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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  January 2018 Revised  May 2018 Published  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, doi: 10.3934/dcdsb.2018268
References:
 [1] S. Bhalekar, Dynamics of fractional order complex Ucar system, Studies in Computational Intelligence, 688 (2017), 747-771. [2] S. Bhalekar, Stability and bifurcation analysis of a generalised scalar delay differential equation, Chaos, 26 (2016), 084306, 7pp. doi: 10.1063/1.4958923. [3] S. Bhalekar, On the Ucar prototype model with incommensurate delays, Signal, Image and Video Processing, 8 (2014), 635-639. doi: 10.1007/s11760-013-0595-2. [4] T. Caraballo, R. Colucci and L. Guerrini, On a predator prey model with nonlinear harvesting and distributed delay, Comm. Pure and Appl. Anal., 17 (2018), 2703-2727. doi: 10.3934/cpaa.2018128. [5] C. W. Eurich, A. Thiel and L. Fahse, Distributed delays stabilize ecological feedback systems, Phys. Rev. Lett., 94 (2005), 158104. doi: 10.1103/PhysRevLett.94.158104. [6] E. Karaoglu, E. Yilmaz and H. Merdan, Hopf bifurcation analysis of coupled two-neuron system with discrete and distributed delays, Nonlinear Dyn, 85 (2016), 1039-1051. doi: 10.1007/s11071-016-2742-0. [7] E. Karaoglu, E. Yilmaz and H. Merdan, Stability and bifurcation analysis of two-neuron network with discrete and distributed delays, Neurocomputing, 182 (2016), 102-110. doi: 10.1016/j.neucom.2015.12.006. [8] C. Li, X. Liao and J. Yu, Hopf bifurcation in a prototype delayed system, Chaos, Solitons and Fractals, 19 (2004), 779-787. doi: 10.1016/S0960-0779(03)00206-6. [9] X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64 (2016), 63-69. doi: 10.1016/j.automatica.2015.10.002. [10] X. Li and S. Song, Stabilization of delay systems: Delay-dependent impulsive control, IEEE Transactions on Automatic Control, 62 (2017), 406-411. doi: 10.1109/TAC.2016.2530041. [11] X. Li and J. Cao, An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE Transactions on Automatic Control, 62 (2017), 3618-3625. doi: 10.1109/TAC.2017.2669580. [12] N. MacDonald, Time Lags in Biological Systems, Springer, New York, 1978. [13] N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge University. 1989. [14] A. Matsumoto and F. Szidarovszky, Delay dynamics in a classical IS-LM model with tax collections, Metroeconomica, 67 (2016), 667-697. doi: 10.1111/meca.12128. [15] A. Matsumoto and F. Szidarovszky, Dynamic monopoly with multiple continuously distributed time delays, Mathematics and Computers in Simulation, 108 (2015), 99-118. doi: 10.1016/j.matcom.2014.01.003. [16] A. Matsumoto and F. Szidarovszky, Boundedly rational monopoly with single continuously distributed time delay, Nonlinear Economic Dynamics and Financial Modelling, Essays in Honour of Carl Chiarella, May 2014, Pages 83-107. [17] A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981. [18] A. Ucar, A prototype model for chaos studies, International Journal of Engineering Science, 40 (2001), 251-258. doi: 10.1016/S0020-7225(01)00060-X. [19] A. Ucar, On the chaotic behaviour of a prototype delayed dynamical system, Chaos, Solitons and Fractals, 16 (2003), 187-194.

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References:
 [1] S. Bhalekar, Dynamics of fractional order complex Ucar system, Studies in Computational Intelligence, 688 (2017), 747-771. [2] S. Bhalekar, Stability and bifurcation analysis of a generalised scalar delay differential equation, Chaos, 26 (2016), 084306, 7pp. doi: 10.1063/1.4958923. [3] S. Bhalekar, On the Ucar prototype model with incommensurate delays, Signal, Image and Video Processing, 8 (2014), 635-639. doi: 10.1007/s11760-013-0595-2. [4] T. Caraballo, R. Colucci and L. Guerrini, On a predator prey model with nonlinear harvesting and distributed delay, Comm. Pure and Appl. Anal., 17 (2018), 2703-2727. doi: 10.3934/cpaa.2018128. [5] C. W. Eurich, A. Thiel and L. Fahse, Distributed delays stabilize ecological feedback systems, Phys. Rev. Lett., 94 (2005), 158104. doi: 10.1103/PhysRevLett.94.158104. [6] E. Karaoglu, E. Yilmaz and H. Merdan, Hopf bifurcation analysis of coupled two-neuron system with discrete and distributed delays, Nonlinear Dyn, 85 (2016), 1039-1051. doi: 10.1007/s11071-016-2742-0. [7] E. Karaoglu, E. Yilmaz and H. Merdan, Stability and bifurcation analysis of two-neuron network with discrete and distributed delays, Neurocomputing, 182 (2016), 102-110. doi: 10.1016/j.neucom.2015.12.006. [8] C. Li, X. Liao and J. Yu, Hopf bifurcation in a prototype delayed system, Chaos, Solitons and Fractals, 19 (2004), 779-787. doi: 10.1016/S0960-0779(03)00206-6. [9] X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica, 64 (2016), 63-69. doi: 10.1016/j.automatica.2015.10.002. [10] X. Li and S. Song, Stabilization of delay systems: Delay-dependent impulsive control, IEEE Transactions on Automatic Control, 62 (2017), 406-411. doi: 10.1109/TAC.2016.2530041. [11] X. Li and J. Cao, An impulsive delay inequality involving unbounded time-varying delay and applications, IEEE Transactions on Automatic Control, 62 (2017), 3618-3625. doi: 10.1109/TAC.2017.2669580. [12] N. MacDonald, Time Lags in Biological Systems, Springer, New York, 1978. [13] N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge University. 1989. [14] A. Matsumoto and F. Szidarovszky, Delay dynamics in a classical IS-LM model with tax collections, Metroeconomica, 67 (2016), 667-697. doi: 10.1111/meca.12128. [15] A. Matsumoto and F. Szidarovszky, Dynamic monopoly with multiple continuously distributed time delays, Mathematics and Computers in Simulation, 108 (2015), 99-118. doi: 10.1016/j.matcom.2014.01.003. [16] A. Matsumoto and F. Szidarovszky, Boundedly rational monopoly with single continuously distributed time delay, Nonlinear Economic Dynamics and Financial Modelling, Essays in Honour of Carl Chiarella, May 2014, Pages 83-107. [17] A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981. [18] A. Ucar, A prototype model for chaos studies, International Journal of Engineering Science, 40 (2001), 251-258. doi: 10.1016/S0020-7225(01)00060-X. [19] A. Ucar, On the chaotic behaviour of a prototype delayed dynamical system, Chaos, Solitons and Fractals, 16 (2003), 187-194.
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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