Tracking Nonlinear Oscillations with Time-Delayed Feedback


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We demonstrate a method for tracking the onset of oscillations (Hopf bifurcation) in nonlinear dynamical systems. Our method does not require a mathematical model of the dynamical system but instead relies on feedback controllability. This makes the approach potentially applicable in an experiment. The main advantage of our method is that it allows one to vary parameters directly along the stability boundary. In other words, there is no need to observe the transient oscillations of the dynamical system for a long time to determine their decay or growth. Moreover, the procedure automatically tracks the change of the critical frequency along the boundary and is able to continue the Hopf bifurcation curve into parameter regions where other modes are unstable.We illustrate the basic ideas with a numerical realization of the classical autonomous dry friction oscillator.



Edited by:

Patrick Sean Keogh




J. Sieber and B. Krauskopf, "Tracking Nonlinear Oscillations with Time-Delayed Feedback", Applied Mechanics and Materials, Vols. 5-6, pp. 417-424, 2006

Online since:

October 2006




[1] Y.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer Verlag, 2004. third edition.

[2] E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang. AUTO97, Continuation and bifurcation software for ordinary differential equations, (1998).

[3] A. Blakeborough, M.S. Williams, A.P. Darby, and D.M. Williams. Philosophical Transactions of the Royal Society of London A, 359: 1869-1891, (2001).

[4] R. Horv´ath. Master's thesis, University of Technology and Economics, Budapest, http: /www. auburn. edu/∼horvaro/index2. html, (2000).

[5] G. St´ep´an and T. Insperger. Research on delayed dynamical systems in Budapest. Dynamical Systems Magazine, 2004. http: /www. dynamicalsystems. org/ma/ma/display?item=85.

[6] J. Sieber and B. Krauskopf. BCANM Preprint, University of Bristol, 2006. http: /hdl. handle. net/ 1983/399.

[7] K. Pyragas. Phys. Rev. Lett., 86(11): 2265-2268, (2001).

[8] Y.N. Kyrychko, K.B. Blyuss, A. Gonzalez-Buelga, S.J. Hogan, and D.J. Wagg. Proc. Roy. Soc. London A, 462: 1271-1294, (2005).