THE ALPHA GROUP DYNAMIC MAPPING

This paper investigates the dynamical behavior of a system of ordinary differential equations (ODEs) governed by a matrix that representsthe division in the algebra of the Alpha group. As the system evolves,the matrix induces topological transitions in geometric spaces, controlledby a rotational parameter. Numerical simulations are performed usinga fourth-order Runge-Kutta method implemented in Python. The re-sults reveal the emergence of topological nodes, the existence of criticalpoints at which the rotation between dividing planes transitions from 0 toπ/2 radians. Near zero radians, the system exhibits a Euclidean geomet-ric structure, while rotations close to π/2 define an Alpha Group space.At these nodes, the matrix-driven ODE system undergoes qualitative dy-namic changes, reflecting distinct topological behaviors. The Alpha Groupmatrix is interpreted as a generator of symmetry transformations, poten-tially analogous to gauge fields under local or global symmetries. Thiswork provides a computational framework for exploring dynamic topolo-gies, attractors at infinity, and internal coherence in hyper-complex vectorspaces.