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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

Tomás Caraballo 1, , Renato Colucci 2, and  Luca Guerrini 3,

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.

Keywords: Differential equation with delay, Hopf bifurcation, qualitative behaviour of solutions, distributed time delay.
Mathematics Subject Classification: 35B32, 74H65, 34K60.
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
References:
[1]

S. Bhalekar, Dynamics of fractional order complex Ucar system, Studies in Computational Intelligence, 688 (2017), 747-771.   Google Scholar

[2]

S. Bhalekar, Stability and bifurcation analysis of a generalised scalar delay differential equation, Chaos, 26 (2016), 084306, 7pp. doi: 10.1063/1.4958923.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. MacDonald, Time Lags in Biological Systems, Springer, New York, 1978. Google Scholar

[13]

N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge University. 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981.  Google Scholar

[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.  Google Scholar

[19]

A. Ucar, On the chaotic behaviour of a prototype delayed dynamical system, Chaos, Solitons and Fractals, 16 (2003), 187-194.   Google Scholar

show all references

References:
[1]

S. Bhalekar, Dynamics of fractional order complex Ucar system, Studies in Computational Intelligence, 688 (2017), 747-771.   Google Scholar

[2]

S. Bhalekar, Stability and bifurcation analysis of a generalised scalar delay differential equation, Chaos, 26 (2016), 084306, 7pp. doi: 10.1063/1.4958923.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

N. MacDonald, Time Lags in Biological Systems, Springer, New York, 1978. Google Scholar

[13]

N. MacDonald, Biological Delay Systems: Linear Stability Theory, Cambridge University. 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981.  Google Scholar

[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.  Google Scholar

[19]

A. Ucar, On the chaotic behaviour of a prototype delayed dynamical system, Chaos, Solitons and Fractals, 16 (2003), 187-194.   Google Scholar

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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