Recurrent equations with sign and Fredholm alternative

Abstract

We prove that a Fredholm-type Alternative holds for recurrent equations with sign, extending a previous result by Cieutat and Haraux in [3]. Moreover, we show that this can be seen a particular case of [1] and we provide a solution to an interesting question raised by Hale in [6]. Finally we characterize the existence of exponential dichotomies also in the nonrecurrent case.

Li–Yorke chaos in nonautonomous Hopf bifurcation patterns—I

“…There are well known examples of quasi-periodic functions e 0 : R → R giving rise to a hull Ω and a map e with these characteristics, as those described by Johnson in [22] and Ortega and Tarallo in [40]. And recently Campos et al [10] have proved that there exist functions e : Ω → R with the required properties whenever the (minimal and uniquely ergodic) flow on Ω is not periodic. The conclusion is that the carried-on analysis provides a pattern of nonautonomous Hopf bifurcation, in which a extremely high degree of complexity is possible.…”

“…In both cases, (Ω, σ, R) is a minimal quasi periodic but not periodic flow. And recently Campos et al [10] have shown the existence of elements of R(Ω) whenever the base flow is nonperiodic, uniquely ergodic and minimal.…”