The Alpha Group Dynamic Mapping

Abstract: 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.