Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique

Maina Martha Wangeci; Samuel Mutua; Nicholas Mwilu Mutothya

doi:doi:10.11648/j.ajam.20261405.16

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Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique

Maina Martha Wangeci^*

, Samuel Mutua

, Nicholas Mwilu Mutothya

Published in American Journal of Applied Mathematics (Volume 14, Issue 5)

Received: 14 August 2026 Accepted: 31 August 2026 Published: 27 September 2026

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Abstract

Chaotic systems, such as the Lorenz and Chen systems, models exhibit high sensitivity to initial conditions. They require fine discretization for accurate long-time integration. Classical time-stepping methods suffer from accumulated truncation errors, whereas, global spectral methods, though highly accurate, yield dense coefficient matrices that are computationally expensive, limiting their efficiency. In this paper we propose an efficient numerical scheme that combines the non-overlapping Domain Decomposition Method (DDM) with the Spectral Relaxation Method (SRM) to solve the Lorenz and Chen systems. The time domain is decomposed into smaller sub intervals, and SRM is applied in each subdomain to linearize the nonlinear terms iteratively, thereby avoiding Taylor series truncation errors while maintaining spectral accuracy. The performance of the proposed Multistage Spectral Relaxation Method (MSRM) is investigated for different step sizes h = 0.01 and h = 0.001 over a long integration time T= 10. Numerical results are compared with the single-domain SRM in terms of accuracy, CPU time and memory usage. The results demonstrate that MSRM achieves comparable accuracy to the single-domain approach while significantly reducing computational time and memory requirements, especially for smaller step sizes. The proposed MSRM provides an effective and efficient framework for the simulation of chaotic systems over a long-time interval.

Published in	American Journal of Applied Mathematics (Volume 14, Issue 5)
DOI	10.11648/j.ajam.20261405.16
Page(s)	331-338
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Domain Decomposition Method, Multistage Spectral Relaxation Method, Lorenz Systems, Chen Systems, Step Size, Computational Efficiency, Chaotic Systems

References

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[2]	Mashuri, A., Adenan, N. H., Abd Karim, N. S., Tho, S. W., & Zeng, Z. (2024). Application of chaos theory in different fields-A literature review. Journal of Science and Mathematics Letters, 12(1), 92-101. https://repositori.mypolycc.edu.my/jspui/handle/123456789/7278
[3]	Motsa, S. S., Dlamini, P. G., & Khumalo, M. (2012). Solving Hyperchaotic Systems Using the Spectral Relaxation Method. Abstract and Applied Analysis, 2012, 1-18. https://doi.org/10.1155/2012/203461
[4]	Biswas, H. R., Hasan, M. M., & Bala, S. K. (2018). Chaos theory and its applications in our real life. Barishal University Journal Part, 1(5), 123-140.
[5]	Shen, B. W., Pielke Sr, R. A., Zeng, X., Baik, J. J., Faghih-Naini, S., Cui, J., & Atlas, R. (2021). Is weather chaotic? Coexistence of chaos and order within a generalized Lorenz model. Bulletin of the American Meteorological Society, 102(1), E148-E158.
[6]	Saberi Nik, H., & Rebelo, P. (2014). Multistage spectral relaxation method for solving the hyperchaotic complex systems. The Scientific World Journal, 2014. https://doi.org/10.1155/2014/943293
[7]	Agiza, H. N. (2002). Controlling chaos for the dynamical system of coupled dynamos. Chaos, Solitons & Fractals, 13(2), 341-352. https://doi.org/10.1016/S0960-0779(00)00234-4
[8]	Chen, J. H., & Chen, W. C. (2008). Chaotic dynamics of the fractionally damped van der Pol equation. Chaos, Solitons & Fractals, 35(1), 188-198. https://doi.org/10.1016/j.chaos.2006.05.010
[9]	Jafari, H., & Firoozjaee, M. A. (2010). Multistage homotopy analysis method for solving nonlinear integral equations. Applications and Applied Mathematics: An International Journal (AAM), 5(3), 4.
[10]	Batiha, B., Noorani, M. S. M., Hashim, I., & Ismail, E. S. (2007). The multistage variational iteration method for a class of nonlinear system of ODEs. Physica Scripta, 76(4), 388. https://doi.org/10.1088/0031-8949/76/4/018
[11]	Evirgen, F., & Özdemir, N. (2011). Multistage adomian decomposition method for solving NLP problems over a nonlinear fractional. https://doi.org/10.1115/1.4002393
[12]	Trefethen, L. N. (2000). Spectral methods in MATLAB. Society for industrial and applied mathematics.
[13]	Motsa, S. S., Magagula, V. M., & Sibanda, P. (2017). The multi‐domain spectral relaxation method for chaotic systems of ordinary differential equations. In Chaotic Systems: Dyn Algo Sync (pp. 29-52). Nova Science Publishers, Inc.
[14]	Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of atmospheric sciences, 20(2), 130-141. https://doi.org/10.1177/0309133315623099
[15]	Chen, G., & Ueta, T. (1999). Yet another chaotic attractor. International Journal of Bifurcation and chaos, 9(07), 1465-1466. https://doi.org/10.1142/S0218127499001024
[16]	Sooraksa, P., & Chen, G. (2018). Chen system as a controlled weather model—physical principle, engineering design and real applications. International Journal of Bifurcation and Chaos, 28(04), 1830009. https://doi.org/10.1142/S0218127418300094

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APA Style

Wangeci, M. M., Mutua, S., Mutothya, N. M. (2026). Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique. American Journal of Applied Mathematics, 14(5), 331-338. https://doi.org/10.11648/j.ajam.20261405.16

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ACS Style

Wangeci, M. M.; Mutua, S.; Mutothya, N. M. Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique. Am. J. Appl. Math. 2026, 14(5), 331-338. doi: 10.11648/j.ajam.20261405.16

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AMA Style

Wangeci MM, Mutua S, Mutothya NM. Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique. Am J Appl Math. 2026;14(5):331-338. doi: 10.11648/j.ajam.20261405.16

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@article{10.11648/j.ajam.20261405.16,
  author = {Maina Martha Wangeci and Samuel Mutua and Nicholas Mwilu Mutothya},
  title = {Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {5},
  pages = {331-338},
  doi = {10.11648/j.ajam.20261405.16},
  url = {https://doi.org/10.11648/j.ajam.20261405.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261405.16},
  abstract = {Chaotic systems, such as the Lorenz and Chen systems, models exhibit high sensitivity to initial conditions. They require fine discretization for accurate long-time integration. Classical time-stepping methods suffer from accumulated truncation errors, whereas, global spectral methods, though highly accurate, yield dense coefficient matrices that are computationally expensive, limiting their efficiency. In this paper we propose an efficient numerical scheme that combines the non-overlapping Domain Decomposition Method (DDM) with the Spectral Relaxation Method (SRM) to solve the Lorenz and Chen systems. The time domain is decomposed into smaller sub intervals, and SRM is applied in each subdomain to linearize the nonlinear terms iteratively, thereby avoiding Taylor series truncation errors while maintaining spectral accuracy. The performance of the proposed Multistage Spectral Relaxation Method (MSRM) is investigated for different step sizes h = 0.01 and h = 0.001 over a long integration time T= 10. Numerical results are compared with the single-domain SRM in terms of accuracy, CPU time and memory usage. The results demonstrate that MSRM achieves comparable accuracy to the single-domain approach while significantly reducing computational time and memory requirements, especially for smaller step sizes. The proposed MSRM provides an effective and efficient framework for the simulation of chaotic systems over a long-time interval.},
 year = {2026}
}

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TY  - JOUR
T1  - Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique
AU  - Maina Martha Wangeci
AU  - Samuel Mutua
AU  - Nicholas Mwilu Mutothya
Y1  - 2026/09/27
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261405.16
DO  - 10.11648/j.ajam.20261405.16
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 331
EP  - 338
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261405.16
AB  - Chaotic systems, such as the Lorenz and Chen systems, models exhibit high sensitivity to initial conditions. They require fine discretization for accurate long-time integration. Classical time-stepping methods suffer from accumulated truncation errors, whereas, global spectral methods, though highly accurate, yield dense coefficient matrices that are computationally expensive, limiting their efficiency. In this paper we propose an efficient numerical scheme that combines the non-overlapping Domain Decomposition Method (DDM) with the Spectral Relaxation Method (SRM) to solve the Lorenz and Chen systems. The time domain is decomposed into smaller sub intervals, and SRM is applied in each subdomain to linearize the nonlinear terms iteratively, thereby avoiding Taylor series truncation errors while maintaining spectral accuracy. The performance of the proposed Multistage Spectral Relaxation Method (MSRM) is investigated for different step sizes h = 0.01 and h = 0.001 over a long integration time T= 10. Numerical results are compared with the single-domain SRM in terms of accuracy, CPU time and memory usage. The results demonstrate that MSRM achieves comparable accuracy to the single-domain approach while significantly reducing computational time and memory requirements, especially for smaller step sizes. The proposed MSRM provides an effective and efficient framework for the simulation of chaotic systems over a long-time interval.
VL  - 14
IS  - 5
ER  -

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Author Information

Maina Martha Wangeci

Department of Mathematics, Statistics and Physical Sciences, Taita Taveta University, Voi, Kenya

Contact Email

http://orcid.org/0009-0008-9524-8602
Samuel Mutua

Department of Mathematics, Statistics and Physical Sciences, Taita Taveta University, Voi, Kenya

Contact Email

http://orcid.org/0000-0002-0061-813X
Nicholas Mwilu Mutothya

Department of Mathematics, Statistics and Physical Sciences, Taita Taveta University, Voi, Kenya

Contact Email

http://orcid.org/0000-0001-7907-048X

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Plain Text BibTeX RIS

APA Style

Wangeci, M. M., Mutua, S., Mutothya, N. M. (2026). Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique. American Journal of Applied Mathematics, 14(5), 331-338. https://doi.org/10.11648/j.ajam.20261405.16

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ACS Style

Wangeci, M. M.; Mutua, S.; Mutothya, N. M. Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique. Am. J. Appl. Math. 2026, 14(5), 331-338. doi: 10.11648/j.ajam.20261405.16

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AMA Style

Wangeci MM, Mutua S, Mutothya NM. Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique. Am J Appl Math. 2026;14(5):331-338. doi: 10.11648/j.ajam.20261405.16

Copy | Download

@article{10.11648/j.ajam.20261405.16,
  author = {Maina Martha Wangeci and Samuel Mutua and Nicholas Mwilu Mutothya},
  title = {Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique},
  journal = {American Journal of Applied Mathematics},
  volume = {14},
  number = {5},
  pages = {331-338},
  doi = {10.11648/j.ajam.20261405.16},
  url = {https://doi.org/10.11648/j.ajam.20261405.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261405.16},
  abstract = {Chaotic systems, such as the Lorenz and Chen systems, models exhibit high sensitivity to initial conditions. They require fine discretization for accurate long-time integration. Classical time-stepping methods suffer from accumulated truncation errors, whereas, global spectral methods, though highly accurate, yield dense coefficient matrices that are computationally expensive, limiting their efficiency. In this paper we propose an efficient numerical scheme that combines the non-overlapping Domain Decomposition Method (DDM) with the Spectral Relaxation Method (SRM) to solve the Lorenz and Chen systems. The time domain is decomposed into smaller sub intervals, and SRM is applied in each subdomain to linearize the nonlinear terms iteratively, thereby avoiding Taylor series truncation errors while maintaining spectral accuracy. The performance of the proposed Multistage Spectral Relaxation Method (MSRM) is investigated for different step sizes h = 0.01 and h = 0.001 over a long integration time T= 10. Numerical results are compared with the single-domain SRM in terms of accuracy, CPU time and memory usage. The results demonstrate that MSRM achieves comparable accuracy to the single-domain approach while significantly reducing computational time and memory requirements, especially for smaller step sizes. The proposed MSRM provides an effective and efficient framework for the simulation of chaotic systems over a long-time interval.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - Numerical Investigation of Solutions of Chaotic Systems Via Domain Decomposition Technique
AU  - Maina Martha Wangeci
AU  - Samuel Mutua
AU  - Nicholas Mwilu Mutothya
Y1  - 2026/09/27
PY  - 2026
N1  - https://doi.org/10.11648/j.ajam.20261405.16
DO  - 10.11648/j.ajam.20261405.16
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 331
EP  - 338
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20261405.16
AB  - Chaotic systems, such as the Lorenz and Chen systems, models exhibit high sensitivity to initial conditions. They require fine discretization for accurate long-time integration. Classical time-stepping methods suffer from accumulated truncation errors, whereas, global spectral methods, though highly accurate, yield dense coefficient matrices that are computationally expensive, limiting their efficiency. In this paper we propose an efficient numerical scheme that combines the non-overlapping Domain Decomposition Method (DDM) with the Spectral Relaxation Method (SRM) to solve the Lorenz and Chen systems. The time domain is decomposed into smaller sub intervals, and SRM is applied in each subdomain to linearize the nonlinear terms iteratively, thereby avoiding Taylor series truncation errors while maintaining spectral accuracy. The performance of the proposed Multistage Spectral Relaxation Method (MSRM) is investigated for different step sizes h = 0.01 and h = 0.001 over a long integration time T= 10. Numerical results are compared with the single-domain SRM in terms of accuracy, CPU time and memory usage. The results demonstrate that MSRM achieves comparable accuracy to the single-domain approach while significantly reducing computational time and memory requirements, especially for smaller step sizes. The proposed MSRM provides an effective and efficient framework for the simulation of chaotic systems over a long-time interval.
VL  - 14
IS  - 5
ER  -

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