normal forms.
A way to obtain simpler normal forms (initiated in [8][9][10][11]) is based on the structure of the solutions set of the homological equation. If the kernel of the homological operator is nontrivial, then the homological equation has infinitely many solutions that depend on arbitrary parameters, and one could select them in order to obtain additional simplifications in the higher-order normal form terms. In this context, a difficulty arises because determining the above-mentioned arbitrary parameters can lead to nonlinear equations. In [8,[12][13][14][15], only linear procedures are used in the further simplification procedure, and the resulting normal form is called the mth order normal form, Poincaré renormalized form or unique normal form. In [16], the simplest normal form obtained by linear procedures is referred to as the pseudo-hypernormal form, in contrast to the simplest normal form, called the hypernormal form, which a priori involves nonlinear procedures. Moreover, in [14], it is shown that this simplest normal form under conjugation can be obtained through linear procedures.
Another means of obtaining further simplifications in the classical normal form refers to the kind of expansion used for the vector field. Instead of the Taylor series, one can work in the framework of graduated Lie algebras (see [7,[17][18][19]), where the vector field is expanded as the sum of quasi-homogeneous terms, as is usually done in the blow-up techniques for determining the topological type of a singularity (see, e.g., [1,18,20]). The use of quasi-homogeneous expansions gives rise to a theory, similar to the classical one, but now the homological equation depends on the lowest-degree quasi-homogeneous term (called the principal part) of Fpxq, which plays the role of the linear part.
Another possibility of obtaining further simplifications in the classical normal form is based on the use of transformations not only in the state variables but also in the time (i.e., one can use equivalence instead of conjugation). This kind of transformation does not change the orbit's structure; only the speed along the trajectories can vary.
This idea can be formalized by defining some homological operators that take into account the effect of time-reparametrizations (see [21]).
The main goal of this paper is to show that the simplest normal form using equivalence (called the orbital hypernormal form) can be characterized by means of linear procedures, with reduced time-reparametrizations and near-identity transformations. Although there are many works devoted to the analysis of hypernormal forms, they are restricted to cases of specific singularities (see [22][23][24][25][26][27]). Here, we present a general approach, valid for any singularity, based on Lie transformations and using restricted operators, which means minimizing the computational effort in the calculation of hypernormal forms.
The orbital hypernormal form is of primary importance in several problems of great interest in the qualitative theory of dynamical systems. In fact, it is unique (if the complementary subspaces to the range of the homological operator are fixed) and it determines the invariants of the vector field. For instance, in the analysis of the center problem for the nilpotent singularity carried out in Proposition 6, we use an orbital hypernormal form (see Theorem 5), which shows the invariants that prevent the center conditions. Moreover, the orbital hypernormal form is the starting point in the study of local bifurcations in linear degeneracies such as saddle-node-Hopf, Hopf-Hopf and triple-zero cases (see [28]), as well as their nonlinear degenerate cases (see [29][30][31]). It is of great interest in the study of bifurcations in control systems (see [26,32]), in the study of the orbital reversibility problem, because the orbital hypernormal form uses as evidence the invariants that prevent this kind of symmetry (see [33][34][35][36][37]), as well as in the study of the center and integrability problems (see [38][39][40][41]).
There are different methods for the effective computation of normal forms. One method is the straightforward use of the near-identity transformations. Another one, which appears to be computationally more efficient, is the use of Lie transforms (see, e.g., [7,10,16,42,43]), where the change in variables is understood as the flow of the autonomous system generated by a vector field called the generator of the change. In the present work, we use this last approach. All the results presented here must be understood in a formal sense, because we will not address the convergence problem for the normal forms.
Summarizing, in what follows, we give a rough description of the contents and main results of this paper. In Section 2, we present the definitions and properties related to quasi-homogeneous vector fields, with special emphasis on the Lie formalism, where our analysis lies.
In Section 3, we present the basic ideas of the classical orbital normal form theory and define the concept of the orbital hypernormal form. The main result of this section is Theorem 2, which states that the computation of orbital normal forms can be carried out taking reduced time-reparametrizations and generators, which is of primary interest in the applications, because a drastic reduction in the computational effort is achieved.
In Section 4, we present a procedure to obtain an orbital hypernormal form for a given vector field. Although this procedure is essentially nonlinear, we show that the simplest normal form is characterized by means of a suitable linear homological operator (see Theorem 3).
In Section 5, we show that orbital normal forms agree with orbital hypernormal forms (i.e., no further simplifications are possible) if the kernel of the orbital homological operator is trivial. Finally, in Section 6, we give some results to compute orbital hypernormal forms in the case of planar systems, which are applied to the analysis of a Takens-Bogdanov singularity.

Basic Definitions and Technical Tools
Recall that a function f of n variables is quasi-homogeneous of type t " pt 1 , . . . , t n q P N n and degree k if f pε t 1 x 1 , . . . , ε t n x n q " ε k f px 1 , . . . , x n q. The vector space of quasi-homogeneous functions of type t and degree k will be denoted by P t k . A vector field F " pF 1 , . . . , F n q T is said to be quasi-homogeneous of type t and degree k if F j P P t k`t j for j " 1, . . . , n. We will denote by Q t k the vector space of quasi-homogeneous vector fields of type t and degree k.
If we denote E " diag`ε t 1 , . . . , ε t n˘, then: Expanding the vector field of system (1) as the sum of quasi-homogeneous terms of type t, we can write the above system as 9 x " Fpxq " F r pxq`F r`1 pxq`¨¨¨, where F k P Q t k for all k. The lowest-degree quasi-homogeneous term F r ‰ 0 (where r P Z) is the principal part of the vector field F with respect to the type t. Taking the type t " p1, . . . , 1q, Taylor expansions of vector fields are used. Instead, a subtle selection of the type t will help us in the normalization procedure because, by using adequate quasi-homogeneous terms, we manage monomials with different homogeneous degrees but the same quasi-homogeneous degree. In particular, the homological operator is not only based on the linear part of the vector field, as in the classical normal form theory, and we could take advantage of the nonlinear terms of the vector field F, which allows further simplifications in the classical normal form, following the former idea of Takens [10].
There are two basic tools in the characterization of the transformed vector field by formal equivalence. The first one is the Lie product, defined by where F, G are smooth vector fields. Its is well-known that it is a bilinear and antisymmetric operation on the space Q t k . The second tool arises when we perform a time-reparametrization depending on the state variables dt dT " 1`µpxq, where µp0q " 0. In this case, the transformed vector field is the original one multiplied by 1`µ.
To take into account the effect of both time-reparametrization and transformations in the state variables, it is enough to combine the above tools. In this respect, it is easy to show that rµF, Gs " p∇µ¨GqF`µ rF, Gs, for any smooth scalar function µ and vector fields F, G. Moreover, from the above equality, it can be easily shown that rµF, Fs " p∇µ¨FqF, and p∇µ¨GqF " rµF, Gs`rµG, Fs´p∇µ¨FqG. (5) Throughout this paper, we will use quasi-homogeneous expansions truncated to some quasi-homogeneous degree. Given a vector field G " G r`Gr`1`¨¨¨e xpanded in quasi-homogeneous terms, we define its quasi-homogeneous k-jet by J k pGq " G r`Gr`1`¨¨¨`Gk .
Sometimes, we need to pick-up the k-degree quasi-homogeneous term of a vector field. As we have already done, we use subscripts to denote its projection on the space of quasihomogeneous vector fields. For instance, rF, Gs k denotes the k-degree quasi-homogeneous term of the Lie product.
There are a number of properties related to the use of quasi-homogeneous expansions, which are proven, e.g., in [17]. Namely, we have that µ k F l P Q t k`l , rF l , G j s P Q t l`j and ∇µ k¨Fl P P t k`l for any µ k P P t k , F l P Q t l , G j P Q t j .

Orbital Normal and Hypernormal Forms
As mentioned before, in this paper, we address the problem of determining the simplest expression to which the n-dimensional system (1) can be reduced by timereparametrizations and near-identity transformations in state variables. Firstly, we recall the basic ideas of the Orbital Normal Form Theory.

Orbital Normal Forms
The classical orbital normal form for system (1) is obtained by splitting the vector field in quasi-homogeneous terms as in (2), and then the simplification procedure, which is performed degree by degree, depends on the principal part F r .
For each k ě 1, the simplifications in the pr`kq-degree quasi-homogeneous term of system (2) are obtained in two steps. Firstly, we reparametrize the time by dt dT " 1`µ k pxq, with µ k P P t k . Then, system (2) becomes dx dT " x 1 " F r pxq`¨¨¨`F r`k´1 pxq`pF r`k pxq`µ k pxq F r pxqq`¨¨¨.
Secondly, we use a near-identity transformation x " y`P k pyq. It is immediate to show that again the transformed system y 1 " Gpyq agrees with the original one up to degree r`k´1, i.e., J r`k´1 pGq " J r`k´1 pFq, and the pr`kq-degree is: where we have introduced the homological operator: As this operator is linear, its range, RangepL k q, is a vector subspace and we can define a co-range (a complementary subspace to the range) of L k in Q t k , which we denote by CorpL k q; that is, Q t k " RangepL k q ' CorpL k q. Then, to simplify the pr`kq-degree quasi-homogeneous term, it is enough to write F r`k " F r r`k`F c r`k where F r r`k P RangepL k q and F c r`k P CorpL k q. By selecting pP k , µ k q satisfying the homological equation L k pP k , µ k q " F r r`k , we can eliminate the part of F r`k belonging to the range of the linear operator L k .
In other words, we achieve G r`k " F r`k´F r r`k " F c r`k , and we can state that this term has been reduced to orbital normal form. The classical orbital normal form theorem arises when we perform formally this procedure for the value k " 1, later for k " 2, and so on. Theorem 1. System (2) can be formally reduced to orbital normal form by a sequence of timereparametrizations and near-identity transformations.

Orbital Hypernormal Forms
The orbital hypernormal form for system (1) is obtained by performing a general time-reparametrization and near-identity transformation, and selecting them to achieve a transformed system that is as simplified as possible. We will see that, in this case, the simplification procedure not only depends on the principal part F r , but also on the higher-order quasi-homogeneous terms.
Next, we use a near-identity transformation and look for the simplest expression that can be obtained. We will introduce the following notation: the transformed of the vector field F of system (1) by a near-identity transformation Φ is denoted by Φ˚F, that is, The orbital normal form procedure tries to simplify, as much as possible, the analytical expression of system (1) using both a nonlinear time-reparametrization dt dT " 1`µpxq and a near-identity transformation Φ. In other words, the goal is to find µ and Φ such that Φ˚pp1`µqFq is as simple as possible.
It is well-known that any near-identity transformation can be understood as the time-1 flow of some autonomous system (see [44]). Namely, any change in variables y " Φpxq can be written as Φpxq " upx, 1q, where u is the solution of the initial value problem:
The vector field U is called a generator of the change. Throughout this article, we will often use generators instead of the change in variables itself. In this case, the transformed vector field is denoted by U˚˚F :" Φ˚F and it can be expressed in terms of nested Lie products (see [7,16,44] and references therein) as U˚˚F " F`rF, Us`1 2! rrF, Us, Us`1 3! rrrF, Us, Us, Us`¨¨¨.
In this context, the orbital normal form procedure consists of determining µ, U such that U˚˚pp1`µqFq is as simple as possible.
From now on, we assume formal expansions for the time-reparametrization and the generator U in quasi-homogeneous terms; that is, Let us introduce the Lie derivative along the principal part F r of the vector field F: This is a linear operator, and then we can define a complement to the range of this operator in P t k , which we denote by This means that On the other hand, we denote by Our first main result states that the computation of orbital hypernormal forms can be achieved taking a reduced time-reparametrization where p µ P À kě1 p P t k and a reduced generator Theorem 2. Let us consider the vector field F given in (2) The proof of the above theorem is presented in Appendix A. Theorem 2 is of primary interest in the applications, because it allows a drastic reduction in the computational effort in the orbital normal form procedure.
For instance, in the analysis of the Hopf normal form, if we take the unit type t " p1, 1q, then the principal part is F 0 " p´y, xq T P Q t 0 , and we have Cor` 2j´1˘" Ker` 2j´1˘" t0u, , for all j ě 1.
Then, the dimension of the subspace p P t k is 0 (if k is odd) or 1 (if k is even), whereas the subspace P t k has dimension k`1. On the other hand, if k is even, p Q t k has dimension 2k`1, whereas the dimension of Q t k is 2k`2 (if k is odd, both spaces have the same dimension). Theorem 2 is also useful in determining the structure of orbital normal forms, because this can be done with reduced generators and reduced time-reparametrizations. In particular, we can restrict the domain of definition of the homological operator given in (6) in the orbital normal form procedure. Proposition 1. Let us consider k P N. Then, because the converse inclusion is trivial. Let us consider U k P Q t k and µ k P P t k . Then, As P t k " Rangep k´r q ' p P t k , we can write µ k " ∇η k´r¨Fr`p µ k for some η k´r P P t k´r and p µ k P p P t k . Then, using (4), we obtain: As δ k´r P Kerp k´r q, we have rF r , δ k´r F r s " 0 (see (4)), and then

Orbital Hypernormal Form Procedure
The orbital hypernormal form procedure consists of determining a generator U P À kě1 Q t k and a time-reparametrization with µ P À kě1 P t k that lead system (2) to its simplest expression. Recall that, from Theorem 2, we can take a reduced generator p U P À kě1 p Q t k and a reduced time-reparametrization with p µ P À kě1 p P t k . The procedure to reduce system (2) to its simplest expression is essentially nonlinear in nature (see (8)). Once this has been done (if it is actually possible), we can state that the system has been reduced to orbital hypernormal form.
The aim of this section is to show that the orbital hypernormal form procedure is feasible and that, in fact, it is essentially linear and can be carried out recursively.
To this end, it is convenient to write the vector field of system (2) as F p0q :" F. Its quasi-homogeneous expansion is where F p0q r`j P Q t r`j , for all j ě 0. In the following subsections, we show how we can simplify as much as possible the quasi-homogeneous terms degree by degree.

Orbital Hypernormal Form of Degree r`1
The first step in the orbital hypernormal form procedure consists of simplifying the pr`1q-th degree quasi-homogeneous term F p0q r`1 , by means of a generator p U 1 P p Q t 1 and a time-reparametrization with p µ 1 P p P t 1 . In this way, the vector field of system (2) is transformed into F p1q :" p U 1˚˚´p 1`p µ 1 qF p0q¯. Its quasi-homogeneous expansion is where F p1q r`j P Q t r`j , for all j ě 1. In particular, the pr`1q-th degree quasi-homogeneous term is given by This fact allows us to introduce the following operator: We observe that we can write F p1q r`1 as We have denoted the above operator by NL to indicate that the operator could be nonlinear (in fact, this happens in the cases that we will present in the following subsections corresponding to higher-degree orbital hypernormal forms). However, in the current case, the quoted operator is linear. Namely, from (8), we have that the pr`1q-th degree quasi- is the homological operator (compare this linear operator with the one defined in (6)).
To reduce pr`1q-th degree quasi-homogeneous term to orbital hypernormal form, we follow the basic idea of the normal form theory. Namely, we consider a complementary subspace Cor´L p1q¯t o the range of the operator L p1q in Q t r`1 , i.e., Then, by splitting F p0q Roughly speaking, the orbital hypernormal form procedure at degree r`1 eliminates in F p0q r`1 the part belonging to Range´L p1q¯a nd then we achieve F p1q r`1 " F p0q,c r`1 P Cor´L p1q¯. Finally, we observe that the operator NL p1q " L p1q depends on F p0q r and we can make explicit this dependence (when necessary) by writing L p1q " L p1q ! F p0q r ) .

Orbital Hypernormal Form of Degree r`2
The second step of the orbital hypernormal form procedure consists of simplifying the pr`2q-th degree quasi-homogeneous term F p1q r`2 of the vector field F p1q . It is done by means of a generator p Since we do not want to modify the pr`1q-th degree term (which has already been simplified in the first step), we choose´p U 1 , p µ 1¯P Ker´NL p1q¯. In this way, the vector field F p1q is transformed into In particular, the pr`2q-th degree quasi-homogeneous term is given by As in the previous case, we define the nonlinear operator The orbital hypernormal form at degree r`2 is obtained by selecting´p U, p µ¯adequately in order to eliminate the part of F p1q r`2 belonging to Range´NL p2q¯. Unfortunately, this is not a feasible task because the above operator is nonlinear. Namely, from (8), we obtain and we can see that p U 1 appears "quadratically" in the last term of the above expression. Therefore, we can define neither complementary subspaces to the range of NL p2q nor the orbital hypernormal form of degree r`2 in a straightforward way. To overcome this difficulty, we notice that Hence, we have As P t 2 " Rangep 2´r q ' p P t 2 , we can writé Observe that η 2´r and p ν 2 depend nonlinearly on p µ 1 and p U 1 . Now, using (4), we can write for some δ 2´r P Kerp 2´r q, p V 2 P p Q t 2 . Therefore, As δ 2´r P Kerp 2´r q, using (4), we obtain r " 0, which leads to the following expression: This fact allows us to define the following linear (homological) operator: Observe that we have where p V 1 " p U 1 , p ν 1 " p µ 1 and p V 2 , p ν 2 have been obtained previously. This means that Range´NL p2q¯Ď Range´L p2q¯.
In fact, in Theorem 3, we will show that Range´NL p2q¯" Range´L p2q¯. Thus, although NL p2q is a nonlinear operator, its range is a subspace of Q t r`2 because it agrees with the range of a linear operator. Hence, we can use the basic ideas of the normal form theory to simplify the pr`2q-order quasi-homogeneous term. Namely, we consider a complement Cor´L p2q¯t o Range´L p2q¯i n Q t r`2 , i.e., Q t r`2 " Range´L p2q¯' Cor´L p2q¯. In this way, we reduce the vector field of system (2) to orbital hypernormal form up to degree r`2 Roughly speaking, the orbital hypernormal form procedure at degree r`2 does not change the quasi-homogeneous term of degree r`1 and eliminates in F p1q r`2 the part belonging to Range´L p2q¯. Then, we achieve F p1q r`1 P Cor´L p1q¯a nd F p2q r`2 " F p1q,c r`2 P Cor´L p2q¯. We finally observe that the operator L p2q depends on F p0q r , F p1q r`1 and we can make explicit this dependence (when necessary) by writing L p2q " L p2q !

Orbital Hypernormal Form of Degree r`N
Let us assume that the vector field of system (2) has been reduced to the following orbital hypernormal form of degree r`N´1: r`j , for all j ě N. Now, we describe the procedure of simplifying the pr`Nq-th degree quasi-homogeneous term F pN´1q r`N´1 of the vector field F pN´1q . We use a generator p The vector field F pN´1q is transformed into F pNq " p U˚˚´p1`p µqF pN´1q¯. Since we do not want to modify the quasi-homogeneous terms having degree less than pr`N´1q (which have already been simplified in the previous steps), we choosé In this way, F pNq agrees with F pN´1q up to degree r`N´1; that is, Moreover, the pr`Nq-th quasi-homogeneous term of F pNq is This suggests that the following nonlinear operator can be defined: The simplification in the r`N degree quasi-homogeneous term is obtained by select-ing´p U, p µ¯adequately in order to eliminate the part of F pN´1q r`N belonging to Range´NL pNq¯. Unfortunately, this is not a feasible task because this is a nonlinear operator. Moreover, as we cannot define complementary subspaces to the range of NL pNq , we cannot define a pr`Nq-order orbital hypernormal form.
To overcome this difficulty, we define the following linear (homological) operator: In Appendix B, we prove the following result.
The above theorem states that Range´NL pNq¯i s a subspace of Q t r`N , because it agrees with the range of the linear operator L pNq . Hence, we can use again the basic ideas of the normal form theory to simplify the pr`Nq-order quasi-homogeneous term. Namely, we consider a complement Cor´L pNq¯t o Range´L pNq¯i n Q t r`N , i.e., Q t r`N " Range´L pNq¯' Cor´L pNq¯.
In this way, the vector field F pN´1q is transformed into where F pNq r`N P Cor´L pNq¯. We notice that the operator L pNq depends on F In summary, a pr`Nq-order orbital hypernormal form for system (2) is where F pkq r`k P Cor´L pkq¯, a complementary subspace to Range´L pkq¯i n Q t r`k , for each k " 1, . . . , N. In this case, L pkq " L pkq

Formal Orbital Hypernormal Form
Let us consider system (2). If the normalization procedure is carried out as described before, first for degree r`1, later for degree r`2, and so on, we obtain a formal orbital hypernormal form for system (2) that corresponds to N " 8.

Definition 1.
A vector field F p8q " ř jě0 F pjq r`j , where F pjq r`j P Q t r`j for j ě 0, is an orbital hypernormal form for system (2) if F pNq r`N P Cor´L pNq¯, for all N P N, where Cor´L pNq¯i s a complementary subspace to Range´L pNq¯i n Q t r`N . In this case, L pNq " L pNq We remark that, if vector field F p8q " ř jě0 F pjq r`j is an orbital hypernormal form for system (2), then we have that F p1q r`1 P Cor´L p1q )¯, and so on.
On the contrary, the vector field F p8q " ř jě0 F pjq r`j is not an orbital hypernormal form for system (2) provided one of the following conditions holds:

etc.
We notice that the orbital hypernormal form procedure provides the simplest analytical expression degree by degree (i.e., no further simplifications are possible).

Orbital Normal Forms vs. Orbital Hypernormal Forms
Obviously, orbital hypernormal forms are simpler than orbital normal forms. Nevertheless, in some situations, classical orbital normal forms agree with orbital hypernormal forms.
The next theorem provides a condition that warrants that the above-mentioned fact occurs.

Theorem 4.
Let us consider the homological operator L k defined in (6) and assume that KerpL k q " tp0, 0qu, for all k P N. Then, classical orbital normal forms agree with orbital hypernormal forms.
The proof of the above theorem is a consequence of Theorem 3 and the following result.

Proposition 2.
Let us assume that KerpL k q " tp0, 0qu, for all k P N. Then, Ker´L pNq¯" tp0, 0qu, and Range´L pNq¯" RangepL N q, for all N P N.
Proof. We use induction on N.
The result for N " 1 is trivial because RangepL 1 q " Range´L p1q¯a nd we assume KerpL 1 q " tp0, 0qu.
Let us assume that the statement is true for N´1, where N ą 1. By the induction hypothesis, we have Ker´L pN´1q¯" tp0, 0qu. Therefore, using Proposition 1, we obtain On the other hand, if Hence, L pNq´p V, p ν¯" L N´p V N , p ν N¯" 0, i.e., p p V N , p ν N q P KerpL N q " tp0, 0qu.

Orbital Hypernormal Forms for Planar Systems
The analysis of normal forms for planar systems and related questions (center problem, integrability, etc.) has been considered in [45]. In this study, a splitting of quasihomogeneous planar vector fields is of great interest. Namely, let us denote the symplectic 2ˆ2 canonical matrix by J "ˆ0´1 1 0˙.
The Hamiltonian vector field defined by a Hamiltonian hpxq, where x " px, yq P R 2 , is denoted by X h " J∇h. Then, any quasi-homogeneous planar vector field F k P Q t k can be univocally written as the sum of a radial vector field and a Hamiltonian vector field: where d k " divpF k q P P t k is the divergence of F k , D 0 " pt 1 x, t 2 yq T P Q t 0 is a radial quasihomogeneous vector field and h k " D 0^Fk P P t k`|t| . Recall that the wedge product of two planar vector fields F, G is defined as F^G " F T JG.
There are two properties that we use in our study of the planar case: the first is Euler's Theorem, which states that ∇µ k¨D0 " k µ k for each µ k P P t k , and also that rF k , D 0 s " kF k , for any F k P Q t k . According to (11), we can write the principal part F r of (2) as F r " X h r`|t|`d r D 0 , where h r`|t| P P t r`|t| , d r P P t r .
Let us denote by r P t k`|t| a complementary subspace to h r`|t| P t k´r in P t k`|t| ; that is, Let also define the linear operator The following result follows from [46] (Theorem 3.18).

Proposition 3.
Let us assume that Ker´r k`|t|¯" t0u. Then, is a complementary subspace to the range of L k . Moreover, KerpL k q " tpη k D 0 ,´rη k q : η k P Kerp k´r qu.
From Proposition 3, we deduce Corollary 1. Let us assume that Kerp k q " Ker´r k`|t|¯" t0u, for all k P N. Then, orbital normal forms agree with orbital hypernormal forms.
We remark that the hypothesis Kerp k q " t0u, for all k P N, holds if, and only if, F r is not polynomially integrable.

A Takens-Bogdanov Singularity
Our goal here is to obtain an orbital hypernormal form for higher-order perturbations of a non-integrable quasi-homogeneous Takens-Bogdanov singularity, which has been analyzed in [46].
According to Propositions 2.5 and 2.13 of the quoted paper, there exists r P N 0 such that the system can be written as 9 x " Fpxq " F r pxq`F r`1 pxq`¨¨¨, (12) and F k P Q t k , for k ě r, being t " p1, r`1q. The principal part is where σ "˘1. As we assume that F r is not integrable, then we have d ‰ 0 if σ "´1, or d R Q X r´1, 1s if σ "`1.
Notice that, for r " 0, the principal part (13) is a vector field associated with a linear system with nonzero trace, and then we deal with a linear focus, node or saddle with nonzero divergence.
Our orbital hypernormal form analysis starts by characterizing the kernel and a complement to the range of the Lie derivative operator (9) associated with the principal part (13).

Proposition 4.
Let us consider k P N, k ą r, and denote by k 1 and k 2 , respectively, the quotient and the rest of the division pk´rq˜pr`1q; that is, k´r " k 1 pr`1q`k 2 , with k 1 , k 2 P N, 0 ď k 2 ď r.
(b) If k 2 " 0, then the complementary subspace to Rangep k´r q is the trivial subspace p P t k " t0u. If k 2 ą 0, then a complementary subspace to Rangep k´r q is p P t k " span ! x k ) .
Proof. As F r is not polynomially integrable, then Kerp k´r q " t0u, and then item (a) holds.
To prove item (b), we first introduce adequate bases for the spaces P t k´r and P t k . We deal with the cases k 2 " 0 and k 2 ą 0 separately.
and a basis of . As dim´P t k´r¯" k 1`1 " dim`P t k˘a nd Kerp k´r q " t0u, we deduce that k´r is onto and then p P t k " t0u.
and a basis of P t k is B k " !
x k 2´1`i pr`1q y k 1`1´i : i " 0, . . . , k 1`1 ) . In this case, we have dim´P t k´r¯" k 1`1 and dim`P t k˘" k 1`2 . Hence, dim´p P t k¯" 1. Next, we determine the matrix of the linear operator k´r with respect to the bases B k´r and B k given before. After some computations, it is easily obtained that k´r´x k 2`i pr`1q y k 1´i¯" pk 2`i pr`1qq x k 2´1`i pr`1q y k 1`1´i`p k´rqd x k 2`r`i pr`1q y k 1´ì σpr`1qpk 1´i q x k 2´1`p i`2qpr`1q y k 1´1´i .
Then, the matrix of the linear operator k´r , associated with the bases given for P t k´r and P t k , is a banded matrix whose non-zero entries are confined to the main diagonal, and to the first and second subdiagonals: where α i " k 2`i pr`1q ‰ 0, for i " 0, . . . , k 1 , β i " pk´rqd, for i " 0, . . . , k 1 , It is a simple matter to show that a complement to the column space of the above matrix is generated by the vector p0, . . . , 0, 1q T . Therefore, is a complementary subspace to Rangep k´r q. Proposition 5. Let us consider k P N, k ą r, and assume that σ "´1, or σ "`1 and |d| ‰ 1`2 pr`1q k´r . Then, Ker´r k`2¯" Cor´r k`2¯" t0u.
Proof. Let us consider the following bases for the subspaces r P t k`2 , r P t r`k`2 : and an arbitrary element r µ k`2 " a 0 x k`2`a 1 x k´r`1 y P r P t k`2 .
Using that y 2 " σx 2r`2´2 h r`|t| , it is a straightforward computation to show that Then, the matrix associated with the linear transformation r k`2 is pk`2q˜d To obtain the result, it is enough to observe that the above matrix is nonsingular if σ "´1 or if σ "`1 and |d| ‰ 1`2 pr`1q k´r .
The next theorem presents an orbital hypernormal form for system (12).
Theorem 5. Let us consider system (12), where F r is given in (13). Let us assume that d P R and also that • σ "´1 and d ‰ 0, or • σ "`1, d R Q X r´1, 1s and |d| ‰ 1`2 pr`1q k´r , for all k P N, k ą r. Then, an orbital hypernormal form is 9 x " F r pxq`8 ÿ j"r`1 jır mod pr`1q where D 0 " px, pr`1qyq T P Q t 0 .
Proof. It is enough to apply Propositions 3-5.
We notice that, if r " 0, then the linearization of system (12) is non-resonant, and the above theorem agrees with the Poincaré Theorem, which states that it is analytically linearizable (see [2]).
The orbital hypernormal form (14) also provides interesting dynamical information for system (12). The next result characterizes the centers of system (12) by means of the orbital hypernormal form (14), which evidences the invariants of the vector field that prevent the center conditions. Proposition 6. The origin for system (12) is a center if, and only if, σ "´1, and the orbital hypernormal form (14) is R x -reversible (i.e., invariant to px, y, tq Ñ p´x, y,´tq).
Proof. The monodromy problem for system (12) has been considered in [47,48], where it is shown that the quoted system is monodromic if, and only if, σ "´1.
The sufficient condition is trivial, because if σ "´1 and the orbital hypernormal form (14) is R x -reversible, then the equilibrium at the origin of system (14) is monodromic and reversible and, consequently, it is a center.
Let us prove the necessary condition. If we assume that the origin of system (12) is a center, then it is monodromic and σ must be equal to´1. Moreover, the origin of the orbital hypernormal form (14) is also a center.
Let us prove by reductio ad absurdum that the orbital hypernormal form (14) is R x -reversible. Let us suppose on the contrary that system (14) is not R x -reversible. Then, there exists j even such that c j ‰ 0 and we denote by j " 2k the lowest index satisfying c 2k ‰ 0. Let us also denote by H the vector field associated with the orbital hypernormal form (14); that is, Notice that the vector field G has a center at the origin because it is monodromic and reversible. On the other hand, after some computations, we obtain H^G "¨G`8 c j x j D 0^Fr " c 2k x 2k D 0^Fr`¨¨¨"´p r`1qc 2k x 2k´x2r`2`y2¯`¨¨¨, where the dots denote higher-order quasi-homogeneous terms. As c 2k ‰ 0, we deduce that H^G is a negative semidefinite function that is nonzero almost everywhere in the neighborhood of the origin. Hence, the origin of H is a focus (stable if c 2k ą 0 or unstable if c 2k ă 0), but this is a contradiction because the origin of the orbital hypernormal form (14) is a center.
As a consequence of the above proposition, we obtain (by using a different approach) the following result of [49].

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Proof of Theorem 2
Let us consider system (2), a near-identity transformation of the state variables associated with a generator U P À kě1 Q t k and a time-reparametrization dt dT " 1`µpxq, with µ P À kě1 P t k . Theorem 2 states that for some p U P À kě1 p Q t k and p µ P À kě1 p P t k . This means that instead of performing on system (2) an arbitrary orbital transformation, we can also do so with a reduced orbital transformation (with the time-reparametrization p µ P À kě1 p P t k and the generator p U P À kě1 p Q t k ). The expression for the transformed vector field U˚˚F by means of the generator U is presented in (7). To write this formula in a compact form, we introduce the following operators: Using the above notation, the expression (8) corresponding to the transformed vector field of F by a generator U and a time-reparametrization dt dT " 1`µpxq reads as U˚˚pp1`µqFq " The proof of Theorem 2 requires the following technical result, which states that the near-identity transformations associated with generators that are multiple of the principal part F r can be avoided in the orbital normal form procedure, since the simplifications obtained through them can also be obtained through time-reparametrizations. Lemma A1. Let us consider the vector field F of system (2). Let also consider α k´r P P t k´r (where k ě r) and µ P À jě1 P t j . Then, there exists q µ " ř jě1 q µ j P À jě1 P t j , such that pα k´r Fq˚˚´p1`µqF¯" p1`q µqF.
Proof. Firstly, we will show using induction that, for each l P N, there exists q µ plq P À jělk P t j satisfying: T plq α k´r F´p 1`µqF¯" q µ plq F, where q µ p1q k "´∇α k´r¨Fr .

Proof of Theorem 2. Let us define
κ " min ! k P N : Proj Kerp k´r qFr pU k q ‰ 0, or Proj Rangep k´r q pµ k q ‰ 0 ) .
If κ " 8, then U P À jě1 p Q t j and µ P À jě1 p P t j , and the result holds trivially taking p U " U, p µ " µ. Let us assume κ ă 8. From the definition of κ, we obtain that U k P p Q k and µ k P p P t k , for k " 1, . . . , κ´1.
We make the following ansatz for p U and p µ: where p U j " U j P p Q t j , µ j " p µ j P p P t j , for j " 1, . . . , κ´1; and we will determine p U j P p Q t j , p µ j P p P t j , for j ě κ, such that equality (A1) holds degree by degree. Firstly, we show how p U κ and p µ κ are obtained by dealing with the κ-degree quasihomogeneous term. Higher-degree terms of p U and p µ can be obtained by repeating the reasoning, and then the proof is completed.
κ " ρ κ´r F r P Kerp κ´r qF r (for some ρ κ´r P Kerp κ´r q) and U p2q κ P p Q t κ .
From Lemma A1, there exists q µ P À jě1 P t j such that where q µ and µ agree up to quasi-homogeneous degree κ´1. Moreover, On the other hand, a generator of the transformation Ψ´1 is´α κ´r F r`¨¨¨, where the dots denote higher-order quasi-homogeneous terms. Using Lemma 2.8 of [35], we obtain that J κ´1 p p Uq " J κ´1 pUq, and

Appendix B. Proof of Theorem 3
Theorem 3 states that Range´NL pNq¯" Range´L pNq¯. This theorem is a consequence of Propositions A1 and A2 below. Their proofs require some technical results.
Lemma A2. Let us consider the vector field F of system (2).
Let also consider U " ř N j"1 U j P À N j"1 Q t j and ν " ř N j"1 ν j P À N j"1 P t j , where N P N, such that J r`N´1´r F, Us`νF¯" 0.
Then, for each µ " ř N j"1 µ j P À N j"1 P t j and l ě 0, we havé Observe that µ plq j does not depend on k, i.e., µ plq j depends univocally on the pj´lq-jet of pU, ν, µq.
Proof. We use induction on l.
For l " 0, the result is trivial becausé where we have introduced µ p0q j " µ j .
Let us consider l ą 0 and assume that the result is true for l´1. Then, As´rF, Us`νF¯r`k " 0 for k " 1, . . . , N´1, we obtain rF, Us r`k´i "´pνFq r`k´i "´k´i ÿ j"1 ν j F r`k´j´i , for i " l, . . . , k´1. Hence, i¨U j´i¯´µ pl´1q i ν j´i¯¸Fr`k´j , and the result is also true for l.
Lemma A3. Let us consider the vector field F of system (2). Let also consider U " ř N j"1 U j P À N j"1 Q t j , µ " ř N j"1 µ j P À N j"1 P t j , where N P N, and denote H " F´U˚˚pp1`µqFq.
Let us assume that ν " ř N j"1 ν j P À N j"1 P t j verifies J r`N´1´r F, Us`νF¯" 0. Then, On the other hand, as we assume that J r`N´1´r F, Us`νF¯" 0, then we obtain rF, Us r`j "´pνFq r`j , for j " 1, . . . , N´1. Therefore, for all l ě 1, we havé T plq U´r F, Us¯¯r`k "´´T plq U pνFq¯r`k, for all k " 1, . . . , N. Lemma A4. Let us consider the vector field F of system (2). Let also consider k, j P N with k ă j, and α k´r P P t k´r . Then, j ÿ i"k´" F r`j´i , α k´r F r`i´k ‰`p ∇α k´r¨Fr`i´k qF r`j´i¯" 0.

Consequently,
Proof. From (3), we obtain j ÿ i"k´" F r`j´i , α k´r F r`i´k ‰`p ∇α k´r¨Fr`i´k qF r`j´i¯" ∇α k´r¨Fr`j´i qF r`i´k`p ∇α k´r¨Fr`i´k qF r`j´i¯.
In the last line, the first sum is zero because the Lie product is anti-symmetric. Moreover, simplifying the second sum by subtracting out the many self-similar terms, it can be easily proven that it is zero and then the proof is completed.
Lemma A5. Let us consider the vector field F of system (2). Let us also consider V " p P t j , such that J r`N´r F, Vs`νF¯" J r`N´r F, p Vs`p νF¯.
If κpV, νq " N`1, then the result holds trivially by taking p V " V, p ν " ν, because V P À N j"1 p Q t j and ν P À N j"1 p P t j . In the case κpV, νq ă N`1, we will show that there exist q V " ř N j"1 q V j P À N j"1 Q t j and q ν " ř N j"1 q ν j P À N j"1 P t j , verifying κp q V, q νq ą κpV, νq, and J r`N´" F, q V ı`q νF¯" J r`N´r F, Vs`νF¯.
Once we prove this, the result is obtained by repeating the reasoning on q V, q ν and so on, until we finally reach p V and p ν satisfying κp p V, p νq " N`1, which, as mentioned before, p P t j , and J r`N´r F, Vs`νF¯" J r`N´" F, p V ı`p νF¯.
Let us denote κ " κpV, νq. We make the following ansatz for q V and q ν: where q V j " V j P p Q t j , q ν j " ν j P p P t j , for j " 1, . . . , κ´1, and q V j P p Q t j , q ν j P p P t j , for j ě κ, will be determined, indicating that (A5) holds.
To define the κ-degree quasi-homogeneous terms q V κ and q ν κ , we notice that rF, Vs`νF¯r`κ " rF r , V κ s`ν κ F r`κ´1 ÿ j"1´" Using that Q t κ " Kerp κ´r qF r ' p Q t κ , we can write V κ " δ κ´r F r`x W κ , for some δ κ´r P Kerp κ´r q and x W κ P p Q t κ .
Moreover, as p P t κ " Rangep κ´r q ' P t κ , we can write ν κ " ∇η κ´r¨Fr`q ν κ , for some η κ´r P P t κ´r and q ν κ P p P t κ . Let us denote by s P t κ´r a complementary subspace to Kerp κ´r q in P t κ´r (i.e., P t κ´r " s P t κ´r ' Kerp κ´r q). Then, we can write η κ´r " η p1q κ´r`η p2q κ´r , where η p1q κ´r P Kerp κ´r q and η p2q κ´r P s P t κ´r . As η p1q κ´r P Kerp κ´r q, we have κ´r´ηκ´r¯" κ´r´η p2q κ´r¯, and consequently, ν κ " ∇η p2q κ´r¨F r`q ν κ . Let us introduce q V κ " x W κ´η p2q κ´r F r . As η p2q κ´r P s P t κ´r , we have η p2q k´r F r P p Q t κ , and consequently, q V κ P p Q t κ . If we denote α κ´r " η p2q κ´r`δ κ´r , then q V κ " V κ´ακ´r F r P p Q t κ , and q ν κ " ν κ´∇ α κ´r¨Fr P p P t κ .
The next propositions show that Range´NL pNq¯a nd Range´L pNq¯a gree.
Proposition A1. Range´L pNq¯Ď Range´NL pNq¯, for all N P N.
Proof. Let us consider p p V, p νq "´ř N´1 j"1 p V j , ř N´1 j"1 p ν j¯`´p V N , p ν N¯b elonging to the domain of definition of L pNq . Then,´ř N´1 j"1 p V j , ř N´1 j"1 p ν j¯P Ker´L pN´1q¯a nd, consequently, J r`N´1´" F, p V ı`p νF¯" 0.
We first show that that there exist U P À N j"1 Q t j and µ P À N j"1 P t j , such that L pNq´p V, p ν¯"´F´U˚˚´p1`µqF¯¯r`N.
Namely, we take U " p V and µ " ř N j"1 µ j P À N j"1 P t j is defined as follows. From Lemma A3, we havé F´p V˚˚pp1`µqFq¯r`N "´´"F, p V ı`p νF¯r`N´pµ 1´p ν 1 qF r`N´1 N´1 ÿ j"2¨µ j´p ν j`j´1 ÿ l"1 µ plq j‚ F r`N´j .
To complete the proof, we use Theorem 2, which states that F´U˚˚´p1`µqF¯¯r`N "´F´p U˚˚´p1`p µqF¯¯r`N, Moreover, we have J r`N´1´F´p U˚˚´p1`p µqF¯¯" J r`N´1´F´U˚˚´p 1`µqF¯¯" 0.
Hence,´p U, p µ¯belongs to the domain of definition of NL pNq , and then NL pNq´p U, p µ¯" L pNq´p V, p ν¯.
Proposition A2. Range´NL pNq¯Ď Range´L pNq¯, for all N P N.
Proof. Let us consider´p U, p µ¯"´ř N´1 j"1 p U j , ř N´1 j"1 p µ j¯`´p U N , p µ N¯b elonging to the domain of definition of NL pNq . We first show that there exist V P À N j"1 Q t j and ν P À N j"1 P t j such that NL pNq´p U, p µ¯"´´rF, Vs`νF¯r`N.
By selecting ν 2 " p µ 2´p µ p1q 1 P P t 2 , we obtain J r`2´" F, p U ı`ν F¯" 0. It is enough to repeat the reasoning for k " 2, . . . , N, to determine ν " ř N j"1 ν j P À N j"1 P t j such that J r`N´1´r F, Vs`νF¯" 0, satisfying NL pNq p p U, p µq "´´"F, p U ı`ν F¯r`N.
To complete the proof, we use Lemma A5, which states that J r`N´" F, p U ı`ν F¯" J r`N´" F, p V ı`p νF¯, This means that´p V, p ν¯belongs to the domain of definition of L pNq , and then NL pNq´p U, p µ¯" L pNq´p V, p ν¯.