Discrete dynamical systems over finite state spaces
Aravindan Munusamy Kalidhasmk.aravindan@jainuniversity.ac.inDepartment of Mechanical EngineeringFaculty of Engineering and TechnologyJAIN (Deemed-to-be University)Bangalore, Karnataka, 562112, IndiaView full profile → , *Nittin SharmaCorresponding authornittin.sharma.orp@chitkara.edu.inCentre of Research Impact and Outcome (CRIO)Chitkara UniversityRajpura, Punjab, 140417, IndiaView full profile → , Vimal Bibhuvimal.bhibu@niu.edu.inSchool of Engineering & TechnologyNoida International UniversityGreater Noida, Uttar Pradesh, 201312, IndiaView full profile → , Shailesh KulkarniDepartment of Electronics and Telecommunication EngineeringVishwakarma Institute of TechnologyPune, Maharashtra, 411037, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Apr 2026
- Published Online:
- 27 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2819
- Pages:
- 3059–3070
Abstract
Digital, computational and engineering applications use the powerful framework of discrete dynamical systems on finite state spaces to model efficiently processes with deterministic or rule based dynamics. This paper suggests a structured representation which is constituted by finite state representation, state transition functions, a mechanism for discrete time updates, and utilizes them to analyze the evolution of the system from one time instant to the next. A strategy for efficiently encoding the state is suggested that will represent finite configurations, and a design of the transition rule is suggested to provide consistent and deterministic transition of the state. Proposed framework focuses on the simplicity of the computation, its scalability and its reproducibility for finite dimensional systems. The stability of convergence is demonstrated by performance evaluation, as well as the accuracy of the state prediction; the computational complexity is low; and transition management efficiency with different starting conditions is shown. The outcome demonstrates the effectiveness of the proposed method to capture discrete dynamic behavior and suggests its potential applications in different areas like automata theory network modelling, control systems and analysis of algorithms, etc., providing a good foundation for the future continuous-time computational research.
Keywords
Subject Classifications
References
[1] S. Fiori, “Discrete-time dynamical systems on structured state spaces: State-transition laws in finite-dimensional Lie algebras,” Symmetry, vol. 17, no. 3, Art. no. 463 (2025), doi: 10.3390/sym17030463.
[2] E. A. Gospodinova, “Research on the stability model in discrete dynamical systems with the Lorenz attractor and the Kropotov–Pakhomov neural network,” Entropy, vol. 28, no. 1, Art. no. 12 (2026), doi: 10.3390/e28010012.
[3] J. Swid and M. Amini, “Recurrence and entropy for discrete-time deterministic dynamical systems,” Symmetry, vol. 18, no. 5, Art. no. 816 (2026), doi: 10.3390/sym18050816.
[4] L. Albantakis and G. Tononi, “The intrinsic cause-effect power of discrete dynamical systems—From elementary cellular automata to adapting animats,” Entropy, vol. 17, no. 8, pp. 5472–5502 (2015), doi: 10.3390/e17085472.
[5] J. D. Gomez, W. G. P. Mayner, M. Beheler-Amass, G. Tononi, and L. Albantakis, “Computing integrated information (Φ) in discrete dynamical systems with multi-valued elements,” Entropy, vol. 23, no. 1, Art. no. 6 (2021), doi: 10.3390/e23010006.
[6] J. A. Aledo, A. Barzanouni, G. Malekbala, L. Sharifan, and J. C. Valverde, “On the periodic structure of parallel dynamical systems on generalized independent Boolean functions,” Mathematics, vol. 8, no. 7, Art. no. 1088 (2020), doi: 10.3390/math8071088.
[7] Z. He, C. D. Charalambous, and P. A. Stavrou, “A new finite-horizon dynamic programming analysis of nonanticipative rate-distortion function for Markov sources,” in Proc. 2025 European Control Conference (ECC), Thessaloniki, Greece, pp. 749–754 (2025), doi: 10.23919/ECC65951.2025.11186895.
[8] I. Nasim and M. E. Henderson, “Dynamically meaningful latent representations of dynamical systems,” Mathematics, vol. 12, no. 3, Art. no. 476 (2024), doi: 10.3390/math12030476.
[9] B. Xin and Z. Wu, “Projective synchronization of chaotic discrete dynamical systems via linear state error feedback control,” Entropy, vol. 17, no. 5, pp. 2677–2687 (2015), doi: 10.3390/e17052677.
[10] A. B. Rajmane, S. B. Patil, V. G. Kale, S. V. Joshi, D. B. Patil, and T. N. Pawase, “Mathematical models for cognitive systems using differential geometry and reinforcement learning,” Journal of Interdisciplinary Mathematics, vol. 28, no. 6, pp. 2089–2097 (2025), doi: 10.47974/JIM-2350.
[11] M. Malykh, M. Gambaryan, O. Kroytor, and A. Zorin, “Finite difference models of dynamical systems with quadratic right-hand side,” Mathematics, vol. 12, no. 1, Art. no. 167 (2024), doi: 10.3390/math12010167.
[12] U. Aydın, T. Başar, and N. Saldi, “Approximation of discrete-time infinite-horizon mean-field equilibria via finite-horizon mean-field equilibria,” in Proc. 2025 IEEE 64th Conference on Decision and Control (CDC), Rio de Janeiro, Brazil, pp. 305–310 (2025), doi: 10.1109/CDC57313.2025.11312792.
[13] H. H. Saleh, I. Nayak, D.-Y. Na, and F. L. Teixeira, “Charge-conserving time extrapolation of dynamical data in kinetic plasma particle-in-cell simulations,” in Proc. 2025 IEEE CNC-USNC-URSI North American Radio Science Meeting (Joint with AP-S Symposium), Ottawa, ON, Canada, p. 3159 (2025), doi: 10.23919/CNC-USNC-URSI64444.2025.11419982.
