Two-sided bounds for the largest Lyapunov exponent and exponential stability criteria for nonlinear systems with arbitrary delays

Abstract

We consider a system of nonlinear differential equations with a given linear part, a nonlinear term bounded in the norm, and variable concentrated and distributed delays. We find two-sided bounds on the maximal Lyapunov exponent expressed via the norm of the nonlinear term and maxima of the delay functions. For some systems, we find the exact value of this exponent. These results give sufficient (and, in some cases, necessary) conditions for a system's exponential stability which are invariant with respect to the delay. We give examples that illustrate our method.