SIAM REVIEW c  2002 Society for Industrial and Applied Mathematics Vol. 44, No. 1, pp. 74–92 Convergence, Oscillations, and Chaos in a Discrete Model of Combat * Hassan Sedaghat † Abstract. A piecewise smooth mapping of the three-dimensional Euclidean space is derived from a discrete-time model of combat. The mathematical analysis of this mapping focuses on the effects of discontinuity caused by the defender’s withdrawal strategy—a prime component of the original model. Both the asymptotics and the transient behavior are discussed, and all the behavior types noted in the title are established as possible outcomes. Key words. transient behavior, asymptotic behavior, limit cycles, chaos, attrition rates, withdrawal rate AMS subject classifications. 39A11, 37E99, 37N99 PII. S003614450138299X 1. Introduction. Deterministic combat models often serve as foundations upon which more complex war games (often with stochastic add-ons) may be based. In this paper, we study the ground version of a combat model proposed by Epstein in [2]. The salient feature of this model is a withdrawal mechanism that, in addition to conventional warfare, enables one to model specialized types of combat like guerrilla warfare (where the battle front is frequently in motion). Although Epstein gave a full derivation of his model, plus historical background and some numerical simulations, in [2, 3], he did not offer a mathematical analysis. In particular, it is by no means clear from the presentations given in [2] or [3] that the model is capable of exhibiting all the behavior types mentioned in the title of this paper. The ground version of Epstein’s model (i.e., without the air support component) may be formulated as a three-dimensional nonlinear system of difference equations that can be represented by a piecewise smooth map. We wish mainly to examine the consequences of a jump discontinuity in the state space—a phenomenon caused by the defender’s aforementioned withdrawal strategy. The methodology for analyzing the discontinuous mapping is based on a somewhat intuitive approach. When a trajectory jumps back and forth between regions with different dynamical regimes, we say that it is mode-switching. Taking advantage of a damping effect, we show that in spite of mode-switching (seen as damped oscillations in the time series), most trajectories converge to a fixed point of the system (Theorem 1). However, the asymptotic behavior is less predictable in a two-dimensional (invari- ant) subspace where damping is not present. Some aspects of the asymptotic behavior * Received by the editors January 2, 2001; accepted for publication (in revised form) August 23, 2001; published electronically February 1, 2002. http://www.siam.org/journals/sirev/44-1/38299.html † Department of Mathematics, Virginia Commonwealth University, P.O. Box 842014, Richmond, VA 23284-2014 (hsedagha@vcu.edu). 74