# American Institute of Mathematical Sciences

June  2019, 24(6): 2639-2655. doi: 10.3934/dcdsb.2018268

## 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  January 2018 Revised  May 2018 Published  June 2019 Early access  October 2018

Fund Project: 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.

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.

Citation: Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268
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##### References:
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 $P_+ = 1$.
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 $P_+$.
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.
The numerical solution (in red) together with its approximation (in blue) given by (20).
For $m = 1$, the fixed points $P_\pm$ are locally asymptotically stable for all $T\geq0$.
For m = 2 and $T = 0.9<T_*$ the fixed points $P_\pm$ are locally asymptotically stable.
For m = 2 and $T = 2>T_*$ the fixed points $P_\pm$ are unstable and a stable limit cycle appears.
For $m = 3$ and $T = 0.6<T_*$ the fixed points $P_\pm$ are locally asymptotically stabel
For $m = 3$ and $T = 0.7>T_*$ the fixed points $P_\pm$ are unstable and a stable limit cycle appears.
The solution of sytem (34) for $T = 2$ and $\tau = 5$. Numerical simulations suggest the evidence of a chaotic behaviour.
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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