Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback
type: Item
instance of: scholarly article
title: Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback
zbMATH Open document ID: 1247.34120
author: Tibor Krisztin
author: Gabriella Vas
published in: Journal of Dynamics and Differential Equations
publication date: 2012-01-10T00:00:00Z
review text: For scalar delay equations of the type \[ \dot x(t) = -\mu x(t) + f(x(t-1)), \] the global attractor \(\mathcal{A}\) (within the class of solutions with not more than two zeros per time unit, in other words, within the corresponding level set of a zero-counting Liapunov functional) is studied. The situation is more complex than in previous work of Krisztin, Walther and Wu. Two spindle-like three dimensional subsets of \(\mathcal{A}\) (as described in the previous results) exist, joining equilibria \(\xi_{-2} < 0 < \xi_2 \) of the equation. There are also unstable equilibria \(\xi_{-1}, \xi_{1} \) satisfying \(\xi_{-2} < \xi_{-1}< 0 < \xi_{1}< \xi_2 \); these correspond to the zero equilibrium in the quoted earlier work. It is shown that, for suitable \( \mu>0\) and \(C^1\)-nonlinearity \(f\), the global attractor \(\mathcal{A}\) contains, in addition, two periodic orbits with large-amplitude in the sense that their range includes the interval \( (\xi_{-1}, \xi_{1})\). Further, the dynamics on \(\mathcal{A}\) can be described completely. All the heteroclinic connections between different periodic orbits, or between periodic orbits and equilibria, which are not excluded by the discrete Liapunov functional do actually exist. The nonlinearities \(f\) are smoothed step functions, a feature that is probably important for the proofs (in that it allows explicit calculations), but not essential for the results. The given examples contribute much to the deeper understanding of the attractors for this type of infinite-dimensional dynamical systems.
reviewed by: Bernhard Lani-Wayda
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37L25
Mathematics Subject Classification ID: 37L45
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34K21
zbMATH DE Number: 5996177
zbMATH Keywords: delay differential equation
zbMATH Keywords: positive feedback
zbMATH Keywords: global attractor
zbMATH Keywords: large amplitude periodic orbits
zbMATH Keywords: heteroclinic connections
MaRDI profile type: Publication
full work available at URL: s10884-011-9225-2
OpenAlex ID: W4254705978
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DOI: 10.1007/S10884-011-9225-2
description: scientific article; zbMATH DE number 5996177