Upper Semicontinuity of Morse Sets of a Discretization of a Delay-Differential Equation
Upper Semicontinuity of Morse Sets of a Discretization of a Delay-Differential Equation
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DOI：
10.1006/jdeq.1998.3507
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发表时间：
1999
影响因子：
2.4
通讯作者：
Tomáš Gedeon;G. Hines
中科院分区：
文献类型：
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作者：
Tomáš Gedeon;G. Hines
摘要本文考虑一类带负反馈的离散时滞问题x（t）= f（x（t），x（t-1））沿着一类步长为1/ n的时间离散化.在原来的问题中，吸引子承认一个很好的莫尔斯分解。我们证明了离散化问题具有全局吸引子。T. Gedeon和K. Mischaikov（1995，J. Dynamical Differential Equations 7，141-190）指出，这样的吸引子也允许莫尔斯分解。然后证明了关于单个莫尔斯集的某些连续性结果，包括如果f（x，y）= f（y），则单个莫尔斯集在n =∞处是上连续的.
Abstract In this paper, we consider a discrete delay problem with negative feedback x ( t )= f ( x ( t ), x ( t −1)) along with a certain family of time discretizations with stepsize 1/ n . In the original problem, the attractor admits a nice Morse decomposition. We prove that the discretized problems have global attractors. It was proved by T. Gedeon and K. Mischaikov (1995, J. Dynamical Differential Equations 7 , 141–190) that such attractors also admit Morse decompositions. We then prove certain continuity results about the individual Morse sets, including that if f ( x , y )= f ( y ), then the individual Morse sets are upper semicontinuous at n =∞.