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