Large spectral gap induced by small delay and its application to reduction

Shuang Chen; Jun Shen

doi:10.3934/dcds.2020190

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2020, Volume 40, Issue 7: 4533-4564. Doi: 10.3934/dcds.2020190

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Large spectral gap induced by small delay and its application to reduction

Shuang Chen^1, and
Jun Shen^2, ,

1.
Center for Mathematical Sciences, Huazhong University of Sciences and Technology, Wuhan, Hubei 430074, China
2.
School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

^* Corresponding author: Jun Shen, junshen85@163.com
^* Corresponding author: Jun Shen, junshen85@163.com

Received: October 2019

Published: April 2020

This work was partly supported by NSFC grant (No.11831012), the Fundamental Research Funds for the Central Universities (No.YJ201646), International Visiting Program for Excellent Young Scholars of SCU

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Abstract

We consider general linear neutral differential equations with small delays in the view of pseudo exponential dichotomy. For the autonomous case, we first count the eigenvalues in a certain half plane, which generalized the previous works on serval special retarded differential equations. We next establish the existence of a pseudo exponential dichotomy for the nonautonomous case, and prove that the corresponding spectral gap approaches infinity as the delay tends to zero. The proof for this large spectral gap induced by small delay is owing to exact bounds and exponents for pseudo exponential dichotomy. Then based on above results, we give an invariant manifold reduction theorem for nonlinear neutral differential equations with small delays. Finally, our results are applied to a concrete example.

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Mathematics Subject Classification: Primary: 34K06; Secondary: 34D09, 34K19.

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