A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles.
type: Item
instance of: scholarly article
title: A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles.
zbMATH Open document ID: 1044.37039
author: Francesca Mantellini
author: Russell Johnson
published in: Discrete and Continuous Dynamical Systems
publication date: 2003-01-01T00:00:00Z
review text: In this interesting paper, the authors study the phenomenon of stability breakdown for nonautonomous ordinary differential equations whose time dependence is determined by a strictly ergodic flow. More precisely, an approach for a nonautonomous version of the classical transcritical bifurcation is provided. The authors work in a skew-product setting and, therefore, to consider systems of the form \[ y'=\omega(y),\quad x'=| \varepsilon| f(y,x,\varepsilon),\tag{*} \] on \(Y\times{\mathbb R}\). Here, \(Y\) is a compact subset of \({\mathbb R}^m\) invariant under the flow \(\{\tau_t\}\) induced by \(y'=\omega(y)\), \(f:{\mathbb R}^m\times{\mathbb R}\times I\to{\mathbb R}\), \(I\) denotes an open interval containing \(0\), is assumed to possess continuous partial derivatives w.r.t.~\(x\) up to order \(3\), and one has \(f(y,0,\varepsilon)\equiv 0\). Moreover, the hypotheses that the flow \((Y,{\mathbb R})\) is strictly ergodic holds throughout the paper. In order to investigate a transcritical bifurcation for \((\ast)\), i.e., its nonautonomous generalization, the authors write \[ f(y,x,\varepsilon)=a(y,\varepsilon)x+b(y,\varepsilon)x^2+p(y,\varepsilon,x) \] and assume \(\bar{a}(0)=0\), \(\bar{b}(0)<0\) for the mean values \(\bar{a}(\varepsilon),\bar{b}(\varepsilon)\) of \(a,b\), respectively. In a first result, they show that for sufficiently small \(\varepsilon>0\), \(| x_0| \) and arbitrary initial values \(y_0\), the \(\omega\)-limit set of \((y_0,x_0)\) in \(Y\times{\mathbb R}\) contains at least one and at most two minimal subsets, and there are almost automorphic extensions of \((Y,\{\tau_t\})\) [cf. \textit{W.~Shen} and \textit{Y.~Yi}, Mem. Am. Math. Soc. 647 (1998; Zbl 0913.58051)]. Under further assumptions on \(a\), there exists a local attractor \(A_{\varepsilon}\subset Y\times{\mathbb R}\) such that \(A_{\varepsilon}\cap(Y\times\{0\})=\emptyset\), and also a statement on the Hausdorff distance between \(A_{\varepsilon}\) and \(Y\times\{0\}\) holds. These res…
Mathematics Subject Classification ID: 37G35
Mathematics Subject Classification ID: 37B55
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 37C55
zbMATH DE Number: 1890733
zbMATH Keywords: nonautonomous bifurcation
zbMATH Keywords: averaging method
zbMATH Keywords: almost automorphic minimal set
zbMATH Keywords: parameter intermittence
zbMATH Keywords: ergodic flow
zbMATH Keywords: local attractor
zbMATH Keywords: Hausdorff distance
zbMATH Keywords: ODEs with quasiperiodic
reviewed by: Christian Pötzsche
MaRDI profile type: Publication
full work available at URL: dcds.2003.9.209
OpenAlex ID: W2076444295
DOI: 10.3934/DCDS.2003.9.209
description: scientific article; zbMATH DE number 1890733