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.
Similar content being viewed by others
References
Abed, E., Wang, H., Chen, R.: Stabilization of period doubling bifurcations and implicatons for control of chaos. Physica D 70, 154–164 (1994)
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)
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)
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)
De Feo, O., Maggio, G.: Bifurcations in the Colpitts oscillator: from theory to practice. Int. J. Bif. Chaos 13(10), 2917–2934 (2003)
Dercole, F., Kuznetsov, Y.: SlideCont: an Auto97 driver for bifurcation analysis of Filippov systems. ACM Trans. Math. Softw. 31, 95–119 (2005)
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)
Eyert, V.: A comparative study on methods for convergence acceleration of iterative vector sequences. J. Comput. Phys. 124(0059), 271–285 (1996)
Galvanetto, U., Bishop, S.: Dynamics of a simple damped oscillator undergoing stick-slip vibrations. Meccanica 34, 337–347 (1999)
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)
Hassouneh, M., Abed, E.: Border collision bifurcation control of cardiac alternans. Int. J. Bif. Chaos 14(9), 3303–3315 (2004)
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)
Hövel, P., Schöll, E.: Control of unstable steady states by time-delayed feedback methods. Phys. Rev. E 72(046203) (2005)
Kevrekidis, I., Gear, C., Hummer, G.: Equation-free: the computer-aided analysis of complex multiscale systems. AIChE J. 50(11), 1346–1355 (2004)
Kuznetsov, Y.: Elements of Applied Bifurcation Theory, 3rd edn. Springer Verlag, New York (2004)
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)
Langer, G., Parlitz, U.: Robust method for experimental bifurcation analysis. Int. J. Bif. Chaos 12(8), 1909–1913 (2002)
Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170, 421–428 (1992)
Pyragas, K.: Control of chaos via an unstable delayed feedback controller. Phys. Rev. Lett. 86(11), 2265–2268 (2001)
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)
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)
Stépán, G., Insperger, T.: Research on delayed dynamical systems in Budapest. Dynamical Systems Magazine. http://www.dynamicalsystems.org/ma/ma/display?item=85 (2004)
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)
Trefethen, L., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton, NJ (2005)
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)
Yanchuk, S., Wolfrum, M., Hövel, P., Schöll, E.: Control of unstable steady states by long delay feedback. Phys. Rev. E 74(026201) (2006)
Author information
Authors and Affiliations
Corresponding author
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
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-007-9217-2