# 数学代考|微分方程代考differential equation作业代写|Formal normal forms

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## 数学代考|微分方程代考differantial equation作业代写|Formal classification theorem

Formal classification of formal vector fields is very much influenced by properties of its principal part, in particular, the linearization matrix $A=\left(\frac{\partial F}{\partial x}\right)(0)$ if the latter is nonzero.
Definition 4.2. An ordered tuple of complex numbers $\lambda=\left(\lambda_{1}, \ldots, \lambda_{n}\right) \in$ $\mathbb{C}^{n}$ is called resonant, if there exist nonnegative integers $\alpha=\left(\alpha_{1}, \ldots, \alpha_{n}\right) \in$ $\mathbb{Z}{+}^{n}$ such that the resonance identity occurs, $$\lambda{j}=\langle\alpha, \lambda\rangle, \quad|\alpha| \geqslant 2,$$
where $\langle\alpha, \lambda\rangle=\alpha_{1} \lambda_{1}+\cdots+\alpha_{n} \lambda_{n}$. The natural number $|\alpha|$ is the order of the resonance.

A square matrix is resonant, if the collection of its eigenvalues is resonant. A formal vector field $F=\left(F_{1}, \ldots, F_{n}\right)$ at the origin is resonant if its linearization matrix $A=\left(\frac{\partial F}{\partial x}\right)(0)$ is resonant.

Though resonant tuples can be dense in some parts of $\mathbb{C}^{n}$ (see $\left.\S 5.1\right)$, their measure is zero.

## 数学代考|微分方程代考DIFFERANTIAL EQUATION作业代写|Induction step: homological equation

The proof of Theorem $4.3$ goes by induction. Assume that the formal vector field $F$ is already partially normalized, and contains no terms of order less than some $m \geqslant 2$ :
$$F(x)=A x+V_{m}(x)+V_{m+1}(x)+\cdots,$$
where $V_{m}, V_{m+1}, \ldots$ are arbitrary homogeneous vector fields of degrees $m, m+1$ etc.

We show that in the assumptions of the Poincaré theorem, the term $V_{m}$ can be removed from the expansion of $F$, i.e., that $F$ is formally equivalent to the formal field $F^{\prime}(x)=A x+V_{m+1}^{\prime}+\cdots$. Moreover, the corresponding conjugacy can be in fact chosen polynomial of the form $H(x)=x+P_{m}(x)$, where $P_{m}$ is a vector polynomial of degree $m$. The Jacobian matrix of such formal morphism is $E+\left(\frac{\partial P_{m}}{\partial x}\right)$.

The conjugacy $H$ with these properties must satisfy the equation (1.24) on the formal level. Keeping only terms of order $\leqslant m$ from this equation and using dots to denote the rest, we obtain
$$\left(E+\frac{\partial P_{m}}{\partial x}\right)\left(A x+V_{m}+\cdots\right)=A(x+P(x))+V_{m}^{\prime}\left(x+P_{m}(x)\right)+\cdots$$
The homogeneous terms of order 1 on both sides coincide. The next nontrivial terms appear in the order $m$. Collecting them, we see that in in order meet the condition $V_{m}^{\prime}=0$, the homogeneous terms $P=P_{m}$ must satisfy the identity
$$\left[\mathbf{A}, P_{m}\right]=-V_{m}, \quad \mathbf{A}(x)=A x$$
where $\mathbf{A}=\mathbf{A}(x)=A x$ is the linear vector field, the principal part of $F$, and the homogeneous vector polynomials $P_{m}$ and $V_{m}$ are considered as vector fields on $\mathbb{C}^{n}$. The left hand side of (4.2) is the commutator, $[\mathbf{A}, P]=\left(\frac{\partial P}{\partial x}\right) A x-A P(x)$.

Conversely, if the condition (4.2) is satisfied by $P_{m}$, the polynomial map $H(x)=x+P_{m}(x)$ conjugates $F=\mathbf{A}+V_{m}+\cdots$ with the (formal) vector field $F^{\prime}(x)=\mathbf{A}+\cdots$ having no terms of degree $m$.

## 数学代考|微分方程代考DIFFERANTIAL EQUATION作业代写|Solvability of homological equation

Solvability of the homological equation depends on the properties of the operator of commutation with the linear vector field $\mathbf{A}$.

Let $\mathcal{D}{m}$ be the linear space of all homogeneous vector fields of degree $m$. This linear space has the standard monomial basis consisting of the fields $$F{k \alpha}=x^{\alpha} \frac{\partial}{\partial x_{k}}, \quad k=1, \ldots, n,|\alpha|=m .$$
We shall order elements of this basis lexicographically so that $x_{i}$ precedes $x_{j}$ if $i\cdots>w_{n}$ that are rationally independent. This assignment extends on all monomials and monomial vector fields if the symbol $\frac{\partial}{\partial x_{j}}$ is assigned the weight $-w_{j}$. Now the monomial vector fields can be arranged in the decreasing order of their weights: the independence condition guarantees that any two different monomials have different weights.
The operator
$$\operatorname{ad}{A}: P \mapsto[\mathbf{A}, P], \quad\left(\operatorname{ad}{A} P\right)(x)=\left(\frac{\partial P}{\partial x}\right) \cdot A x-A P(x),$$
preserves the space $\mathcal{D}_{m}$ for any $m \in \mathbb{N}$.

$$其中⟨一种,λ⟩=一种1λ1+⋯+一种nλn. 自然数|一种|是共振的阶数。 如果其特征值的集合是共振的，则方阵是共振的。一个正式的向量场F=(F1,…,Fn)如果它的线性化矩阵在原点是共振的一种=(∂F∂X)(0)是共振的。 虽然共振元组在某些部分可能是密集的Cn 见 \left.\S 5.1\right见 \left.\S 5.1\right，他们的度量是零。 ## 数学代考|微分方程代考DIFFERANTIAL EQUATION作业代写|INDUCTION STEP: HOMOLOGICAL EQUATION 定理的证明4.3通过感应进行。假设正式向量场F已经部分归一化，并且不包含小于一些的顺序项米⩾2 : F(X)=一种X+在米(X)+在米+1(X)+⋯, 在哪里在米,在米+1,…是度的任意齐次向量场米,米+1等等 我们证明，在 Poincaré 定理的假设中，项在米可以从扩展中删除F，即，那个F形式上等价于形式域F′(X)=一种X+在米+1′+⋯. 此外，相应的共轭实际上可以选择形式的多项式H(X)=X+磷米(X)， 在哪里磷米是一个向量多项式米. 这种形式态射的雅可比矩阵是和+(∂磷米∂X). 共轭H具有这些属性必须满足方程1.24在正式层面。仅保留订单条款⩽米从这个等式并用点表示其余部分，我们得到 (和+∂磷米∂X)(一种X+在米+⋯)=一种(X+磷(X))+在米′(X+磷米(X))+⋯ 两边的 1 阶齐次项重合。下一个重要项按顺序出现米. 收集它们，我们看到为了满足条件在米′=0, 齐次项磷=磷米必须满足恒等式 [一种,磷米]=−在米,一种(X)=一种X 在哪里一种=一种(X)=一种X是线性向量场，主要部分F, 和齐次向量多项式磷米和在米被认为是矢量场Cn. 的左侧4.2是换向器，[一种,磷]=(∂磷∂X)一种X−一种磷(X). 相反，如果条件4.2满足于磷米, 多项式映射H(X)=X+磷米(X)共轭F=一种+在米+⋯与F这r米一种l向量场F′(X)=一种+⋯没有学位条款米. ## 数学代考|微分方程代考DIFFERANTIAL EQUATION作业代写|SOLVABILITY OF HOMOLOGICAL EQUATION 同调方程的可解性取决于具有线性向量场的对易算子的性质一种. 设\mathcal{D}{m} be the linear space of all homogeneous vector fields of degree m. This linear space has the standard monomial basis consisting of the fields$$ F{k \alpha}=x^{\alpha} \frac{\partial}{\partial x_{k}}, \quad k=1, \ldots, n,|\alpha|=m .
$$We shall order elements of this basis lexicographically so that x_{i} precedes x_{j} if i\cdots>w_{n} that are rationally independent. This assignment extends on all monomials and monomial vector fields if the symbol \frac{\partial}{\partial x_{j}} is assigned the weight -w_{j}. Now the monomial vector fields can be arranged in the decreasing order of their weights: the independence condition guarantees that any two different monomials have different weights. The operator$$


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