×

Mode-locking in quasiperiodically forced systems with very small driving frequency. (English) Zbl 1129.37328

Summary: We consider the dynamics of the sine circle map with quasiperiodic forcing and very small forcing frequency. One of our goals is to compare the dynamics of our system with those of the system with larger values of the frequency. In particular we discovered an unusual fine structure of the mode-locking regions which disappears with increase in the forcing frequency. We investigated how this fine structure depends on the system parameters and described the bifurcations which occur on the boundaries of the mode-locking regions.

MSC:

37E05 Dynamical systems involving maps of the interval
37C99 Smooth dynamical systems: general theory
37G99 Local and nonlocal bifurcation theory for dynamical systems
70K50 Bifurcations and instability for nonlinear problems in mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andronov A. A., Theory of Oscillators (1966)
[2] Arnol’d V., Izv. Akad. Nauk SSSR 25 pp 21–
[3] DOI: 10.1063/1.165812 · Zbl 0900.92094 · doi:10.1063/1.165812
[4] Aubry S., Ann. Isr. Phys. Soc. 3 pp 133–
[5] DOI: 10.1103/PhysRevLett.55.2103 · doi:10.1103/PhysRevLett.55.2103
[6] DOI: 10.1103/PhysRevLett.65.533 · doi:10.1103/PhysRevLett.65.533
[7] DOI: 10.1016/0167-2789(95)00205-I · Zbl 0894.58042 · doi:10.1016/0167-2789(95)00205-I
[8] DOI: 10.1016/S0370-1573(97)00055-0 · doi:10.1016/S0370-1573(97)00055-0
[9] Gargner F. M., Phaselock Techniques (1979)
[10] DOI: 10.1016/S0375-9601(99)00260-1 · Zbl 0935.37013 · doi:10.1016/S0375-9601(99)00260-1
[11] DOI: 10.1016/S0167-2789(99)00235-3 · Zbl 0983.37044 · doi:10.1016/S0167-2789(99)00235-3
[12] DOI: 10.1016/0167-2789(84)90282-3 · Zbl 0588.58036 · doi:10.1016/0167-2789(84)90282-3
[13] DOI: 10.1103/PhysRevB.30.172 · doi:10.1103/PhysRevB.30.172
[14] DOI: 10.1016/0167-2789(94)90061-2 · Zbl 0807.58031 · doi:10.1016/0167-2789(94)90061-2
[15] DOI: 10.1103/PhysRevLett.56.1183 · doi:10.1103/PhysRevLett.56.1183
[16] DOI: 10.1088/0305-4470/23/8/006 · Zbl 0715.58025 · doi:10.1088/0305-4470/23/8/006
[17] DOI: 10.1016/S0167-2789(97)00160-7 · Zbl 0925.58057 · doi:10.1016/S0167-2789(97)00160-7
[18] Nishikawa T., Phys. Rev. 54 pp 6114–
[19] DOI: 10.1142/S0218127401004029 · Zbl 1091.37507 · doi:10.1142/S0218127401004029
[20] Pikovsky A., J. Phys. A 15 pp 5209–
[21] DOI: 10.1063/1.166074 · Zbl 1055.37519 · doi:10.1063/1.166074
[22] DOI: 10.1103/PhysRevLett.79.4127 · doi:10.1103/PhysRevLett.79.4127
[23] DOI: 10.1103/PhysRevLett.83.4530 · doi:10.1103/PhysRevLett.83.4530
[24] DOI: 10.1016/0167-2789(87)90229-6 · Zbl 0612.58030 · doi:10.1016/0167-2789(87)90229-6
[25] DOI: 10.1142/S0218127402005376 · doi:10.1142/S0218127402005376
[26] DOI: 10.1016/S0167-2789(97)00168-1 · Zbl 0925.58056 · doi:10.1016/S0167-2789(97)00168-1
[27] DOI: 10.1103/PhysRevLett.77.5039 · doi:10.1103/PhysRevLett.77.5039
[28] DOI: 10.1103/PhysRevA.45.5394 · doi:10.1103/PhysRevA.45.5394
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.