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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 |
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.
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. |
[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. Google Scholar |
[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. 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. |
[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. Google Scholar |
[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. Google Scholar |








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