Baigent, S;
Hou, Z;
Elaydi, S;
Balreira, EC;
Luís, R;
(2023)
A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds.
Journal of Difference Equations and Applications
, 29
(5)
pp. 575-591.
10.1080/10236198.2023.2222855.
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A global picture for the planar Ricker map convergence to fixed points and identification of the stable unstable manifolds.pdf - Published Version Download (2MB) | Preview |
Abstract
A quadratic Lyapunov function is demonstrated for the non-invertible planar Ricker map (Formula presented.) which shows that for (Formula presented.), and (Formula presented.) all orbits of the planar Ricker map converge to a fixed point. We establish that for 0<r, s<2, whenever a positive equilibrium exists and is locally asymptotically stable, it is globally asymptotically stable (i.e. attracts all of (Formula presented.)). Our approach bypasses and improves on methods that rely on monotonicity, which require (Formula presented.). We also use the Lyapunov function to identify the one-dimensional stable and unstable manifolds when the positive fixed point exists and is a hyperbolic saddle.
| Type: | Article |
|---|---|
| Title: | A global picture for the planar Ricker map: convergence to fixed points and identification of the stable/unstable manifolds |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1080/10236198.2023.2222855 |
| Publisher version: | https://doi.org/10.1080/10236198.2023.2222855 |
| Language: | English |
| Additional information: | © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way. The terms on which this article has been published allow the posting of the Accepted Manuscript in a repository by the author(s) or with their consent. |
| Keywords: | Planar Ricker model, non-invertible map, Lyapunov function, global stability, stable and unstable manifolds |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10173841 |
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