Skip to main content
SpringerLink
Log in

Control based bifurcation analysis for experiments

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We introduce a method for tracking nonlinear oscillations and their bifurcations in nonlinear dynamical systems. Our method does not require a mathematical model of the dynamical system nor the ability to set its initial conditions. Instead it relies on feedback stabilizability, which makes the approach applicable in an experiment. This is demonstrated with a proof-of-concept computer experiment of the classical autonomous dry-friction oscillator, where we use a fixed time step simulation and include noise to mimic experimental limitations. For this system we track in one parameter a family of unstable nonlinear oscillations that forms the boundary between the basins of attraction of a stable equilibrium and a stable stick-slip oscillation. Furthermore, we track in two parameters the curves of Hopf bifurcation and grazing-sliding bifurcation that form the boundary of the bistability region.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abed, E., Wang, H., Chen, R.: Stabilization of period doubling bifurcations and implicatons for control of chaos. Physica D 70, 154–164 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baba, N., Amann, A., Schöll, E., Just, W.: Giant improvement of time-delayed feedback control by spatio-temporal filtering. Phys. Rev. Lett. 89(7), 074,101 (2002)

    Article  Google Scholar 

  3. di Bernardo, M., Feigin, M., Hogan, S., Homer, M.: Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical systems. Chaos Solitons Fractals 10, 1881–1908 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Blakeborough, A., Williams, M., Darby, A., Williams, D.: The development of real-time substructure testing. Phil. Trans. R. Soc. London A 359, 1869–1891 (2001)

    Article  Google Scholar 

  5. De Feo, O., Maggio, G.: Bifurcations in the Colpitts oscillator: from theory to practice. Int. J. Bif. Chaos 13(10), 2917–2934 (2003)

    Article  MATH  Google Scholar 

  6. Dercole, F., Kuznetsov, Y.: SlideCont: an Auto97 driver for bifurcation analysis of Filippov systems. ACM Trans. Math. Softw. 31, 95–119 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Yu.A., Sandstede, B., Wang, X.: In: AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont). Computer Science Concordia University, Montreal, Canada. Available: http://cmvl.cs.concordia.ca/ (1997)

  8. Eyert, V.: A comparative study on methods for convergence acceleration of iterative vector sequences. J. Comput. Phys. 124(0059), 271–285 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Galvanetto, U., Bishop, S.: Dynamics of a simple damped oscillator undergoing stick-slip vibrations. Meccanica 34, 337–347 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gauthier, D., Sukow, D., Concannon, H., Socolar, J.: Stabilizing unstable periodic orbits in a fast diode resonator using continuous time-delay autosynchronization. Phys. Rev. E 50(3), 2343–2346 (1994)

    Article  Google Scholar 

  11. Hassouneh, M., Abed, E.: Border collision bifurcation control of cardiac alternans. Int. J. Bif. Chaos 14(9), 3303–3315 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Horváth, R.: Experimental investigation of excited and self-excited vibration. Master's thesis, University of Technology and Economics, Budapest, http://www.auburn.edu /~horvaro/index2.html (2000)

  13. Hövel, P., Schöll, E.: Control of unstable steady states by time-delayed feedback methods. Phys. Rev. E 72(046203) (2005)

    Article  Google Scholar 

  14. Kevrekidis, I., Gear, C., Hummer, G.: Equation-free: the computer-aided analysis of complex multiscale systems. AIChE J. 50(11), 1346–1355 (2004)

    Article  Google Scholar 

  15. Kuznetsov, Y.: Elements of Applied Bifurcation Theory, 3rd edn. Springer Verlag, New York (2004)

    MATH  Google Scholar 

  16. Kyrychko, Y., Blyuss, K., Gonzalez-Buelga, A., Hogan, S., Wagg, D.: Real-time dynamic substructuring in a coupled oscillator-pendulum system. Proc. Roy. Soc. London A 462, 1271–1294 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Langer, G., Parlitz, U.: Robust method for experimental bifurcation analysis. Int. J. Bif. Chaos 12(8), 1909–1913 (2002)

    Article  Google Scholar 

  18. Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)

    Article  Google Scholar 

  19. Pyragas, K.: Control of chaos via an unstable delayed feedback controller. Phys. Rev. Lett. 86(11), 2265–2268 (2001)

    Article  Google Scholar 

  20. Sieber, J., Krauskopf, B.: Control-based continuation of periodic orbits with a time-delayed difference scheme. Int. J. Bif. Chaos (in press). (http://hdl.handle.net/1983/399)

  21. Siettos, C., Maroudas, D., Kevrekidis, I.: Coarse bifurcation diagrams via microscopic simulators: a state-feedback control-based approach. Int. J. Bif. Chaos 14(1), 207–220 (2004)

    Article  MATH  Google Scholar 

  22. Stépán, G., Insperger, T.: Research on delayed dynamical systems in Budapest. Dynamical Systems Magazine. http://www.dynamicalsystems.org/ma/ma/display?item=85 (2004)

  23. Trefethen, L.: Finite difference and spectral methods for ordinary and partial differential equations. Unpublished text, available at http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/pdetext.html (1996)

  24. Trefethen, L., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, NJ (2005)

    MATH  Google Scholar 

  25. Unkelbach, J., Amann, A., Just, W., Schöll, E.: Time-delay autosynchronization of the spatiotemporal dynamics in resonant tunneling diodes. Phys. Rev. E 68(026204) (2003)

    Article  Google Scholar 

  26. Yanchuk, S., Wolfrum, M., Hövel, P., Schöll, E.: Control of unstable steady states by long delay feedback. Phys. Rev. E 74(026201) (2006)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Engineering, King’s College, University of Aberdeen, Aberdeen, AB24 3UE, UK

    Jan Sieber

  2. Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, Queen’s Building, University of Bristol, Bristol, BS8 1TR, UK

    Bernd Krauskopf

Authors
  1. Jan Sieber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernd Krauskopf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Sieber.

Additional information

PACS 05.45-a, 02.30.Oz, 05.45.Gg

Mathematics Subject Classification (2000) 37M20, 37G15, 37M05

The research of J.S. was supported by EPSRC grant GR/R72020/01, and that of B.K. by an EPSRC Advanced Research Fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sieber, J., Krauskopf, B. Control based bifurcation analysis for experiments. Nonlinear Dyn 51, 365–377 (2008). https://doi.org/10.1007/s11071-007-9217-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-007-9217-2

Keywords

Search

Navigation