SECOND-ORDER NORMAL FORMS FOR N-DIMENSIONAL SYSTEMS WITH A NILPOTENT POINT

Normal forms theory is one of the most powerful tools for the study of nonlinear differential equations, in particular, for stability and bifurcation analysis. Many works paid attention to normal forms associated with nilpotent Jacobian where the critical eigenvalues have algebraic multiplicity k (k > 1) and geometric multiplicity one, and in particular, the case k > 2 is more complicated for determining unfolding. Despite a lot of theoretical results on nilpotent normal forms have been obtained, computation developing can not satisfy practical applications. To our knowledge, no results have been reported on the computation of explicit formulas of the nilpotent normal forms for k > 3 with perturbation parameters. The main difficulty is how to determine the complementary spaces of the Lie transformation. In this paper, we achieve the following results. (1) A simple dimension formula for the complementary space of the Lie transform; (2) a simple direct method to determine a basis of the complementary spaces; (3) a simple direct method to determine the projection of any vector to the complementary spaces. Using this method, the second-order normal forms for any n-dimensional nilpotent systems can be given easily. As an illustrative application, the normal forms for the vector field with triple-zero or four-fold zero singularity and functional differential equation with a triple-zero singularity are presented, and explicit formulas for the normal form coefficients with three or four unfolding parameters are obtained.


Introduction
Studying dynamical systems with multiple zero critical singularity is not only theoretically significant but also important in real applications. When the Jacobian of a dynamical system evaluated at a critical point contains one or two zero eigenvalues, the so-called simple zero or double zero bifurcation may occur. A nilpotent singularity corresponding to a double-zero eigenvalue with geometric multiplicity one is known as codimension-2 Bogdanov-Takens (B-T) singularity, which can yield homoclinic orbits to saddle equilibria near the critical point. Since Bogdanov [20] and Takens [2] obtained the normal forms of B-T bifurcation and gave a very detailed bifurcation analysis, many works have been done in this area(e.g. see [1,4,9,13,14,19,21,24] and references therein). The triple-zero eigenvalue with geometric multiplicity one called codimension-3 singularity has also been considered, see Ref. [3,7,15,17,22,23].
There are few studies of codimension-4 or higher codimension problems with non-semisimple nilpotent singularities, perhaps due to the relative rarity of higher codimension singularities in ordinary differential equation (ODE) models. However, in delay differential equations (DDEs) higher codimension singularities seem to occur more frequently.
The method of normal forms provides a powerful tool in finding a simple form which keeps the fundamental dynamics of the original system unchanged [6]. For a practical system, not only the possible qualitative dynamical behavior of the system is of concern, but also the quantitative relationship between the normal forms and the original system needs to be established. For a general singular vector field with non-semisimple nilpotent singularities, the computation of the normal forms is very complicated. In particular, finding the explicit formulas of normal forms in terms of the original systems coefficients with nilpotent singularities is very difficult. Therefore, the crucial part in computing a normal form is the computational efficiency in finding the normal forms coefficients. In this study, we consider the following vector field.ẋ

1)
where J is the canonical Jordan nilpotent form, and F i (x) represents the ith−degree homogeneous polynomial in the Taylor expansion of F (x). Introducing the coordinate transformation, x = y + h 2 (y), (1.2) where h 2 (y) denotes the 2nd-degree homogeneous polynomial in y, and substituting (1.2) into (1.1) yieldṡ y = Jy + Jh 2 (y) − Dh 2 (y)Jy + F 2 (y) + F 3 (y) + · · · + F r−1 (y) + O(|y| r ). (1. 3) The basic ideal of normal forms method is to choose a specific form for h 2 (y) so as to simplify the 2nd-degree terms as much as possible.
Let H 2 n be the linear space of 2nd-degree homogeneous polynomials. Further, we introduce the following linear map of H 2 n into H 2 n : h 2 (y) −→ Dh 2 (y)Jy − Jh 2 (y).
Due to its presence in Lie algebra theory, this map is often denoted as L Nn (h 2 (y)) = −(Dh 2 (y)Jy − Jh 2 (y)), and is called Lie bracket operation. Assume that H 2 n can be (non-uniquely) decomposed as H 2 n = L Nn (H 2 n ) ⊕ W, where W is a complementary spaces of H 2 n .
The purpose of normal forms method is to choose h 2 (y) so that only the terms in W are retained. We denote these terms by F r 2 (y). Thus, (1.3) can be simplified toẏ = Jy + F r 2 (y) +F 3 (y) + · · · +F r−1 (y) + O(|y| r ), (1.4) and so the second-order terms have been simplified.
To determine the nature of the second-order terms that cannot be eliminated (i.e., F r 2 (y)), we must investigate the space complementary to L Nn (H 2 n ). Solving this problem involves the following three main tasks: (1) determining the dimension of the complementary space of the Lie transform; (2) determining the basis of the complementary space of the Lie transform; and (3) determining the projection of any vector in H 2 n to the complementary space. Using this method in this paper, the second-order normal forms for any ndimensional nilpotent systems can be given easily. We will present detailed steps to show how to fulfill these tasks. Our goal is to analyze the codimension-n(n > 2) singularity corresponding to n-zero eigenvalues with geometric multiplicity one.
In Section 2, we define Lie bracket operation (Choquet-Bruhat et al. Ref. [5]). Using the linear transformation L Nn we determine the dimensions of the complementary space of L Nn (H 2 n ) and obtain the basis of the complementary space of L Nn (H 2 n ), which is the key step in calculating the non-semisimple normal form and the explicit coefficients of the normal form. More precisely, we present the following results in this section.
(1) A simple dimension formula for the complementary spaces of the Lie transform; (2) a simple direct method to determine the basis of the complementary spaces; and (3) a simple direct method to determine the projection of any vector in H 2 n to the complementary spaces.
In Section 3, we obtain results for the complementary space of L Nn (H 2 n+p ) with p parameters.
In Sections 4, 5 and 6, as an illustrative application, the normal forms for the vector field and functional differential equation with triple-zero and four-fold zero singularity are considered using the results of section 3. We derive the explicit normal form for the triple-zero and four-fold zero singularity, which are of primary importance in applications. On the one hand, we can determine the terms that are inessential in determining the dynamical and bifurcation behaviors of the system. On the other hand, as we can compute the normal form coefficients, we can identify the parameter values for which nonlinear degeneracies take place. Near these critical parameter values, more complicated bifurcation phenomena can occur.

Complementary Space of L N n (H n )
Let m, n be positive integers, R the real number field. We denote H 2 n the following space of 2nd degree homogeneous polynomials with n variables: where X = (x 1 , x 2 , . . . , x n ) T ; P (x) = (P 1 (x), P 2 (x), · · · P n (x)) T .
. ., f m, n(n+1) 2 = x n x n e m , m = 1, 2, . . . , n, consisting of a standard orthogonal basis of H 2 n , called a natural basis. Further, let For example, when n = 3, h = 4, we have Then the following result is obvious : Define the linear transformation L Nn on H 2 n by N n as follows: In this section, we investigate the dimensions of the complementary space of L Nn (H 2 n ). It is easy to see that the following lemma is true.
Now, we prove the following result. (1) If 2 ≤ h ≤ n, then L Nn is an injective linear mapping from V h to V h+1 .
and h is an odd number. In this case, we can write Since = 0, and f (X) = 0 follows. Case 3. 3 ≤ h ≤ n, and h is an even number. In this case, we can write which yields a 1 = a 2 = . . . = a h 2 = 0, and so f (X) = 0. (2) For n + 1 ≤ h ≤ 2n, we can prove that for any g(X)e 1 ∈ V h+1 , there exists f (X)e 1 ∈ V h such that L Nn (f (X)e 1 ) = g(X)e 1 . Case 4. n + 1 ≤ h ≤ 2n − 1, and h is an odd number. In this case, 2 ≤ h − n + 1 ≤ n, we write Choose a h−n , a h−n+1 , . . . , a h−1 2 as follows: Then taking yields f (X)e 1 ∈ V h and so L Nn (f (X)e 1 ) = g(X)e 1 .
The following theorem provides a general formula for determining the dimension of the complementary space (L Nn (H 2 n )) c . Theorem 2.4. Suppose n ≥ 2. The dimension of any complementary space L Nn (H 2 n ) c is given by is the matrix of L Nn on the standard basis of H 2 n , . Simplifying the matrix M by elementary column transformation yields If n ≡ 0 (mod 4), denote n = 4k + 4. Then If n ≡ 1 (mod 4), denote n = 4k + 1. Then, we obtain If n ≡ 2 (mod 4), denote n = 4k + 2. Then, we have The following example illustrates Theorem 2.4.
In the following, a method to construct a complementary space to L Nn (H 2 n ), and a very simple algorithm to calculate the projection from H 2 n to the complementary space are presented. We have the following theorem.

Proof. By Theorem 2.4, we obtain that
is the matrix of L Nn on the standard basis Then, L Nn (H 2 n ) = span(f 1,1 , . . . ,f 1,s , . . . ,f n,1 , . . . ,f n,s ). It follows from the proof of Theorem 2.4, that and so obtain Hence, Therefore, . . , ξ t ) as an orthonormal basic system solution for the homogeneous linear equation, In this case, A n Y 0 + BZ 0 = β implies Z 0 = B T β for any β ∈ R s . Example 2.10. For n = 2, we have s = 3, t = 2 and and hence , solving the equation: we have

Complementary Space of L N n (H 2 n+p ) with Parameters
Now we want to extend the normal form techniques to systems with parameters. The goal is to transform the system into normal form near the fixed point in both phase space and parameter space.
Obviously, H 2 n+p is a real inner product space with dim H 2 n+p = 1 2 (n + p)(n + p + 1).
Moreover, the following constitutes a standard orthogonal basis of H 2 n+p , called a natural basis. Define the linear transformation L Nn on H 2 n+p by N n , which is defined in (2.1), as follows: Proof. Let A be the matrix of L Nn | U1 on the standard basis of U 1 . Then, where A 1 is the matrix of L Nn | W on the standard basis of W = span{f11, f12, . . . , f1s}, and A 2 is the matrix of L Nn | V on the standard basis of V = span{f1,s+1, f1,s+2, . . . , f1,t}. Simplifying the matrix M by elementary column transformation, we have so we obtain Then, with a similar proof to that for Theorem 2.4, we can prove that , let A be the matrix of L Nn | U1 on the standard basis f 1,1 , . . . , f 1,q , t = q − rankA n , matrix B ∈ R q×t satisfying rankB = t and rank(A n . . .B) = q. Further, if (g 1 (X), g 2 (X) . . . , g t (X)) = (f n,1 , f n,2 , . . . , f n,q )B, and W = span{g 1 (X), g 2 (X), . . . , g t (X)} ⊆ U n , then , a i2 , . . . , a iq ) T , i = 1, 2, . . . , n.
Proof. The proof is similar to that for Theorem 2.8, and thus omitted.

Example 3.4. For n = p = 3, we have
Since Then, .
we solve the equation:

Normal form of a DDE system associated with triple-zero singularity
In this section, we present the normal form of a functional differential equation associated with a trip-zero singularity. Let us consider an abstract retarded functional differential equation with parameters in the phase space C = C([−τ, 0]; R n ), described bẏ under which system (5.1) can be rewritten aṡ Then, the linear homogeneous retard functional differential equation (5.2) can be written asu where L is a bounded linear operator and satisfies Here, η(θ)(θ ∈ [−τ, 0]) is an n × n matrix function of bounded variation. Let A 0 be the infinitesimal generator such that and its adjoint is given by Define the bilinear form between C and In the following, we will consider the case for which L has a triple-zero eigenvalue and all other eigenvalues have negative real parts.
Let Λ be the set of eigenvalues with zero real part and P be the generalized eigenspace associated with Λ which has a triple-zero eigenvalue and P * the space adjoint with P . Then, C can be decomposed as with dim p = 3. Choose the bases Φ and Ψ for P and P * respectively such that where I is the m × m identity matrix and B =      Following the ideas in [9], we consider the enlarged phase space BC, Then, the elements of BC can be expressed as ψ = φ + x 0 α, φ ∈ C, α ∈ R n and where I is the identity operator on C. The space BC has the norm | φ + u 0 α |= | φ | c + | α | R n . Then, the continuous projection π : BC → P, defined by allows us to decompose the enlarged phase space BC = P ⊕ Kerπ. Let u = Φx + y.

Conclusion
In this paper, we have derived second-order explicit formulas of the normal forms associated with nilpotent critical points. As an application, the explicit formulas have been obtained for normal forms with unfolding associated with a triple-zero and a four-fold zero singularity in vector field and retarded functional differential equations. The formulas obtained in this paper can be easily implemented using a computer algebra system such as Maple or Mathematica.