标题： 阿尔法组动态映射 显示英文标题

标题： The Alpha Group Dynamic Mapping

摘要： 本文研究由表示Alpha群代数中分划的矩阵所控制的常微分方程(ODE)系统的动力学行为。 随着系统演化，矩阵在几何空间中引发拓扑转换，这些转换由旋转参数控制。 数值模拟使用在Python中实现的四阶龙格-库塔方法进行。 结果揭示了拓扑节点的出现，以及在分割平面之间的旋转从0到$\pi/2$弧度过渡的临界点的存在。 在接近零弧度时，系统表现出欧几里得几何结构，而接近$\pi/2$的旋转定义了一个Alpha群空间。 在这些节点处，由矩阵驱动的ODE系统经历定性动态变化，反映了不同的拓扑行为。 Alpha群矩阵被解释为对称变换的生成器，可能类似于在局部或全局对称性下的规范场。 这项工作提供了一个计算框架，用于探索动态拓扑、无限远处的吸引子以及超复向量空间中的内部一致性。 显示英文摘要

摘要： This paper investigates the dynamical behavior of a system of ordinary differential equations (ODEs) governed by a matrix that represents the division in the algebra of the Alpha group. As the system evolves, the matrix induces topological transitions in geometric spaces, controlled by a rotational parameter. Numerical simulations are performed using a fourth-order Runge-Kutta method implemented in Python. The results reveal the emergence of topological nodes, the existence of critical points at which the rotation between dividing planes transitions from 0 to $\pi/2$ radians. Near zero radians, the system exhibits a Euclidean geometric structure, while rotations close to $\pi/2$ define an Alpha Group space. At these nodes, the matrix-driven ODE system undergoes qualitative dynamic changes, reflecting distinct topological behaviors. The Alpha Group matrix is interpreted as a generator of symmetry transformations, potentially analogous to gauge fields under local or global symmetries. This work provides a computational framework for exploring dynamic topologies, attractors at infinity, and internal coherence in hyper-complex vector spaces.