Linearization and Holder continuity for nonautonomous systems

[摘要] We consider a nonautonomous system (x) triple over dot = A(t)x + f (t, x, y), (y) triple over dot = g(t, y) and give conditions under which there is a transformation of the form H(t, x, y) =(x+h(t, x, y), y) taking its solutions onto the solutions of the partially linearized system (x) triple over dot = A(t)x, (y) triple over dot = g(t, y). Shi and Xiong [28] proved a special case where g(t, y) was a linear function of yand (x) triple over dot = A(t)x had an exponential dichotomy. Our assumptions on Aand fare of the general form considered by Reinfelds and Steinberga [25], which include many of the generalizations of Palmer's theorem proved by other authors. Inspired by the work of Shi and Xiong, we also prove Holder continuity of H and its inverse in xand y. Again the proofs are given in the context of Reinfelds and Steinberga but we show what the results reduce to when (x) triple over dot = A(t)xis assumed to have an exponential dichotomy. The paper is concluded with the discrete version of the results. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-10-05

[关键词] Nonautonomous dynamics;Linearization;Grobman-Hartman theorem