Spatial distributed Lanchester model considering nonlinear dynamics

Nikita D. Borisov, Vladimir V. Nefedov

Abstract


The paper proposes a spatially distributed Lanchester model for simulating the dynamics of military conflicts, accounting for diffusion and reaction processes in a two-dimensional domain with obstacles. The model describes the evolution of two troop groups, denoted as U(x, y, t) and V(x, y, t), through a system of nonlinear partial differential equations with diffusion coefficients and time-modulated reaction terms. Interaction occurs after the loss of communication with the central command post, when the main strike and force concentration have already been completed. In other words, the modeled event represents a local interaction implemented according to a pre-agreed tactical plan. The study examines the impact of three types of obstacles—square, circular, and smooth convex—modeling impassable areas that affect the spatial dynamics of wave fronts. The numerical solution of the system is implemented using the alternating direction method and the finite element method with time discretization based on the Crank-Nicolson scheme, ensuring stability through adaptive mesh refinement near obstacles. Initial conditions reflect the distribution of troops along the front line with homogeneous Dirichlet boundary conditions applied. The model's stability is confirmed by Lyapunov exponent analysis, and sensitivity to parameters is investigated to assess their impact on system dynamics. The results of numerical experiments illustrate the influence of obstacle geometry on troop concentrations, with a quantitative analysis of the proportions of areas dominated by forces over time.

Full Text:

PDF (Russian)

References


Shumov V. V., Korepanov V. O., «Mathematical Models of Combat and Military Operations», UBS 105, 15—30, 2023 [in Russian]

Novikov D. A., «Hierarchical Models of Military Operations», UBS 37, 25—62, 2012 [in Russian]

Krasnoshchekov P. S., Petrov A. A., «Principles of Model Construction», Mosk. Gos. Univ., Moscow, 1983 [in Russian]

Atkinson M. P., Kress M., MacKay N., «Targeting, Deployment, and Loss-Tolerance in Lanchester Engagements», Oper. Res. 69 (1), 53-68, 2020. Doi: 10.1287/opre.2020.1992

Chung D., Jeong B., «Analyzing Russia--Ukraine War Patterns Based on Lanchester Model Using SINDy Algorithm», J. Defense Model. Simul., 2023. Doi: 10.1177/15485129231176766

Kostić M., Jovanović A., «Lanchester’s Differential Equations as Operational Command Decision Making Tools», Mil. Tech. Courier 68 (3), 523—540, 2020. Doi: 10.5937/vojnotethnicski-343844057

Kress M., «Lanchester Models for Irregular Warfare», Math. Comput. Model. 52 (1--2), 72—80, 2010. Doi: 10.1016/j.mcm.2010.02.005

González E., Villena M., «Spatial Lanchester Models», Eur. J. Oper. Res. 210 (3), 706—715, 2011. Doi: 10.1016/j.ejor.2010.11.002

Cangiotti N., Capolli M., Sensi M., «A Generalization of Unaimed Fire Lanchester’s Model in Multi-battle Warfare», Appl. Math. Comput. 416, 2022. Doi: 10.1016/j.amc.2021.126741

Keane T., «Partial Differential Equations versus Cellular Automata for Modelling Combat,», J. Defense Model. Simul. 8 (2), 65—74, 2011. Doi: 10.1177/1548512910393302

Belavin V. A., Kurdyumov S. P., «Regimes with Sharpening in Demographic Systems», Vychisl. Mat. 3, 45—56, 2000) [in Russian]

Galaktionov V. A., Kurdyumov S. P., Samarsky A. A., «Quasilinear Heat Conduction Equation with a Source: Sharpening Regimes, Localization, and Exact Solutions», Uspekhi Nauki i Tekhniki 2, 23—34, 1986 [in Russian]

Arnold V. I., «Theory of Catastrophes», Nauka, Moscow, 1990 [in Russian]

Marchuk G. I., «Splitting Methods», Nauka, Moscow, 1988 [in Russian]

Samarsky A. A., «Theory of Difference Schemes», Nauka, Moscow, 2001 [in Russian]

Gallagher J. G., «Finite Element Method», Nauka, Moscow, 1984, [in Russian]

Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P., «Numerical Recipes: The Art of Scientific Computing», Cambridge University Press, Cambridge, 2007

Henry D., «Geometric Theory of Semilinear Parabolic Equations», Mir, Moscow, 1985; Springer, Berlin, 1981

Fradkov A., «Dynamics and Stability under Iterated Sanctions and Counter-sanctions», Autom. Remote Control 82 (3), 401—415, 2021. Doi: 10.1134/S0005117921030045

Fryaznov I. V., Bakirova M. I., «On Economical Difference Schemes for the Heat Conduction Equation in Curvilinear Coordinates», Zh. Vychisl. Mat. Mat. Fiz. 12 (4), 987—996, 1972 [in Russian]

Liu Y., Zhang X., Du H., Wang G., Zeng D., «Construction and Simulation of Lanchester Battle Equations Based on Space-based Information Support», J. Phys.: Conf. Ser. 1955, 2021. Doi: 10.1088/1742-6596/1955/1/012089

Korepanov V. O., Chkhartishvili A. G., Shumov V. V., «Basic Models of Combat Operations», UBS 103, 40—77, 2023 [in Russian]

Kurdyumov S. P., Malinetsky G. G., Potapov A. B., «Regimes with Sharpening in Two-Component Media», Mat. Model. 1, 12—25, 1989 [in Russian]

McCartney M., «Battling with Lanchester’s Equations in the Classroom», Int. J. Math. Educ. Sci. Technol. 54 (3), 451—461, 2023. Doi: 10.1080/0020739X.2021.2022230

Richardson L. F., «Mathematical Theory of War», J. Am. Stat. Assoc. 52 (280), 497—510, 1957. Doi: 10.1080/01621459.1957.10501410

Yoo B. J., «Statistical Review and Explanation for Lanchester Model», Int. J. Appl. Math. Stat. 59 (4), 123—135, 2020

Arnold V. I., «Hard and Soft Mathematical Models», MCNMO, Moscow, 2004) [in Russian].

Zang W. B., «Synergetic Economics: Time and Change in Nonlinear Economic Theory», Springer, Berlin, 1999


Refbacks

  • There are currently no refbacks.


Abava  Кибербезопасность ИТ конгресс СНЭ

ISSN: 2307-8162