## 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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