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Off-center reflections: caustics and chaos. (English) Zbl 1041.37018

Summary: We study the possible link between the dynamics of a certain family of circle maps and the caustics of their iterates. The maps are defined by off-center reflections in a mirrored circle; they can also be regarded as perturbed rotations. Some of our experimental observations can be justified rigorously: for example, a lower bound is given for the number of cusps and the mode-locking behavior is studied. Symplectic topology is a particularly useful tool in this study.

MSC:

37E10 Dynamical systems involving maps of the circle
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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References:

[1] Arnold V. I., Izv. Akad. Nauk SSSR Ser. Mat. 25 pp 21– (1961)
[2] DOI: 10.1007/978-1-4684-0147-9
[3] Arnold V. I., Uspekhi Mat. Nauk 38 (4) pp 189– (1983)
[4] Arnold V. I., Univ. Lecture Series 5, in: Topological invariants of plane curves and caustics (1994)
[5] Arnold V. I., Uspekhi Mat. Nauk 51 (1) pp 3– (1996)
[6] Au T. K. K., ”The dynamics of off-center reflection” (1999) · Zbl 1042.37029
[7] Bak P., Directions in chaos 2 pp 16– (1988)
[8] Bruce J. W., Quart. J. Math. Oxford Ser. (2) 35 pp 243– (1984) · Zbl 0547.58013
[9] Bruce J. W., Proc. London Math. Soc. (3) 50 (3) pp 571– (1985) · Zbl 0546.58010
[10] Bruce J. W., Amer. Math. Monthly 88 (9) pp 651– (1981) · Zbl 0473.53002
[11] Bruce J. W., Proc. Roy. Soc. London Ser. A 381 pp 83– (1982) · Zbl 0496.58002
[12] Devaney R. L., An introduction to chaotic dynamical systems,, 2. ed. (1989) · Zbl 0695.58002
[13] Ding E. J., Phys. D 32 (1) pp 153– (1988) · Zbl 0667.58026
[14] Feudel U., Phys. D 88 (3) pp 176– (1995) · Zbl 0894.58042
[15] Giblin P. J., Quart. J. Math. Oxford Ser. (2) 37 (145) pp 17– (1986) · Zbl 0588.53002
[16] Herman M.-R., Geometry and topology (Rio de Janeiro, 1976) pp 271– (1977)
[17] Herman M.-R., Inst. Hautes Études Sci. Publ. Math. 49 pp 5– (1979) · Zbl 0448.58019
[18] Jensen M. H., Phys. Rev. A (3) 30 (4) pp 1960– (1984)
[19] Kaneka K., Prog. Theoretical Phys. 72 (6) pp 1089– (1984) · Zbl 1074.37512
[20] Piña E., Phys. Rev. A (3) 34 (1) pp 574– (1986)
[21] Tabachnikov S. L., Uspekhi Mat. Nauk 45 (1) pp 191– (1990)
[22] Tabachnikov S., Amer. Math. Monthly 102 (10) pp 912– (1995) · Zbl 0842.53003
[23] Yau S.-T., Partial differential equations on manifolds (Los Angeles, 1990) 1 pp 1– (1993)
[24] Zheng W. M., Internat. J. Modem Phys. B 5 (3) pp 481– (1991)
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