Controlling chaos. (English) Zbl 0964.37501

Ott, Edward; Grebogi, Celso; Yorke, James A.

Phys. Rev. Lett. 64, No. 11, 1196-1199 (1990); Erratum, No. 23, 2837 (1990).

Summary: The authors show that one can convert a chaotic attractor to any one of a large number of possible attracting time-periodic motions by making only small time-dependent perturbations of an available system parameter. The method utilizes delay coordinate embedding, and so is applicable to experimental situations in which apriori analytical knowledge of the system dynamics is not available. Important issues include the length of the chaotic transience preceding the periodic motion, and the effect of noise. A numerical example is given.

MSC:

37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C25	Periodic solutions to ordinary differential equations
37C70	Attractors and repellers of smooth dynamical systems and their topological structure

Keywords:

chaotic attractor

References:

[1]	C. Grebogi, Phys. Rev. A 37 pp 1711– (1988)
[2]	C. Grebogi, Phys. Rev. A 36 pp 3522– (1987)
[3]	D. Auerbach, Phys. Rev. Lett. 58 pp 2387– (1987)
[4]	H. Hata, Prog. Theor. Phys. 78 pp 511– (1987)
[5]	A. Katok, Publ. Math. IHES 51 pp 137– (1980)
[6]	R. Bowen, Trans. Am. Math. Soc. 154 pp 377– (1971)
[7]	E. Ott, in: Chaos: Proceedings of a Soviet-American Conference (1990)
[8]	F. Takens, in: Dynamical Systems and Turbulence (1981)
[9]	N. H. Packard, Phys. Rev. Lett. 45 pp 712– (1980)
[10]	G. H. Gunaratne, Phys. Rev. Lett. 63 pp 1– (1989)
[11]	C. Grebogi, Phys. Rev. Lett. 57 pp 1284– (1986)
[12]	P. Romeiras, Phys. Rev. A 36 pp 5365– (1987)
[13]	A. Hubler, Helv. Phys. Acta 62 pp 343– (1989)
[14]	T. B. Fowler, IEEE Trans. Autom. Control 34 pp 201– (1989) · Zbl 0677.93072