2022/01/31 by Weijie Lu, Lu, Weijie, Manuel Pinto +3
Engineering · Immunology and Microbiology · #Classical Analysis and ODEs (math.CA) #Control and Stability of Dynamical Systems #FOS: Mathematics #Macrophage Migration Inhibitory Factor #Phagocytosis and Immune Regulation
paper · pdf · doi:10.48550/arxiv.2201.12945
openalex publication_date 2022/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Hartman-Grobman theorem states that there is a homeomorphism H sending the solutions of the nonlinear system onto those of its linearization under suitable assumptions. Many mathematicians have made contributions to prove Hölder continuity of the homeomorphisms. However, is it possible to improve the Hölder continuity to Lipschitzian continuity? This paper gives a positive answer. We formulate the first result that the homeomorphism is Lipschitzian, but not C1, while its inverse is merely Hölder continuous, but not Lipschitzian. It is interesting that the regularity of the homeomorphism is different from its inverse. Moreover, some illustrative examples are presented to show the effectiveness of our results. Further, motivated by our example, we also propose a conjecture, saying, the regularity of the homeomorphisms is sharp and it could not be improved any more.