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Spectrums of almost periodic linear systems

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Abstract

In this paper we will prove that the almost periodic linear system has only the point spectrum. At first we will introduce the concepts of upper and lower characteristic exponents and uniform upper and uniform lower characteristic exponents. Then we will prove that these two concepts are the same in the case of almost periodic linear systems. Furthermove we will prove that for any almost periodic linear system

$$\frac{{dx}}{{dt}} = A(t)x$$

there is the fundamental matrixX(t)=(x 1(t),...,x n (t)) with

$$\mathop {\lim }\limits_{t - s \to \infty } \frac{1}{{t - s}}\log \left( {\left\| {x_i (t)} \right\|/\left\| {x_i (s)} \right\|} \right) = \lambda _i , i = 1,2, \cdots ,n,$$

which implies (0) has only the point spectrum.

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Zhen-Sheng, L. Spectrums of almost periodic linear systems. Japan J. Appl. Math. 6, 491 (1989). https://doi.org/10.1007/BF03167890

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  • DOI: https://doi.org/10.1007/BF03167890

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