We present a computational investigation of a commutator-driven iterative dynamical system defined on the Heisenberg group. The scheme is constructed by repeatedly updating a group element through multiplication with its commutator relative to a fixed element, thereby inducing a nonlinear evolution governed by the underlying non-abelian structure. Using a minimal implementation in Python, we generate trajectories from prescribed initial conditions and examine their evolution over discrete iterations. Our numerical experiments reveal a consistent and robust linear growth pattern in the group norm of the iterates, characterized by an affine growth. This behavior persists across multiple initial configurations, indicating that the observed dynamics are not incidental but intrinsic to the commutator-driven update mechanism. Further analysis shows that the growth is primarily driven by accumulation in the central component of the Heisenberg group, reflecting the non-commutative coupling embedded in the group law. The results highlight a clear departure from classical contractive or fixed-point iterative schemes, instead demonstrating a deterministic growth regime arising from algebraic structure. The simplicity of the computational framework, combined with the clarity of the emergent linear law, makes this system a useful prototype for exploring nonlinear dynamics on non-abelian groups. These findings open pathways for extending commutator-based iterative methods to more general Lie groups and for investigating their potential applications in computational and theoretical physics.
Essien* et al. (Sun,) studied this question.