On the zeros of exponential polynomials |
| |
Authors: | Cerino E Avellar Jack K Hale |
| |
Institution: | 1. Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912 U.S.A.;2. Universidade Federal de São Carlos, Departamento de Matematica, 13560-São Carlos, São Paulo, Brazil |
| |
Abstract: | Suppose r = (r1, …, rM), rj ? 0, γkj ? 0 integers, k = 1, 2, …, N, j = 1, 2, …, M, γk · r = ∑jγkjrj. The purpose of this paper is to study the behavior of the zeros of the function h(λ, a, r) = 1 + ∑j = 1Naje?λγj · r, where each aj is a nonzero real number. More specifically, if , we study the dependence of . This set is continuous in a but generally not in r. However, it is continuous in r if the components of r are rationally independent. Specific criterion to determine when are given. Several examples illustrate the complicated nature of . The results have immediate implication to the theory of stability for difference equations x(t) ? ∑k = 1MAkx(t ? rk) = 0, where x is an n-vector, since the characteristic equation has the form given by h(λ, a, r). The results give information about the preservation of stability with respect to variations in the delays. The results also are fundamental for a discussion of the dependence of solutions of neutral differential difference equations on the delays. These implications will appear elsewhere. |
| |
Keywords: | |
本文献已被 ScienceDirect 等数据库收录! |
|