# American Institute of Mathematical Sciences

October  1999, 5(4): 813-824. doi: 10.3934/dcds.1999.5.813

## Inertial manifolds with and without delay

 1 Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Received  December 1997 Revised  October 1998 Published  July 1999

This article discusses the relationship between the inertial manifolds "with delay" introduced by Debussche & Temam, and the standard definition. In particular, the "multi-valued" manifold of the same paper is shown to arise naturally from the manifolds "with delay" when considering issues of convergence as the delay time tends to infinity. This leads to a new characterisation of the multi-valued manifold, which allows a fuller understanding of its structure.
Citation: James C. Robinson. Inertial manifolds with and without delay. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 813-824. doi: 10.3934/dcds.1999.5.813
 [1] E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics & Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010 [2] Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249 [3] Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557 [4] Alice Le Brigant. Computing distances and geodesics between manifold-valued curves in the SRV framework. Journal of Geometric Mechanics, 2017, 9 (2) : 131-156. doi: 10.3934/jgm.2017005 [5] Ronny Bergmann, Raymond H. Chan, Ralf Hielscher, Johannes Persch, Gabriele Steidl. Restoration of manifold-valued images by half-quadratic minimization. Inverse Problems & Imaging, 2016, 10 (2) : 281-304. doi: 10.3934/ipi.2016001 [6] Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993 [7] Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97. [8] Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839 [9] Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020213 [10] M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403 [11] Alexander Nabutovsky and Regina Rotman. Lengths of geodesics between two points on a Riemannian manifold. Electronic Research Announcements, 2007, 13: 13-20. [12] George Osipenko. Linearization near a locally nonunique invariant manifold. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 189-205. doi: 10.3934/dcds.1997.3.189 [13] Pei Yean Lee, John B Moore. Gauss-Newton-on-manifold for pose estimation. Journal of Industrial & Management Optimization, 2005, 1 (4) : 565-587. doi: 10.3934/jimo.2005.1.565 [14] Ariadna Farrés, Àngel Jorba. On the high order approximation of the centre manifold for ODEs. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 977-1000. doi: 10.3934/dcdsb.2010.14.977 [15] Jan Prüss, Gieri Simonett. On the manifold of closed hypersurfaces in $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5407-5428. doi: 10.3934/dcds.2013.33.5407 [16] Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873 [17] Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285 [18] Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks & Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021 [19] Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161 [20] Tyrus Berry, Timothy Sauer. Consistent manifold representation for topological data analysis. Foundations of Data Science, 2019, 1 (1) : 1-38. doi: 10.3934/fods.2019001

2019 Impact Factor: 1.338