L.A.S. and negative Schwarzian derivative do not imply G.A.S. in Clark's equation
type: Item
instance of: scholarly article
title: L.A.S. and negative Schwarzian derivative do not imply G.A.S. in Clark's equation
zbMATH Open document ID: 1373.39013
author: Enrique Parreño
author: Víctor Jiménez López
published in: Journal of Dynamics and Differential Equations
publication date: 2016-06-27T00:00:00Z
review text: The paper deals with stability problems for the equilibrium of the Clark difference equation \[ x_{n+1} = \alpha x_n + (1-\alpha)h(x_{n-k}) \] with \(h:I\mapsto I\) being unimodal or decreasing with negative Schwarzian derivative defined by \[ \displaystyle{(Sh)(z) = {{h'''(z)}\over{h'(z)}} - {{3}\over{2}}\left( {{h''(z)}\over{h'(z)}}\right)^2}. \] The paper introduces the so-called class \(S\) of functions of real variable (not to mistake for the \(S\) (schlicht) class of functions of complex variable from the Bieberbach conjecture). It is proved via the Neimark-Sacker bifurcation theory that ``the conjecture of local asymptotic stability (L.A.S.) (stability by the first approximation) implying global asymptotic stability (G.A.S.) of the equilibrium'' fails for \(k\geq 3\) -- some sui-generis delay dependent global asymptotic stability.
reviewed by: Vladimir Răsvan
Mathematics Subject Classification ID: 39A30
Mathematics Subject Classification ID: 39A28
Mathematics Subject Classification ID: 39A22
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 6598709
zbMATH Keywords: global asymptotic stability
zbMATH Keywords: local asymptotic stability
zbMATH Keywords: Neimark-Sacker bifurcation
zbMATH Keywords: Clark difference equation
zbMATH Keywords: negative Schwarzian derivative
Wikidata QID: Q115383493
MaRDI profile type: Publication
full work available at URL: s10884-016-9525-7
OpenAlex ID: W2295862449
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cites work: Local stability implies global stability for the 2-dimensional Ricker map
cites work: Bifurcation analysis on a discrete model of Nicholson's blowflies
cites work: Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems 2. The smooth case
cites work: Topics from One-Dimensional Dynamics
cites work: A delayed-recruitment model of population dynamics, with an application to baleen whale populations
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cites work: Global attractors for difference equations dominated by one-dimensional maps
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cites work: Convergence to equilibria in discrete population models
cites work: Stability of a class of delay-difference equations
cites work: Sensitive dependence to initial conditions for one dimensional maps
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cites work: Global attractivity and presistence in a discrete population model*
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cites work: Wright type delay differential equations with negative Schwarzian.
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cites work: On the Asymptotic Stability of a Class of Linear Difference Equations
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DOI: 10.1007/S10884-016-9525-7
description: scientific article; zbMATH DE number 6598709