Characteristic trajectories of generalized Lanchester equations

Abstract

Generalized Lanchester-type differential equations are used to model attrition processes. This system of non-linear equations has multiple equilibrium solutions, which can be determined by a numerical technique called the Continuation Method when the problem's dimensionality is moderate. System dynamics are investigated and shown to depend critically on a domain of attraction defined by a tube which connects the non-negative equilibrium points and contains the dominant eigenvector at those points. Principles are presented and illustrated for mapping NM-dimensional systems into equivalent two-dimensional systems. This capability is especially important when aggregating subsystems in multi-level systems modeling. It is shown that the two-dimensional Lanchester systems have only four distinct modes of behavior, depending on the number of real positive equilibrium points that they have. A method is described and illustrated for reallocating attrition as state variables approach zero in order to guarantee their non-negativity.