Equilibrium solutions, stabilities and dynamics of Lanchester's equations with optimization of initial force commitments.

Abstract

Generalized Lanchester- type differential equations are used to study combat processes. This system of non-linear equations has multiple equilibrium solutions which can be determined by a numerical technique called the Continuation Method. Useful properties pertaining to neighborhood stability are derived by considering the lowest-dimensional (1*1) problem. A new set of parameters based on the system asymptotes is defined and used to characterize stability. System dynamics are investigated using phase trajectories which are found to depend on the domains of attraction and stabilities of surrounding equilibria. The effect of varying initial force levels (X,Y) is studied by calculating an objective function which is the difference of the losses at the end of a multistage battle simulation. Based on the minimax theorem, a set of mixed strategies for (X,Y) can be found. For highly unstable warfare with large war resources, instability can be used to influence battle outcome.