LYAPUNOV AND SACKER-SELL SPECTRAL INTERVALS

Abstract. In this work, we show that for linear upper triangular systems of dierential equations, we can use the diagonal entries to obtain the Sacker and Sell, or Exponential Dichotomy, and also {under some restrictions{ the Lyapunov spectral intervals. Since any bounded and continuous coecient matrix function can be smoothly transformed to an upper triangular matrix function, our results imply that these spectral intervals may be found from scalar homogeneous problems. In line with our previous work [8], we emphasize the role of integral separation. Relationships between dierent spectra are shown, and examples are used to illustrate the results and dene types of coecient matrix functions that lead to continuous Sacker-Sell spectrum and/or continuous Lyapunov spectrum. Key words. Exponential dichotomy, Sacker-Sell spectrum, Lyapunov exponents, integral separation. AMS subject classications. 35A, 65L 1. Introduction. Since the thesis of Lyapunov more than one hundred years ago, see [16], dierent characteri-zations of spectra for linear nonautonomous systems of dierential equations have been proposed. These spectra also characterize stability properties of a solution trajectory of a nonlinear system, via linearization about the solution trajectory itself