Oscillatory integral characterizations of uniform exponential dichotomy for periodic evolution families

Abstract

Uniform exponential dichotomy is a fundamental notion in the qualitative theory of nonautonomous differential equations and evolution families. In this paper, we establish a new characterization of uniform exponential dichotomy for linear periodic nonautonomous systems in terms of bounded operator-valued oscillatory integral operators. More precisely, we show that the associated evolution family admits a uniform exponential dichotomy if and only if two families of forward and backward oscillatory integrals remain uniformly bounded with respect to the oscillation frequency. The analysis combines Floquet theory with a reduction to the monodromy operator and discrete spectral techniques, thereby linking continuous-time dynamics to the spectral properties of the corresponding discrete evolution. The proposed characterization provides an alternative to classical spectral and Floquet criteria and is well suited to operator-theoretic and perturbation methods. The theoretical results are complemented by illustrative examples and counterexamples demonstrating the sharpness of the assumptions. As applications, we derive robustness results under small periodic perturbations and establish bounded solvability criteria for periodically forced linear systems.