Skip to main content
SpringerLink
Log in

Periodic External Input Tunes the Stability of Delayed Nonlinear Systems: From the Slaving Principle to Center Manifolds

  • Conference paper
  • First Online:
Selforganization in Complex Systems: The Past, Present, and Future of Synergetics

Part of the book series: Understanding Complex Systems ((UCS))

  • 1243 Accesses

Abstract

The work illustrates a recent analysis technique that demonstrates that external periodic input affects the stability of the time-averaged nonlinear dynamics of a delayed system. At first, the article introduces the fundamental elements of delayed differential equations and then applies these to a nonlinear delayed problem close to a transcritical bifurcation. We observe a shift of stability in the system induced by the fast periodic driving.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Longtin, A., Milton, J.G.: Modelling autonomous oscillations in the human pupil light reflex using nonlinear delay-differential equations. Bull. Math. Biol. 51, 605–624 (1989)

    Article  MATH  Google Scholar 

  2. Glass, L., Mackey, M.: From Clocks to Chaos: The Rhythms of Life. Princeton University Press, Princeton (1988)

    MATH  Google Scholar 

  3. Boulet, J., Balasubramaniam, R., Daffertshofer, A., Longtin, A.: Stochastic two delay-differential model of delayed visual feedback effects on postural dynamics. Phil. Trans. Royal Soc. A 368, 423–438 (2010)

    Article  MATH  Google Scholar 

  4. Hutt, A., Bestehorn, M., Wennekers, T.: Pattern formation in intracortical neuronal fields. Network: Comput. Neural Syst. 14, 351–368 (2003)

    Article  Google Scholar 

  5. Atay, F.M., Hutt, A.: Neural fields with distributed transmission speeds and constant feedback delays. SIAM J. Appl. Dyn. Syst. 5, 670–698 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coombes, S., Owen, M.: Bumps, breathers, and waves in a neural network with spike frequency adaptation. Phys. Rev. Lett. 94, 148102 (2005)

    Article  Google Scholar 

  7. Franovic, I., Todorovic, K., Vasovic, N., Buric, N.: Spontaneous formation of synchronization clusters in homogenous neuronal ensembles induced by noise and interaction delays. Phys. Rev. Lett. 108, 094101 (2012)

    Article  Google Scholar 

  8. Campbell, S.A.: Calculating center manifolds for delay differential equations using maple. In: Delay Differential Equations: Recent Advances and New Directions. Springer, Berlin (2008)

    Google Scholar 

  9. Redmond, B., LeBlanc, V., Longtin, A.: Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry. Physica D 166, 131–146 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wischert, W., Wunderlin, A., Pelster, A., Olivier, M., Groslambert, J.: Delay-induced instabilities in nonlinear feedback systems. Physical Review E 49, 203–219 (1994)

    Article  MathSciNet  Google Scholar 

  11. Schanz, M., Pelster, A.: Synergetic system analysis for the delay-induced Hopf bifurcation in the Wright equation. SIAM J. Applied Dynamical Systems 2, 277–296 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Haken, H.: Synergetics. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  13. Schoener, G., Haken, H.: The slaving principle for Stratonovich stochastic differential equations. Z. Phys. B 63, 493–504 (1986)

    Article  Google Scholar 

  14. Chicone, C., Latushkin, Y.: Center manifolds for infinite dimensional nonautonomous differential equations. J. Diff. Eqs. 141, 356–399 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Boxler, P.: A stochastic version of center manifold theory. Prob. Theory Relat. Fields 83, 509 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  16. Xu, C., Roberts, A.: On the low-dimensional modelling of Stratonovich stochastic differential equations. Physica A 225, 62–80 (1996)

    Article  MathSciNet  Google Scholar 

  17. Hutt, A., Longtin, A., Schimansky-Geier, L.: Additive global noise delays turing bifurcations. Phys. Rev. Lett. 98, 230601 (2007)

    Article  Google Scholar 

  18. Hale, J., Lunel, S.: Introduction to functional differential equations. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  19. Quesmi, R., Babram, M.A., Hbid, M.: A maple program for computing a terms of a center manifold, and element of bifurcations for a class of retarded functional differential equations with Hopf singularity. Appl. Math. Comp. 175, 932–968 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Campbell, S.A., Belair, J.: Analytical and symbolically-assisted investigations of Hopf bifurcations in delay-differential equations. Can. Appl. Math. Quart. 3, 137–154 (1995)

    MathSciNet  MATH  Google Scholar 

  21. Faria, T., Magalhaes, L.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Diff. Eqs. 122, 281 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Asl, F., Ulsoy, A.: Analysis of a system of linear delay differential equations. J. Dyn. Syst. Meas. Cont. 125, 215–223 (2003)

    Article  Google Scholar 

  23. Li, J., Hansen, C.: Forced phase-locked response of a nonlinear system with time delay after Hopf bifurcation. Chaos Solit. Fract. 25, 461–473 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu, J., Chung, K.: Effects of time delayed position feedback on a van der Pol–Duffing oscillator. Physica D 180, 17–39 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Frank, T.D., Beek, P.J.: Stationary solutions of linear stochastic delay differential equations: Applications to biological systems. Phys. Rev. E 64, 021917 (2001)

    Article  Google Scholar 

  26. Guillouzic, S., L’Heureux, I., Longtin, A.: Small delay approximation of stochastic delay differential equation. Phys. Rev. E 59, 3970 (1999)

    Article  MATH  Google Scholar 

  27. Yeganefar, N., Pepe, P., Dambrine, M.: Input-to-state stability of time-delay systems: A link with exponential stability. IEEE Transactions On Automatic Control 53, 1526–1531 (2008)

    Article  MathSciNet  Google Scholar 

  28. Cox, S., Roberts, A.: Center manifolds of forced dynamical systems. J. Austral. Math. Soc. Ser. B 32, 401–436 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Amann, A., Schoell, E., Just, W.: Some basic remarks on eigenmode expansions of time-delay dynamics. Physica A 373, 191–202 (2007)

    Article  Google Scholar 

  30. Lefebvre, J., Hutt, A., LeBlanc, V.G., Longtin, A.: Reduced dynamics for delayed systems with harmonic or stochastic forcing. Chaos 22, 043121 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hutt, A., Longtin, A., Schimansky-Geier, L.: Additive noise-induced turing transitions in spatial systems with application to neural fields and the Swift-Hohenberg equation. Physica D 237, 755–773 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA CR Nancy - Grand Est, 615 Rue du Jardin Botanique, 54602, Villers-les-nancy, France

    Axel Hutt

  2. Département des Neurosciences Fondamentales, Université de Genève, CMU, 1 Rue Michel Servet, 1211, Genève 4, Switzerland

    Jérémie Lefebvre

Authors
  1. Axel Hutt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jérémie Lefebvre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Axel Hutt .

Editor information

Editors and Affiliations

  1. Institut für Theoret. Physik I, Universität Stuttgart Institut für Theoret. Physik I, Stuttgart, Germany

    Günter Wunner

  2. Fachbereich Physik, Forschungszentrum OPTIMAS, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Axel Pelster

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Hutt, A., Lefebvre, J. (2016). Periodic External Input Tunes the Stability of Delayed Nonlinear Systems: From the Slaving Principle to Center Manifolds. In: Wunner, G., Pelster, A. (eds) Selforganization in Complex Systems: The Past, Present, and Future of Synergetics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-27635-9_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-27635-9_2

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-27633-5

  • Online ISBN: 978-3-319-27635-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation