zbMATH — the first resource for mathematics

Kneading plane of the circle map. (English) Zbl 0817.58030
This paper investigates the structure of the kneading plane for circle maps, and a method for its construction is shown.
Reviewer: R.Cowen (Nairobi)

37D99 Dynamical systems with hyperbolic behavior
Full Text: DOI
[1] Glass, L.; Perez, R., Fine structure of phase locking, Phys. rev. lett., 48, 1772, (1982)
[2] Perez, R.; Glass, L., Bistability, period doubling bifurcations, and chaos in a periodically forced oscillator, Phys. lett., 90A, 441, (1982)
[3] Glass, L.; Guevara, M.; Ahrier, A.; Perez, R., Bifurcation and chaos in a periodically stimulated cardiac oscillator, Physica, 7D, 89, (1983)
[4] Hoppensteadt, F.; Keener, J., Phase locking of biological clocks, J. math. biol., 15, 339, (1982) · Zbl 0489.92006
[5] Jensen, M.; Bak, P.; Bohr, T., Transition to chaos by interaction of resonances in dissipative systems. I. circle maps, Phys. rev., A30, 1960, (1984)
[6] Coullet, P.; Tresser, C.; Arneodo, A., Transition to chaos for double periodic flows, Phys. lett., 77A, 327, (1980)
[7] Feigenbaum, M.; Kadanoff, L.; Shanker, S., Quasiperiodicity in dissipative systems: a renormalization group analysis, Physica, 5D, 370, (1982)
[8] Shanker, S., Scaling behavior in a map of a circle to itself: empirical results, Physica, 5D, 405, (1982)
[9] Ostlund, S.; Rand, D.; Sethna, J.; Siggia, E., Universal properties of the transition from quasiperiodicity to chaos in dissipative systems, Physica, 8D, 303, (1983) · Zbl 0538.58025
[10] Newhouse, S.; Palis, J.; Takens, F., Bifurcations and stability of families of diffeomorphisms, Pub. math. I.H.E.S., 57, 5, (1983) · Zbl 0518.58031
[11] Ito, R., Rotation sets are closed, Math. proc. camb. phil. soc., 89, 107, (1981) · Zbl 0484.58027
[12] Chenciner, A.; Gambaudo, J.M.; Tressor, C., Une remarque sur la structure des endomorphisms degreé 1 du cercle, C.R. acad. sci. Paris, 299 I, 253, (1984)
[13] Boyland, P.L., Bifurcations of circle maps: Arnold tongues, Commun. math. phys., 106, 353, (1986) · Zbl 0612.58032
[14] Mackay, R.S.; Tresser, C., Transition to topological chaos for circle maps, Physica, 19D, 206, (1986) · Zbl 0596.58027
[15] Milnor, J.; Thurston, W., On iterated maps of the interval, () · Zbl 0664.58015
[16] Guckenheimer, J., Bifurcation of dynamical systems, () · Zbl 0205.54102
[17] Collet, P.; Eckmann, J.P., Iterated maps on the interval as dynamical systems, (1980), Birkhaüser · Zbl 0458.58002
[18] Metropolis, N.; Stein, M.L.; Stein, P.R., On finite limit sets for transformations on the unit interval, J. comb. theory, 15, 25, (1973) · Zbl 0259.26003
[19] Derrida, B.; Gervois, A.; Pomeau, Y., Iteration of endomorphisms on the real axis and representation of numbers, Ann. inst H. poicaré, 29, 305, (1979) · Zbl 0416.28012
[20] Universal metric properties of bifurcations of endomorphisms, J. phys., A12, 269, (1979) · Zbl 0416.28011
[21] MacKay, R.S.; Tresser, C., Some flesh on the skeleton: the bifurcation structure of bimodal maps, Physica, 27D, 412, (1987) · Zbl 0626.58038
[22] Bernhardt, C., Rotation intervals of a class of endomorphisms of the circle, Proc. lond. math. soc. III ser., 45, 258, (1982) · Zbl 0458.58020
[23] Zheng, W.M., Symbolic dynamics for the circle map, Int. J. mod. phys., B5, 481, (1991)
[24] Ostlund, S.; Kim, S.H., Renormalization of quasiperiodic mapping, Phys. scripta, 9, 193, (1985) · Zbl 1063.37526
[25] Zheng, W.M., Applied symbolic dynamics for the Lorenz-like map, Phys. rev., A42, 2076, (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.