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A study on global stabilization of periodic orbits in discrete-time chaotic systems by using symbolic dynamics. (English) Zbl 1340.37008
Summary: In this report, a control method for the stabilization of periodic orbits for a class of one- and two-dimensional discrete-time systems that are topologically conjugate to symbolic dynamical systems is proposed and applied to a population model in an ecosystem and the Smale horseshoe map. A periodic orbit is assigned as a target by giving a sequence in which symbols have periodicity. As a consequence, it is shown that any periodic orbits can be globally stabilized by using arbitrarily small control inputs. This work is a new attempt to systematically design a control system based on symbolic dynamics in the sense that one estimates the magnitude of control inputs and analyzes the Lyapunov stability.
MSC:
37B10 Symbolic dynamics
74H65 Chaotic behavior of solutions to dynamical problems in solid mechanics
93D15 Stabilization of systems by feedback
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[1] Birkhoff, G. D.: Dynamical Systems. American Mathematical Society, New York 1927. · Zbl 0992.37001 · doi:10.1002/zamm.19280080636
[2] Bollt, E. M., Dolnik, M.: Encoding information in chemical chaos by controlling symbolic dynamics. Phys. Rev. E 55 (1997), 6, 6404-6413. · doi:10.1103/physreve.55.6404
[3] Corron, N. J., Pethel, S. D.: Experimental targeting of chaos via controlled symbolic dynamics. Phys. Lett. A 313 (2003), 192-197. · doi:10.1016/s0375-9601(03)00754-0
[4] Glenn, C. M., Hayes, S.: Targeting Regions of Chaotic Attractors Using Small Perturbation Control of Symbolic Dynamics. Army Research Laboratory Adelphi MD 1996, No. ARL-TR-903.
[5] Hadamard, J.: Les surfaces à curbures opposés et leurs lignes géodesiques. J. Math. Pure Appl. 5 (1898), 27-73. · JFM 29.0522.01
[6] Hayes, S., Grebogi, C., Ott, E.: Communicating with chaos. Phys. Rev. Lett. 70 (1993), 20, 3031-3034. · doi:10.1103/physrevlett.70.3031
[7] Lai, Y.-C.: Controlling chaos. Comput. Phys. 8 (1994), 1, 62-67. · Zbl 1047.93522 · doi:10.1063/1.4823262
[8] Levinson, N.: A second order differential equation with singular solutions. Ann. of Math. (2) 50 (1949), 1, 126-153. · Zbl 0045.36501 · doi:10.2307/1969357
[9] Lind, D.: Multi-dimensional symbolic dynamics. Symbolic Dynamics ans its Applications (S. G. Williams, Proc. Symp. Appl. Math. 60 (2004), 61-80. · Zbl 1070.37009 · doi:10.1090/psapm/060/2078846
[10] May, R. M.: Simple mathematical models with very complicated dynamics. Nature 261 (1976), 459-467. · Zbl 0527.58025 · doi:10.1038/261459a0
[11] Morse, M., Hedlund, G. A.: Symbolic dynamics. Amer. J. Math. 60 (1938), 815-866. · Zbl 0063.04115 · doi:10.1215/S0012-7094-44-01101-4
[12] Moser, J.: Stable and Random Motions in Dynamical Systems. Princeton University Press, 1973. · Zbl 0991.70002
[13] Ott, E., Grebogi, C., Yorke, J. A.: Controlling chaos. Phys. Rev. Lett. 64 (1990), 11, 1196-1199. · Zbl 0964.37502 · doi:10.1103/PhysRevLett.69.3479
[14] Robinson, C.: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press, 1999. · Zbl 0914.58021
[15] Robinson, E. A., Jr.: Symbolic dynamics and tiling of \({\mathbb{R}}^d\). Symbolic Dynamics ans its Applications (S. G. Williams, Proc. Symp. Appl. Math. 60 (2002), 81-120. · doi:10.1090/psapm/060/2078847
[16] Smale, S.: Diffeomorphism with many periodic points. Differential and Combinatorial Topology (S. S. Cairns, Princeton University Press 1963, pp. 63-80.
[17] Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, 1991. · Zbl 1027.37002 · doi:10.1007/b97481
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