# The Recognition of the Bifurcation Problem with Trivial Solutions

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- (i)
- A list of normal forms, with some properties that all bifurcation problems up to the given codimension are equivalent to one of them.
- (ii)
- Constructing and analyzing the universal unfolding of the normal forms.
- (iii)
- The solutions to the recognition problem for the normal forms.

## 2. Basic Concepts and Preliminaries

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 3. Intrinsic Submodule

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

## 4. Statement of the Main Result

**Proposition**

**3.**

- (a)
- $\mathcal{S}(g)=\mathcal{S}(h)$, if g is t-equivalent to h.
- (b)
- $\mathcal{S}(h)$ is an intrinsic submodule of finite codimension.
- (c)
- $$\mathcal{S}(h)=\sum _{\alpha =({\alpha}_{1},{\alpha}_{2})}\{<{x}^{{\alpha}_{1}}{\lambda}^{{\alpha}_{2}}>\{x\}|{D}^{\alpha}f(0,0)\ne 0\}.$$

**Proof.**

**Theorem**

**4.**

- (a)
- ${D}^{\alpha}f(0,0)=0$ for every monomial ${x}^{{\alpha}_{1}}{\lambda}^{{\alpha}_{2}}x\in \mathcal{S}{(h)}^{\perp}$.
- (b)
- ${D}^{\alpha}f(0,0)\ne 0$ for each intrinsic generator ${x}^{{\alpha}_{1}}{\lambda}^{{\alpha}_{2}}x$ of $\mathcal{S}(h)$.

**Proof.**

- (a)
- It is proved immediately by contradiction.
- (b)
- By Proposition 3(c), the result is clear. ☐

**Definition**

**3.**

**Lemma**

**3.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Proposition**

**4.**

- (a)
- If $p\in \mathcal{P}(h)$ and g is equivalent to h, then $g+p$ is equivalent to g.
- (b)
- If $T(h)$ has a finite codimension, then $\mathcal{P}(h)$ is an intrinsic submodule of ${\epsilon}_{x,\lambda}\{x\}$ with a finite codimension.

**Proof.**

**Definition**

**4.**

**Proposition**

**5.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**5.**

## 5. Examples

**Example**

**1.**

**Proof.**

**Example**

**2.**

**Proof.**

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Li, Y.; Huang, D.; Pei, D.
The Recognition of the Bifurcation Problem with Trivial Solutions. *Symmetry* **2019**, *11*, 935.
https://doi.org/10.3390/sym11070935

**AMA Style**

Li Y, Huang D, Pei D.
The Recognition of the Bifurcation Problem with Trivial Solutions. *Symmetry*. 2019; 11(7):935.
https://doi.org/10.3390/sym11070935

**Chicago/Turabian Style**

Li, Yanqing, Dejian Huang, and Donghe Pei.
2019. "The Recognition of the Bifurcation Problem with Trivial Solutions" *Symmetry* 11, no. 7: 935.
https://doi.org/10.3390/sym11070935