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The Lanchester square-law model extended to a (2,2) conflict. (English) Zbl 0797.92031

This paper concerns the Lanchester model of conflict \(\dot z=Az\), where \(z\) is a \(2n\)-vector and \(A\) is a \(2n\times 2n\) matrix of the form \(({0\atop -b}{-a\atop 0})\). The authors consider the inverse problem, which is to calculate the parameter-matrices \(a,b\) by observing suitable positions of \(z\). Thus, when \(n=1\), just two observations suffice, since the trajectories are level curves of the Hamiltonian \(a(y^ 2-y_ 0^ 2)-b(x^ 2-x^ 2_ 0)\).
This ‘Hamiltonian’ approach is then followed for the case \(n=2\), and the authors produce transformations of the variables to obtain ‘decoupled’ equations with simple solutions. From this, it follows that just three observations of \(z\) will suffice to calculate \(a\) and \(b\), provided a certain determinant is zero. Two numerical examples are discussed in detail. (Presumably there are some interesting canonical forms and bifurcations here, but the paper does not discuss them).

MSC:

91D99 Mathematical sociology (including anthropology)
34A30 Linear ordinary differential equations and systems
37-XX Dynamical systems and ergodic theory
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