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

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.

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

 Citation: • • 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 $P_+ = 1$.

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 $P_+$.

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.

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

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

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

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

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

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

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

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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