Abstract
A problem of stabilization of unstable cycles is solved in this paper using delay feedback at the example of a model equation with cubic nonlinearity. We consider the case when, in a problem without control, exactly one multiplicator of the cycle is located outside the unit circle. The delay time is selected proportional to the period of the initial cycle so that, in the problem with control, the initial solution should still exists. The D-partition is obtained for the plane of the complex coefficient of the delayed control. The main result consists in analytically found conditions for parameters of the delayed feedback control (coefficient and delay time) under which the initial periodic solution becomes stable. The necessary and sufficient conditions for proper parameters of the problems under which the stabilization problem is solvable are also determined. As a consequence, the problem of stability of the Stuart-Landau equation cycle is completely solved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Schuster, H.G., Handbook of Chaos Control, Weinheim: Wiley, 1999.
Gauthier, D.J., Resource letter: Controlling chaos, Am. J. Phys., 2003, vol. 71, pp. 750–759.
Pyragas, K., Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 1992, vol. 170, pp. 421–428.
Nakajima, H. and Ueda, Y., Limitation of generalized delayed feedback control, Physica D, 1998. Vol. 111, pp. 143–150.
Nakajima, H., On analytical properties of delayed feedback control of chaos, Phys. Lett. A, 1997, vol. 232, pp. 207–210.
Fiedler, B., Flunkert, V., Georgi, V., Hövel, P., and Schöll, E., Refuting the odd number limitation of time-delayed feedback control, Phys. Rev. Lett., 2007, vol. 98, p. 114101.
Fiedler, B., Flunkert, V., Georgi, M., Hövel, P., and Schöll, E., Beyond the odd-number limitation of time-delayed feedback control, in Handbook of Chaos Control, Schöll, E., Ed., Weinheim: Wiley, 2008, pp. 73–84.
Flunkert, V., Delay-Coupled Complex Systems and Applications to Lasers, Berlin: Springer-Verlag, 2011.
Bruno, A.D., The Local Method of Nonlinear Analysis of Differential Equations, Berlin: Springer-Verlag, 1989.
Arnold, V.I., Additional Chapters of Theory of Ordinary Differential Equations, Moscow: Nauka, 1978.
Kashchenko, I.S., Dynamics of an equation with a large coefficient of delay control, Doklady Mathem., 2011, vol. 83, pp. 258–261.
Kashchenko, A.A., Stability of the simplest periodic solutions in the Stuart-Landau equation with large delay, Autom. Control Compt. Sci., 2013, vol. 47, pp. 566–570.
Glazkov, D.V. and Kaschenko, S.A., Local dynamics of DDE with large delay in the vicinity of the self-similar cycle, Model. Anal. Inform. Sist., 2010, vol. 17, no. 3, pp. 38–47.
Kashchenko, I.S. and Kashchenko, S.A., Asymptotic of difficult spatiotemporal structures in systems with big delay, News of Higher Education “Applied Nonlinear Dynamics”, 2008, vol. 16, pp. 137–146.
Glazkov, D.V., Local dynamics of an equation with long delay feedback, Model. Anal. Inform. Sist., 2011, vol. 18, no. 1, pp. 75–85.
Glyzin, S.D., Dynamic properties of the simplest finite-difference approximations of the “reaction-diffusion” boundary value problem, Diff. Equat., 1997, vol. 33, pp. 808–814.
Glyzin, S.D., Difference approximations of “reaction-diffusion” equation on a segment, Model. Anal. Inform. Syst., 2009, vol. 16, no. 3, pp. 96–116.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.G. Bogaevskaya, I.S. Kashchenko, 2014, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2014, No. 1, pp. 53–65.
About this article
Cite this article
Bogaevskaya, V.G., Kashchenko, I.S. Influence of delayed feedback control on the stability of periodic orbits. Aut. Control Comp. Sci. 48, 477–486 (2014). https://doi.org/10.3103/S0146411614070037
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0146411614070037