On the ultradifferentiable normalization

We show the theory of the formal ultradifferentiable normalization. The tools utilized here are KAM methods and Contraction Mapping Principle in the Banach space fixed with weighted norms.


Introduction
The normal form theory founded for dynamical systems by Poincaré is widely used to analyze local dynamical properties, whose main idea is to simplify systems to proper forms. In the modern language, we seek the simple representation of the equivalent class (X , · )/ ∼, where X is the set of the (local) one-parameter(two-parameters) transformation groups maybe fulfilling some special structures, · is the norm or topology and ∼ refers to the admissible transformations preserving such structures. Many researchers including Dulac, Sternberg, Chen, Birkhoff, Il'yashenko devote their great efforts to the development of the theory. Nowadays, it plays an important role in the study of bifurcations and stabilities and is also widely applied to celestial mechanics, biomathematics, control theory, and so on.
First, let us recall some classical results in the normal form theory. Due to celebrated Poincaré-Dulac reductions, many nonlinear terms can be eliminated by formal coordinates substitutions. Beyond this, in the Poincaré domain systems are always analytically conjugated to their normal forms. However, in the Siegel domain more complicated phenomena arise. Without any small divisor, the analytical normalization was shown in [1] for systems preserving special structures. Then under Siegel's and Bruno's small divisor conditions the corresponding results of the analytical linearization were confirmed. Moreover, by the hyper-bolic condition results about C 1 and C ∞ linearization were got by Hartman and Sternberg, together with the C k and C ∞ normalization by Sternberg and Chen, respectively. See [3] for more details.
Recently, to bridge the gap between analytical and C ∞ conjugations, the ultradifferentiable topology, which is the refinement of C ∞ topology and was once introduced in [10] by Rudin, emerges before our eyes. In the Gevrey smooth case, it was proved in [11] that the Gevrey-α smooth vector fields can be changed into their normal forms by the Gevrey-(α + μ + 1) smooth coordinates substitutions at the origin for the hyperbolic liner part and the Siegel type small divisor condition with the index μ, which was improved into an accurate one in [13] for the diagonal linear part. Meanwhile, more degenerated formal Gevrey-α vector fields were studied in [6] and see also [2] for the Gevrey linearization. Furthermore, the Siegel-Sternberg linearization theorem for ultradifferentiable systems was given by [7]. So, the task of the work is to explore the theorems about the ultradifferentiable normalization.
In general, the ultradifferentiable functions belong to the C ∞ functional class but their derivatives have certain norm controls with respect to the order. More precisely, for the given positive sequence {m(t)} t∈N , the function f is ultradifferentiable on the set U in terms of the wights m(t), provided that for any compact set K ⊆ U , there exist positive constants M and C such that where U is an open set in R d . They are also called to be local Denjoy-Carleman classes of Roumieu type. As shown in [7,11], the whole proofs of convergence of the conjugations shall contain two steps. The one inherited from the C ∞ topology is to utilize the path method to confirm the Chen type theorem that the formal conjugations lead to the real ones by the hyperbolic non-degenerated condition. The other as the extension of analytic topology is to get the loss of smoothness sharper in the formal norms. Here we restrict our attention on the part of formal conjugations. Denote C d [[x]] by the formal Taylor series with complex vector-valued coefficients in C d . Then the function f ∈ C d [[x]] is formal ultradifferentiable with the weight function E(t), provided that there exist positive constant r and Here e j refers to the unit vector with the j-th component 1. The above means that the logs of the weights are convex with respect to t. Particularly, the Gevrey-α smooth function f satisfying f (0) = 0 is of formal Gevrey-α + 1 with weights E(t) = e −(α+1)t ln t . We now generalize results in [7,11] to deal with formal ultradifferentiable normalization by all kinds of small divisor conditions. Consider the system then the small divisor condition is given by : [2, ∞) → R + is the increasing continuous function. By properly choosing the constant c and extending its definition domain, technically we make that is continuous and increasing in [1, ∞) and (1) = 1. Without concerning the case that λ is in the Poincaré domain, (t) is assumed to be either When A is in the Jordan normal form, the formal normal form of (1.1) is by the Poincaré-Dulac formal normal form reductions. Let The following shows that all small divisors can be formally excluded by the proper loss of the ultradifferentiable smoothness.
Additionally, if system (1.1) can be linearized, the above result is also valid as q → +∞.
The next is generalized from [13] to confirm the stop of the loss of smoothness for the 'larger' small divisor. Consider the following small divisor condition (1.2) fixed with for τ ∈ (0, ∞) and μ > 0. It is of the Liuvillean type fulfilling Let E s (t) = e ω s (t) be the weight function, where the logarithm ω s (t) with s = (s 1 , At last, if there is no small divisor, we show that the Gevrey smoothness also does some good to make the clearer classification of nonlinear terms. When A is in the diagonal form, by (1.3) of condition (1.2) the formal normal form of system (1.1) has such further decomposition where γ = O(||x||) is a scalar-valued function as x → 0, the other resonant terms are in g andĝ = O(||x|| 2 ) as x → 0. Denote the lowest degree of resonant monomials inĝ by Ord(ĝ) = q forĝ(x) = |k|≥ p, jĝ k, j x k e j . Here e j refers to the unit vector with the j-th component 1 as above. By results of the classical textbook [1] system (1.1) can be analytic normalization, provided the original system is analytic andĝ vanishes. Particularly, it was proved in [16] that completely integrable systems, which have sufficiently many independent first integrals and are the partial case of [18], shall obey this convergent criterion. Note again that the formal Gevrey-s function f is just formally ultradifferentiable with weight function E s (t) = e −st ln t . Thus, reviewing it in the ultradifferentiable category, we get the following.

Theorem 1.3 Assume that system (1.1) is formal Gevrey-s, A is in the diagonal form and
there exists a formal Gevery-ŝ coordinates substitution turning system (1.1) into its normal form, whereŝ = max{s, 1 q−1 }. Additionally, ifĝ = 0, the above result is also valid as q → +∞.
Here we do several remarks. First, by proper non-degenerated conditions, which were mentioned by Theorem 8.1 (pp. 24) in [7] for our Theorems 1.1 and 1.2 and Theorem 3.2 (pp. 254) in [11] for our Theorem 1.3, the above formal conjugations can be modified into really ultradifferentiable ones. Next, if the matrix A in system (1.1) has the nilpotent part, similar results as Theorems 1.1 and 1.2 under condition (C2) are valid by using ad −1 A o in (6.1) instead. At last, the case admitting Theorem 1.3 appears at the planar degenerated Hopf and homoclinic bifurcations. Since there occurs no small divisor, we think that Gevrey smoothness is enough to catch the order of focus and saddle points quantitatively.
Due to Taguchi's comments in [12], the purpose of Ramis to use formal power series is to overcome some difficulties, when the usual technique of functional analysis breaks down. In the view of our series results, out of the Poincaré domain, it is enough to apply formal Gevrey conjugacy to classify different resonant terms without any small divisors by Theorem 1.3. When Siegel type small divisors appear, the formal loss of smoothness for the normalization stops in the same formal Gevrey class but with different Gevrey indices by [13]. Now turning to (i) of Theorem 1.2, the formal loss of smoothness for the ultradifferentiable normalization stops in the class with different weight functions, while the slight loss must be allowed in (ii) of Theorem 1.2. At last, the large loss occurs to overcome all kinds of small divisors by Theorem 1.1. Therefore, we think that the ultradifferentiable smoothness is useful to detect fine structures between the analytic and C ∞ normalization.
The rest parts are organized as follows. In Sect. 2, notations, basic definitions, and key lemmas are provided. Then Theorems 1.1 and 1.2 are proved in Sect. 3 by Contraction Mapping Principle, while Sect. 4 contains the proof of Theorem 1.2 via KAM steps. At last, we show the connections between Gevrey conjugations and bifurcations as the application in Sect. 5.

Preliminary
In this part, we provide notations, basic definitions, and key lemmas. All notations using frequently in this part are listed as follows.
Denote C d [[x]] by the formal Taylor series with complex vector-valued coefficients in C d . Then we restrict our focus on the set of formal series Here the logarithm weight function ω(t): [1, ∞) → R − is a C 1 strictly decreasing function such that ω(1) = 0, lim t→+∞ ω(t) = −∞, ω (t) is non-positive and decreasing. Moreover, by setting E(t) = e ω(t) , sometimes we use the notation Additionally, denote two index sets by Then setting f r = (k, j)∈ r f k, j x k e j and f nr = f − f r , we make First of all, we study general properties of the formal space (X , f ω,r ), which is the subspace of formal series from C d [[x]] having the finite · ω,r norm.

Lemma 2.1 The set (X , f ω,r ) is a complete Banach space.
Proof For any f ∈ X we build which yields a complete Banach space l 1 , when it is fixed with the norm f = k, j |f k, j |. So it is equivalent to say that the space (X , f ω,r ) is just weighted l 1 , which confirms the result.
Especially, comparing with the classical formal Gevrey norm we would like to use the following one instead For the completeness of the work, we write details down to confirm the equivalence of these norms.

Lemma 2.2 There exists a positive constant
for all t ∈ N. Therefore, from the above, it leads to s e −s|k| ln |k| e s r |k| ≤ e −s|k| ln |k| e s r |k| .
Then from the below, we obtain that This completes the proof.
By those arguments we take (X , · ω,r ) or (X , · E,r ) fixed with the norm given by (2.1) or (2.2) for formal ultradifferentiable functions, respectively. Additionally, we use · s,r instead for the formal Gevrey-s functional class. Then comes the study of the weight function.
Then the following statements hold.
which implies

5)
where c 1 = e −ω(β) and κ 1 (u) = e ω (u)(β−γ ) . Additionally, assume that ω ∈ C l for l ≥ 2 and there exists a constant M > 0 such that |ω (l) (t)| ≤ M for all t ≥ 1, we have that where κ 2 (u) = e P(u) , Therefore, (2.3) of result (a) is verified, which leads to (2.4) by the fact ω(m) ≤ 0 and taking m = 2. Now consider the function , which is non-negative for t ≤ w/2 and non-positive for t ≥ w/2. Thus we obtain that Using the Taylor expansion with the Lagrange type remainder, we have that for 0 ≤ γ < β and ξ ∈ (γ , β). In another form, the above is which confirms (2.5) of result (b). By additional assumptions we use Similarly, we get the control which proves (2.6).
Next, we deal with norms of multiplicities and compositions of formal functions. As usual, denote by the majorant operator Naturally, for the general f ∈ X , we have that (2.9) In the book [15, Lemma 5.10, pp. 51], the following lemma was mentioned.

Lemma 2.4
Let E ≡ 1 and f , g ∈ X . Then the following statements hold.
Here the notation f g denotes | f k, j | ≤ |g k, j | for all possible k and j.
Lemma 2.5 Assume that ω and E are the same as the ones in Lemma 2.3. Then the following statements hold.
by Lemma 2.4(a). Heref is given by formula (2.9). Then we compare the form of typical terms of f • g with ones of f • g. Arbitrary choosing k and rewriting g k in a precise form we obtain that . This completes the proof.
As usual, ∇ x f is the gradient function of f with respect to the variable x. The following two are key to this part.

Lemma 2.6
Assume that ω and E are the same as the ones in Lemma 2.3. Then we have that Proof Using the fact that sup u≥0 {ue −δu } = (eδ) −1 and by (2.5) of Lemma 2.3(b), directly we have that This completes the proof.
Denote the operator P by , and η ≥ 1. Then the following statements hold.
, and s 2 ≥ s 2,0 > 0 for the preassigned positive constant s 2,0 , then inequality (2.10) is still valid for c 4 Here c 2 (β, γ ) and P(u) are the same as the ones in Lemma 2.3(b).
Especially, when we restrict our focus on the normalization with small divisors of the Diophantine type, the classical Gevrey smoothness is proper. In our category, the key lemma in [13] can be represented as follows.
Moreover, in the general case, we have the following lemma.

Lemma 2.9
Assume that q = Ord( f ) = Ord(g) ≥ 2,ω(t) is given by (1.5), E and E are the same as the ones in Theorem 1.1. Then we have that Here constants c 1 and c 2 are the same as the ones in Lemma 2.3(b).

Contraction mapping principle in the ultradifferentiable normalization
In this part, we provide the proofs of Theorems 1.1 and 1.3 via the Contraction Mapping Principle. First, we introduce a criterion to apply the Contraction Mapping Principle. Let F be a function or map on (X , · E,r ). It was mentioned in [15] that the function or map F is strongly contracting, provided that for h E,r ≤ r and h E,r ≤ r . As usual, O(1) refers to the bounded quantity as r → 0. Set L(·, ··) to be a real bilinear form on (X , · E,r ).
Proof Making h = 0, we note that as r → 0. By careful computations, we have that as r → 0. Moreover, by the following estimation and the similar one these lead to for h E,r ≤ r and h E,r ≤ r as r → 0. So making α + β 1 + β 2 ≥ 0, we complete the proof.

Now consider system
Then doing the coordinates substitution x = y + h(y) to the above, it yields that where g r = g. By simple computations, we obtain that which is equivalent to and g(y) = f (y + h) r .

Denote by the linear operator
whose inverse is x k e j , g = g nr .

Then Eq. (3.2) can be represented as
in the space (X nr , · E,r ) for any f ∈ (X , · E,r ), where the f -shifted map S f is given by and L 1 (·, ··) is the bilinear one as follows The complete subset utilized here iŝ X nr,q,r = {h ∈ X nr,q | h E,r ≤ r }.

Lemma 3.2
Assume that q = Ord( f ) = Ord(g) ≥ 2 and there exists r 0 > 0 such that f E,r 0 < ∞ for E(t) = e ω(t) , then the following statements hold.
(i) The map S f (·) is strongly contracting in (X nr,r , · E,r ).

Proof First, we show that
Then notice that which by Lemma 2.5(b) and 2.9 implies for (d + 2)r ≤ r 0 . So the map Ad −1 A S f (·) nr is strongly contracting. These also imply the strong contraction of S f as takes (1.3), which confirms (i).

Proof of Theorem 1.1
Since q = Ord(g) ≥ 2, by Poincaré-Dulac formal normal form reductions we can set Ord( f ) = Ord(g), provided that q < ∞. To change system (1.1) into its normal forms, it is equivalent to solve h ∈ (X nr,q,r , · E,r ) from functional equation (3.2), whose precise form is (3.4) for any f ∈ (X , · E,r 0 ). It admits a unique solution for the sufficiently small r > 0 by Lemma 3.2(ii). As q → ∞, the part L 1 (·, ··) vanishes. So it completes the proof by Lemma 3.2(i).

Proof of Theorem 1.3
In this part, the norm to character the formal Gevrey-s functional class is written as · s,r for E(t) = e −st ln t . Here we have that either g = γ (x)Ax +ĝ for Ord(ĝ) = q < ∞ or g = γ (x)Ax andĝ = 0 as q → ∞.
When q < ∞, without loss of generality, we can assume that γ satisfies γ (0) = 0, which is a polynomial of degree q − 1 at most. Denote by f q+ = |k|≥q, j f k, j x k e j for f = k, j f k, j x k e j . Then we define that f r ,q+ = f r q+ and f r ,− = f r − f r ,q+ . Submitting g = γ (x)Ax +ĝ into Eq. (3.2) and doing projections, it yields Regard the classical scalar multiplication operator as a bilinear one, which is denoted by for the function g and the map f on a Banach space. As before, the linear operator Ad A and the map S f are the same as the ones given by (3.3) and (3.5), respectively. Note again that the complete subset utilized here iŝ Notice that f (y + h) r ,− = f for any h ∈ X nr,q . So system (3.7) can be rewritten as Here ρ = 1/(1 + γ ),ρ = γρ, and γ is given by f r ,− = γ (y)Ay (3.14) from (3.8).

By Lemma 3.2 the map S f (h) is strongly contracting and so are S f (h) r and P S f (h) r .
So taking h = 0 and comparing the norm of both sides, we obtain that P • f r s,r = |λ 1 |r γ (0) s,r , which implies γ (0) s,r = O(r ) as r → 0. Similarly, we also have that as r → 0. That is to say, γ regarded as the function of h satisfies (3.1) with β = 0. Similarly as shown in the proof of Lemma 3.1, from the above we have that for h s,r ≤ r as r → 0. So there exists r 1 > 0 such that γ (h) s,r ≤ 1/2 for h s,r ≤ r and r ≤ r 1 . By simple computations, we obtain that for h s,r and h s,r ≤ r as r → 0. This completes the proof. Therefore, we rewrite Theorem 1.3 into a precise form and prove it. (i) When 2 ≤ q < ∞, there exists a formal Gevery-ŝ coordinates substitution turning system (1.1) into its normal forms, whereŝ = max{s, 1 q−1 }, (ii) When q → ∞, i.e.ĝ = 0, the above result is valid forŝ = s.

Proof Since the linear operator Ad A has a bounded inverse Ad −1
A by condition (1.3), the key is to verify the strongly contracting of the right sides of Eqs. (3.10) and (3.15).
When 2 ≤ q < ∞, without loss of generality, we can assume that the (q − 1)-th order truncated system of (1.1) is in its normal forms by Poincaré-Dulac formal reductions. So we choose the complete subset (X nr,q,r , · ŝ,r ). At first, we control the norms of ρ and ρ. Similarly as shown in the proof of Lemma 3.3, we can assume that λ 1 = 0 and P is the projection. Thus from (3.14), it yields that P( f r ,− ) = λ 1 y 1 γ (y), which implies for r < r 0 and any possible s, where c 5 = f s,r 0 /(|λ 1 |r 2 0 ). Then taking r ≤ 1/(2c 5 ), the above leads to At last, by Lemma 3.2 the map S f (h) given by (3.5) is strongly contracting, and so is S f (h) nr . Thus T 1 1 given by (3.11) is strongly contracting because that ρ is bounded for r ≤ 1/(2c 5 ). Similarly, T 1 3 given by (3.13) is also strongly contracting by Lemma 3.2(ii) for s ≥ 1/(q − 1). Moreover, notice that T 1 2 by (3.12) is the linear bounded form satisfying So it is also a strong contracting one, i.e. Eq. (3.10) has the unique solution for the sufficiently small r via Contraction Mapping Principle, which confirms (i).
When q → ∞, i.e.ĝ = 0, the subset is (X nr,r , · s,r ) and the bilinear form L 2 (·, ··) satisfies (3.9), which yields α = 0 in Lemma 3.1. Now we do calculations of T 2 1 and T 2 2 one by one. Note that we have known that S f (·) nr is strongly contracting. On the one hand, from another form of (3.16) and Lemmas 3.3 and 3.1, the map L 2 (ĝ(·), S f (·) nr ) is strongly contracting and so is T 2 1 . On the other hand, F 1 =ĝ and F 2 = Ah satisfy condition (3.1) with β = 0. So the map T 2 2 by (3.17) is also strongly contracting by Lemma 3.3 again. Similarly, Eq. (3.15) has a unique solution, which completes the proof of (ii).

KAM steps in the ultradifferentiable normalization
In this part, the proof of Theorem 1.2 will be shown. More precisely, we begin with solving the homological equation, which yields the control of the one-step transformation and the convergence of the final one by KAM methods. Here we follow the scheme shown in [4, pp. 70-72] for our case. Now we rewrite system (1.1) in this form where f nr = f nr and f r = f − f nr . Doing the coordinates substitution x = y + h(y), which satisfies h nr = h, to the above, it yields that where , As usual, [·, ··] is the classical Lie bracket. Define ad F h := [F, h] and simply mark ad Ax · as ad A ·. Under the following conditions of ω s (t) with s = (s 1 , s 2 ) , and s 2 ≥ s 2,0 > 0 for q = Ord( f ) = Ord(g) ≥ 2 and the preassigned positive constant s 2,0 , KAM steps can be applied. (i) Assume that η = 1, δ ≤ 1, and 4ec −1 c 4 r −1 F E s ,r ≤ 1. By condition (C1) we have that (ii) Assume that η > 1 and 4ec −1 c 4 r −1 F E s ,r ≤ 1. By condition (C2) we have that Proof Set F (u) = |k|=u, j F k, j x k e j . Then we have that which yields for s ≥ q and t ≥ q. So, we can obtain that which implies the estimate E s ,re −δ for η = 1 and δ ≤ 1, and E (s 1 ,s 2 +δ) ,r for η > 1 by Lemma 2.7. Denote by · * = · E s ,re −δ and · * = · E (s 1 ,s 2 +δ) ,r , respectively. Choosing a large N and from the above, we have that Making N → ∞ and setting c 5 = 2c −1 c 3 , we complete the proof.
Denote by · * = · E s ,re −δ and · * = · E (s 1 ,s 2 +δ) ,r , · * * = · E s ,re −2δ and · * * = · E (s 1 ,s 2 +2δ) ,r for (C1) and (C2), respectively. The following is the one-step cancellation of the KAM scheme, which is the key of this part. As mentioned before, doing x = y + h(y) to (4.1), it turns to (4.2), where h is the solution of the following homological equation So we can rewrite system (4.1) into and where ρ = r + h E s ,r . Here ∂· and ∂ 2 · denote the Jacobian and Hessian matrix, respectively. Notice that by Lemma 4.1 and condition (4.5), we have and h * ≤ r δ (4.10) by condition (4.6), where · * = · E s ,re −δ and · * = · E (s 1 ,s 2 +δ) ,r for (C1) and (C2), respectively. Moreover, regarding ∂h as an operator on X , it yields that for Ord(g) ≥ q by making μ = 0 in (2.10) of Lemma 2.7(b) and ∂h ≤ e −1 δ −1 r −1 e δ h * ≤ e − 3 4 < 1 (4.12) for any g from Lemma 2.6. Here ∂h = ∂h E s ,re −2δ and ∂h = ∂h E (s 1 ,s 2 +δ) ,re −δ for (C1) and (C2), respectively. Then we can control norms of R i for i = 1, . . . , 4 one by one. For R 1 and R 2 , from inequality (4.9), we notice that This shall lead to by inequality (4.8), (4.5), and (4.9) and by inequality (4.7) and (4.9). Next comes R 3 . On the one hand, from (4.11), it yields that On the other hand, we have that (4.5) and (4.9), where M = i |λ i |. Naturally, we obtain that At last, we deal with R 4 . Note that by (4.11) and (4.12) the Newman series can be applied to get for both · * * and · . The first part can be controlled as follows by (4.11). The second is to use similarly as in the norm estimation of the other hand part of R 3 . The third is While the last one is different, we shall apply (4.12) to get for any r < ρ. So we make f E s ,r = 0 r with 0 sufficiently small and r ≤ 1. Now choose δ n = δ 0 n −2 . Here δ 0 is a small positive parameter determined below for different cases. Take r n = r n−1 e −δ n−1 with r 0 = r under condition (C1). By inductions, we can assume that f (0) = f . Then in the n-th step, it begins with f (n−1) (x) instead of f (x) in system (4.1), goes on with solving homological equation (4.3) by f r = f (n−1) r , h =ĥ n , and f nr = f (n−1) nr in the norm · E s ,r n e −δn under condition (C1) and · E (s 1 ,s 2 +δn ) ,r under condition (C2), respectively, and ends in system (4.4) for f + = f (n) .

Applications
As shown in [7,11] and ours, the theory of ultradifferentiable normal forms can make the small divisor 'visible' in the smooth category. Moreover, we go further to show that it provides quantitative descriptions of nonlinear terms via conjugations, which bridge the gap between 'real' normal forms, the geometry of general leaves, and bifurcations.
We shall indicate that Theorem 1.3 can be applied to characterize the order of planar saddles and foci quantitatively. Consider the real C ∞ vector field where X = (x, y) ∈ R 2 and z ∈ R. At the origin O there is a resonant saddle of the following in proper local coordinates dx dz = px + f 1 (x, y), dy dz = −qy + f 2 (x, y), (5.2) where p and q ∈ N are co-prime, f 1 and f 2 = O(|x| 2 + |y| 2 ) as (x, y) → 0. We call that it is formally non-integrable, associated with its formal normal form dx dz = px + l a l x(x q y p ) l , dy dz = −qy + l b l y(x q y p ) l , provided that there exists l ∈ N such that b l p + a l q = 0. And its order is given by l 0 = min{l | b l p + a l q = 0}. Or the origin O is a fine focus in proper local coordinates as follows dx dz = −y + f 1 (x, y), dy dz = x + f 2 (x, y), (5.3) whose formal normal form is dx dz = −y − y b m (x 2 + y 2 ) m + x a l (x 2 + y 2 ) l , where z ∈ R. Then its order is given by l 0 = min{l | a l = 0}, provided that the system is formally non-integrable. (5.2) or fine focus of form (5.3) with the order l 0 in system (5.1). Then the following statements hold.

Corollary 1 Assume that the origin O is a formally non-integrable resonant saddle of form
(i) When the orign O is a resonant saddle, if system (5.1) is Gevrey-s smooth, there exists a Gevrey-ŝ smooth coordinates substitution, which turns system (5.1) into its normal form, whereŝ = max{s, l 0 ( p+q)+1 l 0 ( p+q) }. (ii) When the orign O is a fine focus, if system (5.1) is of formal Gevrey-s, there exists a formal Gevrey-ŝ coordinates substitution, which turns system (5.1) into its normal form, whereŝ = max{s, 1 2l 0 }.
where x = i |x i | is the classical l 1 norm. So, we obtain that ad N o ≤ (|k| + 1).
Then from ad A = ad D + ad N , it yields that because ad N is also the nilpotent part satisfying ad N ad A (x k e j ) = ad A ad N (x k e j ) = (λ · k − λ j )ad N (x k e j ).