Asymptotic behavior of first order scalar linear autonomous retarded functional di¤erential equations
This paper studies the asymptotic behavior of first order scalar linear autonomous Retarded Functional Differential Equations (RFDE). A spectral decomposition of the solution in terms of the exponential solutions is used to define the Dominant Spectral Component as the sum (with appropriate coefficients) of exponential solutions with characteristic roots of maximal real part, and likewise the Predominant Spectral Components as the sum of the exponential solutions with characteristic roots of maximal real part plus those (if any) with real parts greater than or equal to zero. Exponential bounds on the differences between the Solution and the Dominant Spectral Component (Predominant Spectral Components) provide a framework to investigate the exponential convergence to asymptotic behavior. Exponential solutions in the Dominant Spectral Component with real (complex) characteristic roots give rise to nonoscillatory (oscillatory) asymptotic behavior. Numerics for a simple RFDE illuminates the characteristics of the asymptotic behavior, illustrates how the asymptotic behavior depends on the characteristic roots in the right-most half plane, and demonstrates that the exponential convergence to asymp- totic behavior is fairly rapid in a few delays. Tauberian techniques are used to express the asymptotic behavior of the integral of the solution in terms of its Laplace transform.
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