Abstract
A discrete time two‐nation arms race model involving a piecewise constant nonlinear control function is formulated and studied. By elementary but novel arguments, we are able to give a complete analysis of its asymptotic behavior when the threshold parameter in the control function varies from 0+ to ∞. We show that all solutions originated from positive initial values tend to limit one or two cycles. An implication is that when devastating weapons are involved, “terror equilibrium” can be achieved and escalated race avoided. It is hoped that our analysis will provide motivation for further studying of discrete‐time equations with piecewise smooth nonlinearities.
Highlights
In 1, pages 87–90, a simple dynamical model of a two-nation arms race based on Richardson’s ideas in 2 is explained, and several interesting conclusions are drawn which can be used to explain stable and escalated arms races
An 1 − rA An−1 sABn−1 u, 1.1 where the constant sA measures country A’s distrust of country B in that it reacts to the way B arms itself, rA ∈ 0, 1 is a measure of A’s own economy, and u is the basic annual expenditure
If we assume a similar situation for country B, we have
Summary
In 1, pages 87–90 , a simple dynamical model of a two-nation arms race based on Richardson’s ideas in 2 is explained, and several interesting conclusions are drawn which can be used to explain stable and escalated arms races. We will treat our λ as a bifurcation parameter and distinguish four different cases i λ > 1, ii λ 1, iii 0 < λ < 1 − a, and iv 1 − a ≤ λ < 1, and consider different z−2, z−1 in A, B, C, or D and discuss the precise asymptotic behaviors of the corresponding solutions determined by them.
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