Global Stability of Infinite Systems of Nonlinear Differential Equations and Nonhomogeneous Countable Markov Chains

Countable systems of differential equations $\dot x=f(x)$ in $X\subset l_1$ with bounded Jacobi operators $J(x)=\partial f/\partial x$ are studied. Sufficient conditions of global stability and global asymptotic stability are obtained, where for any $x\in X$ the matrix $J^T(x)$ is the transition-rate matrix for a countable Markov chain and $X$ is a subset of a linear affine variety. Results are applied to two infinite systems arising from the modern queueing theory.