Nonautonomous bifurcation of bounded solutionsII: A Shovel-Bifurcation pattern

Christian Pötzsche

doi:10.3934/dcds.2011.31.941

Article Contents

2011, Volume 31, Issue 3: 941-973. Doi: 10.3934/dcds.2011.31.941

This issue Previous Article On the topology of wandering Julia components Next Article Bifurcations of multiple homoclinics in general dynamical systems

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

Christian Pötzsche^1,

1.
Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, D-85758 Garching

Received: April 30, 2010

Revised: May 31, 2011

Published: August 2011

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Abstract

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.

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Mathematics Subject Classification: 37C60, 39A28, 34C23, 37D10, 37G10.

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