NORMAL FORMS OF NILPOTENT SYSTEM IN $ \mathbb{C}^{2}\times\mathbb{C}^{2}$

Shuyuan Xiao; Shaoyun Shi

doi:10.11948/20220466

2023 Volume 13 Issue 1

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Shuyuan Xiao, Shaoyun Shi. NORMAL FORMS OF NILPOTENT SYSTEM IN $ \mathbb{C}^{2}\times\mathbb{C}^{2}$[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 526-552. doi: 10.11948/20220466

Citation:

Shuyuan Xiao, Shaoyun Shi. NORMAL FORMS OF NILPOTENT SYSTEM IN $ \mathbb{C}^{2}\times\mathbb{C}^{2}$[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 526-552. doi: 10.11948/20220466

NORMAL FORMS OF NILPOTENT SYSTEM IN $ \mathbb{C}^{2}\times\mathbb{C}^{2}$

Shuyuan Xiao¹,
Shaoyun Shi^2, ,

1.
School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China
2.
School of Mathematics, Jilin University, Changchun, 130012, China

Corresponding author: Email: shisy@jlu.edu.cn(S. Shi)

Abstract

Abstract

In this paper, we consider the following nilpotent system:
$\dot{\theta}=\omega+\Theta(\theta, u), \quad\dot{u}=Au+f(\theta, u), $
where $\theta \in \mathbb{C}^{2}$, $u\in\mathbb{C}^{2}$, $\omega=(\omega_{1}, \omega_{2})\in\mathbb{R}^{2}$, $A=\begin{pmatrix}\lambda & 1\\ ~ &\lambda\end{pmatrix}$, $\Theta$ and $f$ are analytic functions and $2\pi-$periodic in each component of the vector $\theta$, $\Theta=O(|u|)$ and $f=O(|u|^{2})$ as $u\rightarrow 0.$ Two kinds of normal forms are presented based on the different small-divisor conditions.
- Normal forms /
- nilpotent system /
- small-divisor
MSC: 37G05

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References

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