Skip to main content
Log in

Characterizing multistability regions in the parameter space of the Mackey–Glass delayed system

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

Proposed to study the dynamics of physiological systems in which the evolution depends on the state in a previous time, the Mackey–Glass model exhibits a rich variety of behaviors including periodic or chaotic solutions in vast regions of the parameter space. This model can be represented by a dynamical system with a single variable obeying a delayed differential equation. Since it is infinite dimensional requires to specify a real function in a finite interval as an initial condition. Here, the dynamics of the Mackey–Glass model is investigated numerically using a scheme previously validated with experimental results. First, we explore the parameter space and describe regions in which solutions of different periodic or chaotic behaviors exist. Next, we show that the system presents regions of multistability, i.e. the coexistence of different solutions for the same parameter values but for different initial conditions. We remark the coexistence of periodic solutions with the same period but consisting of several maximums with the same amplitudes but in different orders. We characterize the multistability regions by introducing families of representative initial condition functions and evaluating the abundance of the coexisting solutions. These findings contribute to describe the complexity of this system and explore the possibility of possible applications such as to store or to code digital information.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. M.C. Mackey, L. Glass, Oscillation and chaos in physiological control systems. Science 197(4300), 287–289 (1977)

    Article  ADS  Google Scholar 

  2. J. Bélair, in Dynamical disease: mathematical analysis of human illness;[the papers are based on a nato advanced research workshop held in Mont Tremblant, Québec, Canada in February 1994]. AIP Press (1995)

  3. D. Biswas, T. Banerjee, Time-Delayed Chaotic Dynamical Systems (Springer, Berlin, 2018)

    MATH  Google Scholar 

  4. J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, vol. 99 (Springer Science & Business Media, Berlin, 2013)

    MATH  Google Scholar 

  5. L. Junges, J.A.C. Gallas, Intricate routes to chaos in the mackey-glass delayed feedback system. Phys. Lett. A 376(18), 2109–2116 (2012)

    Article  ADS  Google Scholar 

  6. J. Losson, M.C. Mackey, A. Longtin, Solution multistability in first-order nonlinear differential delay equations. Chaos Interdiscipl. J. Nonlinear Sci. 3(2), 167–176 (1993)

    Article  MathSciNet  Google Scholar 

  7. B. Mensour, A. Longtin, Controlling chaos to store information in delay-differential equations. Phys. Lett. A 205(1), 18–24 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  8. T.K. Lim, K. Kwak, M. Yun, An experimental study of storing information in a controlled chaotic system with time-delayed feedback. Phys. Lett. A 240(6), 287–294 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  9. B.B. Zhou, R. Roy, Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators. Phys. Rev. E 75(2), 026205 (2007)

    Article  ADS  Google Scholar 

  10. K. Pyragas, Synchronization of coupled time-delay systems: analytical estimations. Phys. Rev. E 58(3), 3067 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  11. M.-Y. Kim, C. Sramek, A. Uchida, R. Roy, Synchronization of unidirectionally coupled mackey-glass analog circuits with frequency bandwidth limitations. Phys. Rev. E 74(1), 016211 (2006)

    Article  ADS  Google Scholar 

  12. E.M. Shahverdiev, R.A. Nuriev, R.H. Hashimov, K.A. Shore, Chaos synchronization between the mackey-glass systems with multiple time delays. Chaos Solit. Fract. 29(4), 854–861 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  13. A. Namajūnas, K. Pyragas, A. Tamaševičius, An electronic analog of the mackey-glass system. Phys. Lett. A 201(1), 42–46 (1995)

    Article  ADS  Google Scholar 

  14. P. Amil, C. Cabeza, A.C. Marti, Exact discrete-time implementation of the mackey–glass delayed model. IEEE Trans. Circ. Syst. II: Express Briefs 62(7), 681–685 (2015)

    MATH  Google Scholar 

  15. P. Amil, C. Cabeza, C. Masoller, A.C. Martí, Organization and identification of solutions in the time-delayed mackey-glass model. Chaos Interdiscipl. J. Nonlinear Sci. 25(4), 043112 (2015)

    Article  MathSciNet  Google Scholar 

  16. H. L. Smith, in An introduction to delay differential equations with applications to the life sciences, volume 57 (Springer, New York, 2011)

  17. A. Bellen, M. Zennaro, Numerical Methods for Delay Differential Equations (Oxford University Press, Oxford, 2013)

    MATH  Google Scholar 

  18. J.G. Freire, Stern–brocot trees in cascades of mixed-mode oscillations and canards in the extended bonhoeffer–van der pol and the fitzhugh–nagumo models of excitable systems. Phys. Lett. A 375(7), 1097–1103 (2011)

    Article  ADS  Google Scholar 

  19. J.G. Freire, C. Cabeza, A. Marti, T. Pöschel, J.A.C. Gallas, Antiperiodic oscillations. Sci. Rep. 3, 1–4 (2013)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Uruguayan institutions Programa de Desarrollo de las Ciencias Básicas (MEC-Udelar, Uruguay) and Comisión Sectorial de Investigación Científica (Udelar, Uruguay) for the grant Física Nolineal (ID 722) Programa Grupos I+D. The numerical experiments presented here were performed at the ClusterUY (site: https://cluster.uy)

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Arturo C. Marti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tarigo, J.P., Stari, C., Cabeza, C. et al. Characterizing multistability regions in the parameter space of the Mackey–Glass delayed system. Eur. Phys. J. Spec. Top. 231, 273–281 (2022). https://doi.org/10.1140/epjs/s11734-021-00353-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjs/s11734-021-00353-0

Navigation