The Extended Alpha Group Dynamic Mapping

This paper investigates the qualitative behavior of a system of ordinary differential equations (ODEs) defined by a matrix operator derived from the algebraic structure of the Alpha Group. The system depends on a rotational parameter that continuously deforms the underlying geometry of the phase space. Using a fourth-order Runge-Kutta numerical scheme, we analyze the evolution of trajectories and identify the presence of critical parameter values at which the system undergoes qualitative transitions. In particular, we observe the emergence of critical dynamical regions associated with changes in the interaction between dynamically defined subspaces. As the rotation parameter varies from $0$ to $\pi/2$, the system transitions from a regime with Euclidean-type geometric behavior to a non-Euclidean configuration induced by the Alpha Group structure. These transitions correspond to changes in stability and global phase space organization, including the formation of invariant structures and attractor-like behavior at infinity. The results suggest that the underlying matrix operator acts as a generator of structured transformations governing the system dynamics. This work provides a computational and qualitative framework for studying parameter-dependent dynamical systems with evolving geometric structure.