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Bifurcations of circle maps: Arnol’d tongues, bistability and rotation intervals. (English) Zbl 0612.58032
This paper studies the bifurcations of families of noninvertible circle maps defined by \(x+w+bp(x)\) (mod 1), where (b,w) are two real parameters, p(x) being a continuous periodic function of period 1, which verifies other hypotheses given in § 3. When \(p(x)\equiv (2\pi)^{-1}\sin (2\pi x)\), a ”standard” map is obtained. More particularly, the author considers the bifurcation curves in the parameter plane (b,w), which are the boundaries of the existence domains of periodic points associated with a rational rotation number. In this case, such domains, with \(b>1\), are the extension (with overlappings) of what he calls ”Arnol’d tongues” (known before Arnol’d in the theory of nonlinear harmonic and subharmonic synchronization of oscillators), obtained when the map is a diffeomorphism \((| b| <1\) and not \(b<1\) as indicated in the text). The case corresponding to irrational rotation numbers is also considered. Theorems give information about the dynamics (bistability and aperiodicity) of the maps for (b,w) belonging to various regions of the parameter plane.
Reviewer: C.Mira

MSC:
37G99 Local and nonlocal bifurcation theory for dynamical systems
57R35 Differentiable mappings in differential topology
70K50 Bifurcations and instability for nonlinear problems in mechanics
47H10 Fixed-point theorems
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[1] Arnol’d, V.I.: Small denominators. I. On the mappings of the circumference onto itself. Trans. Am. Math. Soc. 2nd Ser.46, 213 (1965)
[2] Aronson, D., Chory, M., Hall, G., McGehee, R.: Bifurcation from an invariant circle for two parameter families of maps of the plane: a computer assisted study. Commun. Math. Phys.83, 303 (1982) · Zbl 0499.70034 · doi:10.1007/BF01213607
[3] Bernhardt, C.: Rotation intervals of a class of endomorphisms of the circle. Proc. Lond. Math. Soc., III Ser.45 part 2 (1982) · Zbl 0458.58020
[4] Block, L.: Homoclinic points of mappings of the interval. Proc. Am. Math. Soc.72, 3, 576-580 (1978) · Zbl 0365.58015 · doi:10.1090/S0002-9939-1978-0509258-X
[5] Block, L., Franke, J.: Existence of periodic points for maps ofS 1. Invent. Math.22, 69-73 (1973) · Zbl 0272.58005 · doi:10.1007/BF01425575
[6] Brunovsky, P.: Generic properties of the rotation number of one parameter diffeomorphisms of the circle. Czech. Math. J.24 (99), 74-90 (1974) · Zbl 0308.58007
[7] Collet, P., Eckmann, J.-P.: Iterated maps on the interval as dynamical systems. Boston: Birkhäuser 1980 · Zbl 0458.58002
[8] Coullet, P., Tresser, C., Arneodo, A.: Transition to turbulence for doubly periodic flows. Phys. Lett.77A, (5), 327-331 (1980)
[9] Feigenbaum, M., Kadanoff, L., Shenker, S.: Quasiperiodicity in dissipative systems: a renormalization group analysis. Physica5D, 370-382 (1982)
[10] Glass, L., Guevara, M., Shrier, A., Perez, R.: Bifurcation and chaos in a periodically stimulated cardiac oscillator. Proc. of the Los Alamos Conf. on ?Order in Chaos,? Physica7D, 89-103 (1983)
[11] Greenspan, B., Holmes, P.: Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators. SIAM J. Math. Anal. (to appear) · Zbl 0547.34028
[12] Hall, G.: AC ? Denjoy counterexample. Ergodic Theory Dyn. Syst.1, 261-272 (1981) · Zbl 0501.58035 · doi:10.1017/S0143385700001243
[13] Hall, G.: Personal communication, 1983
[14] Hall, G.: Resonance zones in two-parameter families of circle homeomorphism. MRC Technical Summary Report, May, 1983
[15] Herman, M.: Measure de Lebesque et nombre de rotation. Lecture Notes in Math. Vol.597. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0366.57007
[16] Herman, M.: Sur la conjugaison differentiable des diffeomorphismes du cercle des rotations. Publ. Math., Inst. Hautes Etud. Sci.49, 5-234 (1979) · Zbl 0448.58019 · doi:10.1007/BF02684798
[17] Hoppensteadt, F., Keener, J.: Phase locking of biological clock. J. Math. Biol.15, 339-349 (1982) · Zbl 0489.92006 · doi:10.1007/BF00275692
[18] Ito, R.: Rotation sets are closed. Math. Proc. Camb. Philos. Soc.89, 107-111 (1981) · Zbl 0484.58027 · doi:10.1017/S0305004100057984
[19] Jensen, M., Bak, P., Bohr, T.: Complete devils staircase, fractal dimension, and universality of mode-locking structure in the circle map. Phys. Rev. Lett.50, 1637-1639 (1983) · doi:10.1103/PhysRevLett.50.1637
[20] Kadanoff, L.: Supercritical behavior of an ordered trajectory. J. Stat. Phys.31, 1-27 (1983) · doi:10.1007/BF01010921
[21] Newhouse, S., Palis, J., Takens, F.: Bifurcations and stability of families of diffeomorphisms. Publ. Math., Inst. Hautes Etud. Sci.57, 5-72 (1983) · Zbl 0518.58031 · doi:10.1007/BF02698773
[22] Ostlund, S., Rand, D., Sethna, J., Siggia, E.: Universal properties of the transition from quasiperiodicity to chaos in dissipative systems. Physica8D, 303-342 (1983) · Zbl 0538.58025
[23] Perez, R., Glass, L.: Bistability period doubling bifurcations and chaos in a periodically forced oscillator. Phys. Lett.90A, 441-443 (1982)
[24] Shenker, S.: Scaling behavior in a map of a circle onto itself: empirical results. Physica5D, 405-408 (1982)
[25] Singer, D.: Stable orbits and bifurcations of maps of the interval. SIAM J. Appl. Math.35, 260-267 (1978) · Zbl 0391.58014 · doi:10.1137/0135020
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