Robustness of nonuniform exponential dichotomies in Banach spaces

Abstract

We give conditions for the robustness of nonuniform exponential dichotomies in Banach spaces, in the sense that the existence of an exponential dichotomy for a given linear equation x = A(t)x persists under a sufficiently small linear perturbation. We also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the "angles" between the stable and unstable subspaces. Our proofs exhibit (implicitly) the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions. We emphasize that we do not need the notion of admissibility (of bounded nonlinear perturbations). We also obtain related robustness results in the case of nonuniform exponential contractions. In addition, we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time.

On the robustness of nonuniform exponential trichotomies

“…Our aim is to show that the existence of a nonuniform exponential trichotomy for Eq. (2) persists under sufficiently small C 1 perturbations B(t, λ)x with an exponential decay in time, in such a way that the stable, unstable and center subspaces associated to the nonuniform exponential dichotomies in Eq. (1) are of class C 1 in λ.…”

“…In particular, the problem was discussed by Massera and Schäffer [9] (building on earlier work of Perron [12]; see also [10]), Coppel [7], and in the case of Banach spaces by Dalec'kiȋ and Kreȋn [8], with different approaches and successive generalizations. For more recent works we refer to [2,6,11,13,14] and the references therein. With the exception of [2], all these works consider only the case of uniform exponential behavior.…”

“…For more recent works we refer to [2,6,11,13,14] and the references therein. With the exception of [2], all these works consider only the case of uniform exponential behavior. We refer the reader to [1,3] for related discussions on the ubiquity of the nonuniform exponential behavior, particularly in the context of ergodic theory.…”

C1 regularity of the stable subspaces with a general nonuniform dichotomy

“…We refer the reader to the books [2][3][4][5] for details and references. Inspired both in the classical notion of exponential dichotomy and in the notion of nonuni-formly hyperbolic trajectory introduced by Pesin in [6,7], Barreira and Valls [8][9][10][11] have introduced the notion of nonuniform exponential dichotomies and have developed the corresponding theory in a systematic way for the continuous and discrete dynamics during the last few years. See also the book [12] for details.…”

Robustness of nonuniform dichotomies with different growth rates

“…For more recent work we refer to [9,14,15,17,20,21] and the references therein. We also refer to [4,6] for the study of robustness of nonuniform exponential dichotomies.…”