Nonlinear Dynamics

, Volume 51, Issue 3, pp 365–377 | Cite as

Control based bifurcation analysis for experiments

Original Paper


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.


Bifurcation analysis Numerical continuation Hybrid experiments Hardware-in-the-loop 


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  1. 1.
    Abed, E., Wang, H., Chen, R.: Stabilization of period doubling bifurcations and implicatons for control of chaos. Physica D 70, 154–164 (1994)MATHCrossRefMathSciNetGoogle Scholar
  2. 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)CrossRefGoogle Scholar
  3. 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)MATHCrossRefMathSciNetGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    De Feo, O., Maggio, G.: Bifurcations in the Colpitts oscillator: from theory to practice. Int. J. Bif. Chaos 13(10), 2917–2934 (2003)MATHCrossRefGoogle Scholar
  6. 6.
    Dercole, F., Kuznetsov, Y.: SlideCont: an Auto97 driver for bifurcation analysis of Filippov systems. ACM Trans. Math. Softw. 31, 95–119 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 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: (1997)
  8. 8.
    Eyert, V.: A comparative study on methods for convergence acceleration of iterative vector sequences. J. Comput. Phys. 124(0059), 271–285 (1996)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Galvanetto, U., Bishop, S.: Dynamics of a simple damped oscillator undergoing stick-slip vibrations. Meccanica 34, 337–347 (1999)MATHCrossRefMathSciNetGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 11.
    Hassouneh, M., Abed, E.: Border collision bifurcation control of cardiac alternans. Int. J. Bif. Chaos 14(9), 3303–3315 (2004)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Horváth, R.: Experimental investigation of excited and self-excited vibration. Master's thesis, University of Technology and Economics, Budapest, /~horvaro/index2.html (2000)
  13. 13.
    Hövel, P., Schöll, E.: Control of unstable steady states by time-delayed feedback methods. Phys. Rev. E 72(046203) (2005)CrossRefGoogle Scholar
  14. 14.
    Kevrekidis, I., Gear, C., Hummer, G.: Equation-free: the computer-aided analysis of complex multiscale systems. AIChE J. 50(11), 1346–1355 (2004)CrossRefGoogle Scholar
  15. 15.
    Kuznetsov, Y.: Elements of Applied Bifurcation Theory, 3rd edn. Springer Verlag, New York (2004)MATHGoogle Scholar
  16. 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)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Langer, G., Parlitz, U.: Robust method for experimental bifurcation analysis. Int. J. Bif. Chaos 12(8), 1909–1913 (2002)CrossRefGoogle Scholar
  18. 18.
    Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)CrossRefGoogle Scholar
  19. 19.
    Pyragas, K.: Control of chaos via an unstable delayed feedback controller. Phys. Rev. Lett. 86(11), 2265–2268 (2001)CrossRefGoogle Scholar
  20. 20.
    Sieber, J., Krauskopf, B.: Control-based continuation of periodic orbits with a time-delayed difference scheme. Int. J. Bif. Chaos (in press). (
  21. 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)MATHCrossRefGoogle Scholar
  22. 22.
    Stépán, G., Insperger, T.: Research on delayed dynamical systems in Budapest. Dynamical Systems Magazine. (2004)
  23. 23.
    Trefethen, L.: Finite difference and spectral methods for ordinary and partial differential equations. Unpublished text, available at (1996)
  24. 24.
    Trefethen, L., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, NJ (2005)MATHGoogle Scholar
  25. 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)CrossRefGoogle Scholar
  26. 26.
    Yanchuk, S., Wolfrum, M., Hövel, P., Schöll, E.: Control of unstable steady states by long delay feedback. Phys. Rev. E 74(026201) (2006)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of EngineeringKing’s College, University of AberdeenAberdeenUK
  2. 2.Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering MathematicsQueen’s Building, University of BristolBristolUK

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