Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach

Abstract

For a nonautonomous linear system with nonuniform contraction, we construct a topological conjugacy between this system and an unbounded nonlinear perturbation. This topological conjugacy is constructed as a composition of homeomorphisms. The first one is set up by considering the fact that linear system is almost reducible to diagonal system with a small enough perturbation where the diagonal entries belong to spectrum of the nonuniform exponential dichotomy; and the second one is constructed in terms of the crossing times with respect to unit sphere of an adequate Lyapunov function associated to the linear system.

Higher regularity of homeomorphisms in the Hartman-Grobman theorem for semilinear evolution equations

“…To weaken the Palmer's conditions, Jiang [23] presented a version of Hartman-Grobman theorem by setting that the linear system admits a generalized exponential dichotomy. Huerta [22,9] constructed a topological conjugacy between linear system and an unbounded nonlinear perturbation, while nonautonomous linear system admits a nonuniform contraction. Barreira and Valls [2,3,4,5] proved several versions of Hartman-Grobman theorem in different situations with the assumption that the linear systems admit a nonuniform dichotomy.…”

Higher regularity of homeomorphisms in the Hartman-Grobman theorem and a conjecture on its sharpness

“…An important and interesting problem is the regularity of the linearization, which have greatly attracted many mathematicians' attentions. Among the works on the linearization mentioned above, a lot of papers were devoted to proving the Hölder continuity of the homeomorphisms in the linearization theorem (see Backes et al [13], Barreira and Valls [14,15,16,17], Dragičević et al [36,37], Huerta et al [18,19], Hein and Prüss [11], Jiang [20,21], Pötzche [25], Shi and Zhang [28], Rodrigues and Solà-Morales [39,40], Xia et al [23,29], Zhang et al [42,43,44], Tan [46], Shi and Xiong [47]). For the sake of easier illustration, we restate the Palmer's linearization theorem [12] which has extended the classical Hartman-Grobman theorem ( [4,5]) to the nonautonomous case.…”

“…[14,15,16,17]) or employs the Bellman inequality (see e.g. [11,13,19,23,29,25,47]). However, the disadvantage of the Bellman inequality results in an exponential estimate of the form e αt (α > 0).…”

“…Palmer [12] firstly extended the Hartman-Grobman theorem to the nonautonomous case. In order to weaken Palmer's linearization theorem, various versions of Hartman-Grobman theorem were established, Backes et al [13] (for nonhyperbolic systems), Barreira and Valls [14,15,16,17] (with nonuniform exponential dichotomies), Huerta et al [18,19] (nonuniform exponential contraction), Jiang [20] (generalized exponential dichotomy), Jiang [21] (ordinary dichotomy), Fenner and Pinto [22] and Xia et al [23] (for impulsive systems), Papaschinopoulos [24] (for differential equations with piecewise constant argument), Pötzche [25] (for dynamic equations on time scales), Reinfelds and Sermone [26], Reinfelds and Šteinberga [27] (dynamical equivalence), Shi and Zhang [28] (monograph for linearization), Xia et al [29] (with unbounded nonlinear term).…”

A Hartman-Grobman theorem for algebraic dichotomies

“…In order to weaken the conditions of Palmer's linearization theorem, some improvements were given in Backes et al [23] (without exponential dichotomy), Barreira et al [24,25,26,27] (nonuniform dichotomy), Jiang [28,29] (generalized dichotomy and ordinary dichotomy), Huerta [30,31] (with nonuniform contraction), Papaschinopoulos [32], Pinto et al [33], Zou et al [34] (for differential equations with piecewise constant argument), Pötzche [35] (for dynamic systems on time scales), Fenner and Pinto [36] and Xia et al [37] (for the instantaneous impulsive system).…”