Abstract
A quasi-periodic orbit possesses the properties of both a periodic orbit and a chaotic orbit, which are almost periodic and aperiodic, respectively. Whereas the stabilization of a periodic orbit is widely discussed, that of a quasi-periodic orbit remains an enigma because of its aperiodicity. We focus on a quasi-periodic orbit on an invariant closed curve in discrete-time systems, which is the simplest case of dynamics on a high-dimensional invariant torus. In this case, a quasi-periodic orbit can be characterized by its rotation number, which is reflected in the design of control methods. We apply three control methods, the external force control, the delayed feedback control, and the pole assignment method, to stabilize an unstable quasi-periodic orbit. Although these control methods have been used to stabilize unstable periodic orbits, we show that they are also applicable to unstable quasi-periodic orbits.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bruin, H.: Numerical determination of the continued fraction expansion of the rotation number. Physica D: Nonlinear Phenomena 59(1–3), 158–168 (1992). http://dx.doi.org/10.1016/0167-2789(92)90211-5
Ciocci, M.C., Litvak-Hinenzon, A., Broer, H.: Survey on dissipative KAM theory including quasi-periodic bifurcation theory based on lectures by Henk Broer. In: Montaldi, J., Ratiu, T. (eds.) Peyresq Lectures in Geometric Mechanics and Symmetry. Cambridge University Press, Cambridge (2005)
Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57, 617–656 (1985). doi:10.1103/RevModPhys.57.617
Fujisaka, H., Yamada, T.: Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys. 69(1), 32–47 (1983). doi:10.1143/PTP.69.32
Ichinose, N., Komuro, M.: Delayed feedback control and phase reduction of unstable quasi-periodic orbits. Chaos: Interdisc. J. Nonlinear Sci. 24, 033137 (2014). http://dx.doi.org/10.1063/1.4896219
Kamiyama, K., Komuro, M., Endo, T.: Algorithms for obtaining a saddle torus between two attractors. I. J. Bifurcat. Chaos 23(9), 1330032 (2013). http://dx.doi.org/10.1142/S0218127413300322
Kaneko, K.: Overview of coupled map lattices. Chaos: Interdisc. J. Nonlinear Sci. 2(3), 279–282 (1992). http://dx.doi.org/10.1063/1.165869
Khinchin, A., Eagle, H.: Continued Fractions. Dover Books on Mathematics. Dover Publications, New York (1997)
MacKay, R.: A simple proof of Denjoy’s theorem. Math. Proc. Cambridge Philos. Soc. 103(2), 299–303 (1988)
Ott, E., Grebogi, C., Yorke, J.A.: Controlling Chaos. Phys. Rev. Lett. 64, 1196–1199 (1990). doi:10.1103/PhysRevLett.64.1196
Pavani, R.: The numerical approximation of the rotation number of planar maps. Comput. Math. Appl. 33(5), 103–110 (1997). http://dx.doi.org/10.1016/S0898-1221(97)00023-0
Pecora, L.M., Carroll, T.L., Johnson, G.A., Mar, D.J., Heagy, J.F.: Fundamentals of synchronization in chaotic systems, concepts, and applications. Chaos: Interdisc. J. Nonlinear Sci. 7(4), 520–543 (1997). http://dx.doi.org/10.1063/1.166278
Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Lett. A 170(6), 421–428 (1992). http://dx.doi.org/10.1016/0375-9601(92)90745-8
Veldhuizen, M.V.: On the numerical approximation of the rotation number. J. Comput. Appl. Math. 21(2), 203–212 (1988). http://dx.doi.org/10.1016/0377-0427(88)90268-3
Wonham, W.: On pole assignment in multi-input controllable linear systems. IEEE Trans. Autom. Control 12(6), 660–665 (1967). doi:10.1109/TAC.1967.1098739
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer Japan
About this chapter
Cite this chapter
Ichinose, N., Komuro, M. (2015). Stabilization Control of Quasi-periodic Orbits. In: Aihara, K., Imura, Ji., Ueta, T. (eds) Analysis and Control of Complex Dynamical Systems. Mathematics for Industry, vol 7. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55013-6_8
Download citation
DOI: https://doi.org/10.1007/978-4-431-55013-6_8
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-55012-9
Online ISBN: 978-4-431-55013-6
eBook Packages: EngineeringEngineering (R0)