Stability criteria for off-diagonally monotone nonlinear dynamical systems

Abstract

This paper discusses the global stability of a nonlinear dynamical system \dot{x}=f(x) in which f is a locally Lipschitz continuous off-diagonally monotone function and f(\theta) > \theta . Two results are proved: 1) if f is piecewise-linear function and if -f is an M -function, then a unique equilibrium point exists and it is globally asymptotically stable; 2) if f is a nonlinear function with separate variables in the sense that f is given by f_{1}(x)= \Sigma^{n}_{j-1}f_{ij}(x_j) for all i , and if -f is an M -function satisfying f(x^{\ast})= \theta for some nonnegative vector x^{\ast} , then x^{\ast} is globally asymptotically stable. These results are applied to the stability analyses of a large scale composite system and a compartmental system.