Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis

Rapheal Oladipo Fifelola; Sydney Chinedu Osuala; Adedapo Kehinde Femi

doi:doi:10.11648/j.ijssam.20240903.12

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Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis

Rapheal Oladipo Fifelola^*

, Sydney Chinedu Osuala, Adedapo Kehinde Femi

Published in International Journal of Systems Science and Applied Mathematics (Volume 9, Issue 3)

Received: 13 December 2024 Accepted: 2 January 2025 Published: 7 January 2025

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Abstract

This paper investigates the stability of the Lanchester Ordinary Differential Equation (ODE) in asymmetric warfare, where two forces with differing lethality coefficients engage. The system exhibits marginal stability at the equilibrium point after linearization, characterized by purely imaginary eigenvalues, indicating that the forces are balanced but without a definitive resolution. An energy-based analysis further supports this by identifying a characteristic frequency associated with the interaction of the forces. These findings suggest that asymmetric warfare scenarios are inherently prone to sustained oscillations, reflecting a dynamic equilibrium between the opposing forces. The presence of these oscillations indicates that while the forces may not decisively defeat one another, a long-term balance persists, preventing either side from achieving a clear victory. The results imply that asymmetric warfare is likely to lead to prolonged conflicts with no easy resolution, as the dynamics between the forces result in cyclical patterns of attack and defense. This work highlights the importance of understanding the stability of such systems, providing insights into the potential for sustained conflict when forces are unequal. The study contributes to the broader understanding of conflict dynamics, offering valuable perspectives on how these imbalances affect the course of warfare. The findings could inform military strategy, particularly in planning for engagements where one side holds a clear advantage over the other but still faces persistent resistance.

Published in	International Journal of Systems Science and Applied Mathematics (Volume 9, Issue 3)
DOI	10.11648/j.ijssam.20240903.12
Page(s)	44-54
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), 2025. Published by Science Publishing Group

Keywords

Lanchester Model, Asymmetric Warfare, Stability Analysis, Linearization, Eigenvalues, Marginal Stability, Conflict Dynamics, Oscillations

References

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[2]	W. H. Taylor, “The mathematics of warfare: Lanchester’s Laws,” Naval Research Logistics Quarterly, vol. 30, no. 4, pp. 453-474, 1983.
[3]	R. T. Rowland, “The legacy of Lanchester’s models in modern warfare,” Mathematical and Computer Modelling, vol. 44, no. 7-8, pp. 823-832, 2006. https://doi.org/10.1016/j.mcm.2006.04.020
[4]	P. K. Davis, “The application of Lanchester’s Laws to operational analysis in the battlefield,” Operations Research, vol. 43, no. 6, pp. 978-989, 1995. https://doi.org/10.1287/opre.43.6.978
[5]	P. J. Bracken, “Simulating warfare: The practical applications of Lanchester models,” Journal of Defense Modeling and Simulation, vol. 1, no. 1, pp. 47-57, 1995. https://doi.org/10.1177/154851299500100105
[6]	X. Ji, W. Zhang, F. Xiang, W. Yuan, and J. Chen, “A swarm confrontation method based on Lanchester law and Nash equilibrium,” Electronics, vol. 11, no. 6, pp. 896, 2022. https://doi.org/10.3390/electronics11060896
[7]	I. V. Kotlyarov, “The role of resource allocation in asymmetric conflicts,” Journal of Military Science, vol. 12, no. 3, pp. 235-248, 2021.
[8]	M. Kress, “Lanchester models for modern multi- polar battles,” European Journal of Operational Research, vol. 264, no. 1, pp. 46-54, 2018. https://doi.org/10.1016/j.ejor.2017.07.026
[9]	R. O. Fifelola, U. L. Okafor, and K. F. Adedapo, “Analyzing the Stability of Lanchester Warfare Models for Symmetric Warfare Scenarios,” International Journal of Applied Mathematical Research, vol. 13, no. 2, pp. 69-73, 2024. https://doi.org/10.14419/4j2s5d90
[10]	M. Kress, “Operational analysis in combat models,” Mathematics and Computers in Simulation, vol. 76, no. 1-3, pp. 240-249, 2007.
[11]	C. Spradlin and G. Spradlin, “Extended Lanchester’s equations: Exploring multi-dimensional combat dynamics,” Journal of Mathematical Warfare, vol. 5, no. 1, pp. 1-23, 2007.
[12]	X. Chen, Y. Jing, C. Li, and M. Li, “Warfare command stratagem analysis for winning based on Lanchester attrition models,” Journal of Systems Science and Applied Mathematics, vol. 21, no. 1, pp. 94-105, 2012. https://link.springer.com/article/10.1007/s11518- 011-5177-7
[13]	V. Protopopescu, R. T. Santoro, R. L. Cox, and P. Rusu, “Combat modeling with partial differential equations: The bidimensional case,” Technical Report, 1990. https://doi.org/10.2172/5233113
[14]	M. Kress, “Lanchester models for irregular warfare,” Mathematics, vol. 8, no. 5, pp. 737, 2020. Special Issue on Mathematical Models in Security, Defense, Cyber Security, and Cyber Defense. https://doi.org/10.3390/math8050737
[15]	A. C. Kalloniatis, K. Hoek, M. Zuparic, and M. Brede, “Optimising structure in a networked Lanchester model for fires and manoeuvre in warfare,” Journal of Systems Science and Applied Mathematics, vol. 29, no. 6, pp. 1863-1878, 2020. https://doi.org/10.1080/01605682.2020.1745701

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APA Style

Fifelola, R. O., Osuala, S. C., Femi, A. K. (2025). Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis. International Journal of Systems Science and Applied Mathematics, 9(3), 44-54. https://doi.org/10.11648/j.ijssam.20240903.12

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ACS Style

Fifelola, R. O.; Osuala, S. C.; Femi, A. K. Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis. Int. J. Syst. Sci. Appl. Math. 2025, 9(3), 44-54. doi: 10.11648/j.ijssam.20240903.12

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AMA Style

Fifelola RO, Osuala SC, Femi AK. Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis. Int J Syst Sci Appl Math. 2025;9(3):44-54. doi: 10.11648/j.ijssam.20240903.12

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@article{10.11648/j.ijssam.20240903.12,
  author = {Rapheal Oladipo Fifelola and Sydney Chinedu Osuala and Adedapo Kehinde Femi},
  title = {Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {9},
  number = {3},
  pages = {44-54},
  doi = {10.11648/j.ijssam.20240903.12},
  url = {https://doi.org/10.11648/j.ijssam.20240903.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20240903.12},
  abstract = {This paper investigates the stability of the Lanchester Ordinary Differential Equation (ODE) in asymmetric warfare, where two forces with differing lethality coefficients engage. The system exhibits marginal stability at the equilibrium point after linearization, characterized by purely imaginary eigenvalues, indicating that the forces are balanced but without a definitive resolution. An energy-based analysis further supports this by identifying a characteristic frequency associated with the interaction of the forces. These findings suggest that asymmetric warfare scenarios are inherently prone to sustained oscillations, reflecting a dynamic equilibrium between the opposing forces. The presence of these oscillations indicates that while the forces may not decisively defeat one another, a long-term balance persists, preventing either side from achieving a clear victory. The results imply that asymmetric warfare is likely to lead to prolonged conflicts with no easy resolution, as the dynamics between the forces result in cyclical patterns of attack and defense. This work highlights the importance of understanding the stability of such systems, providing insights into the potential for sustained conflict when forces are unequal. The study contributes to the broader understanding of conflict dynamics, offering valuable perspectives on how these imbalances affect the course of warfare. The findings could inform military strategy, particularly in planning for engagements where one side holds a clear advantage over the other but still faces persistent resistance.},
 year = {2025}
}

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TY  - JOUR
T1  - Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis
AU  - Rapheal Oladipo Fifelola
AU  - Sydney Chinedu Osuala
AU  - Adedapo Kehinde Femi
Y1  - 2025/01/07
PY  - 2025
N1  - https://doi.org/10.11648/j.ijssam.20240903.12
DO  - 10.11648/j.ijssam.20240903.12
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 44
EP  - 54
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20240903.12
AB  - This paper investigates the stability of the Lanchester Ordinary Differential Equation (ODE) in asymmetric warfare, where two forces with differing lethality coefficients engage. The system exhibits marginal stability at the equilibrium point after linearization, characterized by purely imaginary eigenvalues, indicating that the forces are balanced but without a definitive resolution. An energy-based analysis further supports this by identifying a characteristic frequency associated with the interaction of the forces. These findings suggest that asymmetric warfare scenarios are inherently prone to sustained oscillations, reflecting a dynamic equilibrium between the opposing forces. The presence of these oscillations indicates that while the forces may not decisively defeat one another, a long-term balance persists, preventing either side from achieving a clear victory. The results imply that asymmetric warfare is likely to lead to prolonged conflicts with no easy resolution, as the dynamics between the forces result in cyclical patterns of attack and defense. This work highlights the importance of understanding the stability of such systems, providing insights into the potential for sustained conflict when forces are unequal. The study contributes to the broader understanding of conflict dynamics, offering valuable perspectives on how these imbalances affect the course of warfare. The findings could inform military strategy, particularly in planning for engagements where one side holds a clear advantage over the other but still faces persistent resistance.
VL  - 9
IS  - 3
ER  -

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Author Information

Rapheal Oladipo Fifelola

Department of Mathematical Sciences, Nigerian Defence Academy, Kaduna, Nigeria

Contact Email

http://orcid.org/0009-0000-8218-8884
Sydney Chinedu Osuala

Department of Mathematical Sciences, Nigerian Defence Academy, Kaduna, Nigeria

Contact Email
Adedapo Kehinde Femi

Department of Physical and Chemical Sciences, Federal University of Health Sciences, Ila-Orangun, Nigeria

Contact Email

http://orcid.org/0009-0006-1786-5053

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Plain Text BibTeX RIS

APA Style

Fifelola, R. O., Osuala, S. C., Femi, A. K. (2025). Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis. International Journal of Systems Science and Applied Mathematics, 9(3), 44-54. https://doi.org/10.11648/j.ijssam.20240903.12

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ACS Style

Fifelola, R. O.; Osuala, S. C.; Femi, A. K. Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis. Int. J. Syst. Sci. Appl. Math. 2025, 9(3), 44-54. doi: 10.11648/j.ijssam.20240903.12

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AMA Style

Fifelola RO, Osuala SC, Femi AK. Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis. Int J Syst Sci Appl Math. 2025;9(3):44-54. doi: 10.11648/j.ijssam.20240903.12

Copy | Download

@article{10.11648/j.ijssam.20240903.12,
  author = {Rapheal Oladipo Fifelola and Sydney Chinedu Osuala and Adedapo Kehinde Femi},
  title = {Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis},
  journal = {International Journal of Systems Science and Applied Mathematics},
  volume = {9},
  number = {3},
  pages = {44-54},
  doi = {10.11648/j.ijssam.20240903.12},
  url = {https://doi.org/10.11648/j.ijssam.20240903.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20240903.12},
  abstract = {This paper investigates the stability of the Lanchester Ordinary Differential Equation (ODE) in asymmetric warfare, where two forces with differing lethality coefficients engage. The system exhibits marginal stability at the equilibrium point after linearization, characterized by purely imaginary eigenvalues, indicating that the forces are balanced but without a definitive resolution. An energy-based analysis further supports this by identifying a characteristic frequency associated with the interaction of the forces. These findings suggest that asymmetric warfare scenarios are inherently prone to sustained oscillations, reflecting a dynamic equilibrium between the opposing forces. The presence of these oscillations indicates that while the forces may not decisively defeat one another, a long-term balance persists, preventing either side from achieving a clear victory. The results imply that asymmetric warfare is likely to lead to prolonged conflicts with no easy resolution, as the dynamics between the forces result in cyclical patterns of attack and defense. This work highlights the importance of understanding the stability of such systems, providing insights into the potential for sustained conflict when forces are unequal. The study contributes to the broader understanding of conflict dynamics, offering valuable perspectives on how these imbalances affect the course of warfare. The findings could inform military strategy, particularly in planning for engagements where one side holds a clear advantage over the other but still faces persistent resistance.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Stability Analysis of Asymmetric Warfare Dynamics Using the Lanchester Ordinary Differential Equation Model Through Jacobian Linearization and Energy Analysis
AU  - Rapheal Oladipo Fifelola
AU  - Sydney Chinedu Osuala
AU  - Adedapo Kehinde Femi
Y1  - 2025/01/07
PY  - 2025
N1  - https://doi.org/10.11648/j.ijssam.20240903.12
DO  - 10.11648/j.ijssam.20240903.12
T2  - International Journal of Systems Science and Applied Mathematics
JF  - International Journal of Systems Science and Applied Mathematics
JO  - International Journal of Systems Science and Applied Mathematics
SP  - 44
EP  - 54
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20240903.12
AB  - This paper investigates the stability of the Lanchester Ordinary Differential Equation (ODE) in asymmetric warfare, where two forces with differing lethality coefficients engage. The system exhibits marginal stability at the equilibrium point after linearization, characterized by purely imaginary eigenvalues, indicating that the forces are balanced but without a definitive resolution. An energy-based analysis further supports this by identifying a characteristic frequency associated with the interaction of the forces. These findings suggest that asymmetric warfare scenarios are inherently prone to sustained oscillations, reflecting a dynamic equilibrium between the opposing forces. The presence of these oscillations indicates that while the forces may not decisively defeat one another, a long-term balance persists, preventing either side from achieving a clear victory. The results imply that asymmetric warfare is likely to lead to prolonged conflicts with no easy resolution, as the dynamics between the forces result in cyclical patterns of attack and defense. This work highlights the importance of understanding the stability of such systems, providing insights into the potential for sustained conflict when forces are unequal. The study contributes to the broader understanding of conflict dynamics, offering valuable perspectives on how these imbalances affect the course of warfare. The findings could inform military strategy, particularly in planning for engagements where one side holds a clear advantage over the other but still faces persistent resistance.
VL  - 9
IS  - 3
ER  -

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