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The aim of this paper is to study the dynamical behavior of positive solutions for a system of rational difference equations of the following form:un+1=αun-1/β+γvn-2p,vn+1=α1vn-1/β1+γ1un-2p,n=0,1,…, where the parametersα,β,γ,α1,β1,γ1,pand the initial valuesu-i,v-ifori=0,1,2are positive real numbers.

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Several recent results regarding common diagonal Lyapunov solutions are further explored here. The first one, attributed to Redheffer and revisited by Shorten and Narendra, reduces the diagonal stability of a matrix to common diagonal Lyapunov solutions on two matrices of order one less. We present a shorter, purely matrix-theoretic proof of this result along with its extensions. The second one concerns two different necessary and sufficient conditions, due to Oleng, Narendra, and Shorten, for a pair of 2 × 2 matrices to share a common diagonal Lyapunov solution. We show that these two conditions are connected directly to each other.

We investigate the behavior of the solutions of the recursive sequence  xn+1=α+xn-1/xnk,  n=0,1,…, whereα∈[0,∞),k∈(0,∞),and the initial conditionsx-1,x0are arbitrary positive numbers. Included are results that considerably improve those in the recently published paper by Hamza and Morsy (2009).