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This study provides a brief overview of the application of possible modifications of Lanchester-type models, namely, the representation of differential equations of such models in stochastic form. The stochastic setting of differential levels is used in Dynamic models if it is necessary to take into account the influence of random fluctuations (in particular, in radio engineering, thermodynamics, population dynamics models, etc.). As for Lanchester-type models, their stochastic appearance would allow considering the influence of random factors and elements of uncertainty, which are present to a certain extent in any combat operations. At the same time, unlike deterministic models, the numerical solution of systems of stochastic differential equations in such models requires the use of special methods, the choice of a specific one may be based on the requirements for the need to obtain an unambiguous approximate solution, or the probability distribution of the desired quantities. The possibility of obtaining different types of solutions is due to a characteristic feature of the developed methods for numerical integration of stochastic differential equations, namely, the existence of weak and strong approximate methods for solving them. For Lanchester equations, as models for predicting the probable course and results of combat operations, it seems appropriate to obtain a solution precisely in the form of parameters for distributions of random variables, which is possible after processing the results of using weak numerical methods. In addition, such methods are considered easier to implement in practice. Of particular note are the issues of estimating the stability of solutions (in the sense of Lyapunov) of stochastic models. While for Lanchester-type models, approximate practical methods for estimating stability can be considered, especially in relation to the simplest, linear statements of basic equations. The study considers an example of using the stochastic Lanchester-type model based on a system of linear inhomogeneous differential equations, with assumptions about the stability of solutions to the stochastic formulation of such equations. [ABSTRACT FROM AUTHOR]

A popular mathematical model for the formation of an inhomogeneous topography on the surface of a plate (flat substrate) during ion bombardment was considered. The model is described by the Bradley-Harper equation, which is frequently referred to as the generalized Kuramoto-Sivashinsky equation. It was shown that inhomogeneous topography (nanostructures in the modern terminology) can arise when the stability of the plane incident wavefront changes. The problem was solved using the theory of dynamical systems with an infinite-dimensional phase space, in conjunction with the integral manifold method and Poincaré-Dulac normal forms. A normal form was constructed using a modified Krylov-Bogolyubov algorithm that applies to nonlinear evolutionary boundary value problems. As a result, asymptotic formulas for solutions of the given nonlinear boundary value problem were derived. [ABSTRACT FROM AUTHOR]