Equivariant Normal Forms for Parameterized Delay Differential Equations with Applications to Bifurcation Theory  

作　　者：Shang Jiang GUO Yu Ming CHEN Jian Hong WU 

机构地区：[1]College of Mathematics and Econometrics,Hu'nan University [2]Department of Mathematics,Wilfrid Laurier University [3]Department of Mathematics and Statistics,York University

出　　处：《Acta Mathematica Sinica,English Series》2012年第4期825-856,共32页数学学报（英文版）

基　　金：supported by NSFC(Grant No.10971057);the Key Project of Chinese Ministry of Education(Grant No.[2009]41);by Hu'nan Provincial Natural Science Foundation(Grant No.10JJ1001);by the Fundamental Research Funds for the Central Universities,Hu'nan University;supported by NSERC of Canada;by ERA Program of Ontario;supported in part by MITACS,CRC,and NSERC of Canada

摘　　要：In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.

关 键 词：Equivariant delay differential equation normal form EQUILIBRIUM stability BIFURCATION 

分 类 号：O175[理学—数学]