Keywords
Heisenberg group
non-commutative coupling
nonlinear dynamics
numerical experiments
Abstract
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.
References
Aboukhisheen, K. (2025). The variational iteration method: A unified framework for nonlinear differential equations with modern enhancements. Azzaytuna University Journal, 2025, 56.
Ahmad, A. E.-S., & Ahmed, S. A. (2014). Fixed points by certain iterative schemes with applications. Fixed Point Theory and Applications, 2014, 121.
Ahmad, A. E.-S., & Kamal, A. (2009). Construction of fixed points by some iterative schemes. Fixed Point Theory and Applications, 2009, 612491.
Ahmad, F., Samraiz, M., Ahmad, J., & De la Sen, M. (2026). AK-iterative scheme for fixed point approximation in hyperbolic metric spaces and applications in Volterra integral equation. Fixed Point Theory, Algorithms and Applications, 2026, 7.
Akram, S., Junjua, D. U. D., Akram, F. & Khalid, L. (2025). Iterative techniques for nonlinear equations: Addressing multiple roots with unknown multiplicity. In polynomials-Exploring fundamental mathematical expressions. Intech Open.
Alharthi, N. H., Okeke, G. A., Udo, A. V., Alqahtani, R. T., & Yusuf, A. (2026). Novel iterative method for approximation of fixed points of generalized (α, β)-nonexpansive mappings with applications to SEIR epidemic model. Scientific Reports, 16, 11833.
Baillon, J. B. (1975). Un théorème de type ergodique pour les contractions non linéaires. Comptes Rendus de l’Académie des Sciences de Paris, Série A-B, 280, 1511–1514.
Bauschke, H. H., & Combettes, P. L. (2017). Convex analysis and monotone operator theory in Hilbert spaces (2nd ed.). Springer.
Browder, F. E. (1965). Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences of the United States of America, 54(4), 1041–1044.
Chidume, C. E. (2009). Geometric properties of Banach spaces and nonlinear iterations. Springer.
Cordero, A., Torregrosa, J. R., & Vassileva, M. P. (2016). Design, analysis, and applications of iterative methods for solving nonlinear systems. In D. Lee et al. (Eds.), Nonlinear systems: Design, analysis, estimation and control. IntechOpen.
Dehaish, B. A., Khamsi, M. A., & Kozłowski, W. M. (2015). On the convergence of iteration processes for semigroups of nonlinear mappings in modular function spaces. Fixed Point Theory and Applications, 2015, 3.
Goebel, K., & Kirk, W. A. (1990). Topics in metric fixed point theory. Cambridge University Press.
Goel, S. (2023). Fixed point theory and iterative procedures: From classical theorems to modern developments. Asian Journal of Mathematical Sciences, 7(1).
Grau-Sánchez, M., Noguera, M., & Gutiérrez, J. M. (2015). New iterative technique for solving a system of nonlinear equations. Applied Mathematics and Computation, 271, 446–466.
Hall, P. (1934). Nilpotent groups. Proceedings of the London Mathematical Society, 38, 29–95.
Kirk, W. A., & Sims, B. (Eds.). (2001). Handbook of metric fixed point theory. Springer.
Mann, W. R. (1953). Mean value methods in iteration. Proceedings of the National Academy of Sciences, 39(6), 506–510.
Mohamadi, I. (2011). Iterative methods for variational inequalities over the intersection of fixed point sets of nonexpansive semigroups in Banach spaces. Fixed Point Theory and Applications, 2011, Article 620284.
Okorie, C. E., Johnson, B. O., Michael, A. I., & Fidelis, A. T. (2018). Comparison of some iterative methods of solving nonlinear equations. International Journal of Theoretical and Applied Mathematics, 4(2), 22–28.
Robinson, D. J. S. (2004). A course in the theory of groups (2nd ed.). Springer.
Saqib, M., Iqbal, M., Ahmed, S., Ali, S., & Ismaeel, T. (2015). New modification of fixed point iterative method for solving nonlinear equations. Applied Mathematics, 6(11), 1857–1863.
Saqib, M., Radwan, T., & Alharbi, R. (2026). Iterative methods for nonlinear equations in coupled systems using modified homotopy perturbation. Ain Shams Engineering Journal, 17(2), 103965.
Shah, F. A., & Noor, M. A. (2024). Exploring effective iterative methods for nonlinear equations using variational iteration technique. Engineering and Applied Science Letters, 7(1), 1–9.
Shumyatsky, P. (2005). Applications of Lie ring methods to group theory. Journal of Algebra, 292(2), 507–524.
Stoll, M. (1996). On the asymptotics of growth functions of groups. Journal of the London Mathematical Society, 54(1), 67–80.
Wartono, W., & Suryani, I. (2020). The solution of nonlinear parabolic equation using variational iteration method. Jurnal Matematika, Statistika dan Komputasi, 16(3), 287–295.
Xu, H.-K. (2002). Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society, 66(1), 240–256.
Zhang, L., Wu, Q.-B., Chen, M.-H., & Lin, R.-F. (2021). Two new effective iteration methods for nonlinear systems with complex symmetric Jacobian matrices. Computational and Applied Mathematics, 40, 97.
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Copyright (c) 2026 Author