Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study

Tomás Caraballo; Renato Colucci; Luca Guerrini

doi:10.3934/dcdsb.2018268

Article Contents

2019, Volume 24, Issue 6: 2639-2655. Doi: 10.3934/dcdsb.2018268

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Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study

1.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/Tarfia s/n, 41012 Sevilla, Spain
2.
Departament of Engineering, University Niccolò Cusano, Via Don Carlo Gnocchi, 3 00166, Roma, Italy
3.
Department of Management, Polytechnic University of Marche, Piazza Martelli 8, 60121, Ancona (AN), Italy

Received: December 31, 2017

Revised: April 30, 2018

Early access: October 2018

Published: June 2019

This work has been partially supported by FEDER and the Spanish Ministerio de Economía y Competitividad project MTM2015-63723-P and the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492

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Abstract

In this article we consider a model introduced by Ucar in order to simply describe chaotic behaviour with a one dimensional ODE containing a constant delay. We study the bifurcation problem of the equilibria and we obtain an approximation of the periodic orbits generated by the Hopf bifurcation. Moreover, we propose and analyse a more general model containing distributed time delay. Finally, we propose some ideas for further study. All the theoretical results are supported and illustrated by numerical simulations.

Keywords:

Mathematics Subject Classification: 35B32, 74H65, 34K60.

Citation:

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Figure 1. The solution starting on the left hand side of $O$ converges to $P_- = -1$, while that starting on the right hand side of $O$ converges to $P_+ = 1$.

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Figure 2. The solution starting on the left hand side of $O$ converges to a limit cycle around $P_-$, while that starting on the right hand side of $O$ converges to a limit cycle around $P_+$.

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Figure 3. The solution $x(t)$ and the graph of $(x(t),x'(t))$ for $\delta = \varepsilon = 1$ and $\tau = 1.72$. The attractor appears to be chaotic.

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Figure 4. The numerical solution (in red) together with its approximation (in blue) given by (20).

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Figure 5. For $m = 1$, the fixed points $P_\pm$ are locally asymptotically stable for all $T\geq0$.

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Figure 6. For m = 2 and $T = 0.9<T_*$ the fixed points $P_\pm$ are locally asymptotically stable.

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Figure 7. For m = 2 and $T = 2>T_*$ the fixed points $P_\pm$ are unstable and a stable limit cycle appears.

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Figure 8. For $m = 3$ and $T = 0.6<T_*$ the fixed points $P_\pm$ are locally asymptotically stabel

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Figure 9. For $m = 3$ and $T = 0.7>T_*$ the fixed points $P_\pm$ are unstable and a stable limit cycle appears.

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Figure 10. The solution of sytem (34) for $T = 2$ and $\tau = 5$. Numerical simulations suggest the evidence of a chaotic behaviour.

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Figure 11. The solution of sytem (36) for $T = 1.6$ and $\tau = 1.14$. Numerical simulations suggest the evidence of a chaotic behaviour, this is supported by the presence of a strange attractor similar to the famous Lorenz attractor.

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