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Nonautonomous bifurcation of bounded solutions II: A Shovel-Bifurcation pattern

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  • This paper continues our work on local bifurcations for nonautonomous difference and ordinary differential equations. Here, it is our premise that constant or periodic solutions are replaced by bounded entire solutions as bifurcating objects in order to encounter right-hand sides with an arbitrary time dependence.
        We introduce a bifurcation pattern caused by a dominant spectral interval (of the dichotomy spectrum) crossing the stability boundary. As a result, differing from the classical autonomous (or periodic) situation, the change of stability appears in two steps from uniformly asymptotically stable to asymptotically stable and finally to unstable. During the asymptotically stable regime, a whole family of bounded entire solutions occurs (a so-called "shovel"). Our basic tools are exponential trichotomies and a quantitative version of the surjective implicit function theorem yielding the existence of strongly center manifolds.
    Mathematics Subject Classification: 37C60, 39A28, 34C23, 37D10, 37G10.


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