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Recurrence of Invariant Circles in Dissipative and Conservative Maps

  Quasiperiodicity and mode lockings in dissipative systems were extensively studied mainly in the model of the one-dimensional circle map. As a control parameter passes through a critical value, a quasiperiodic transition to chaos occurs through the break-up of the invariant circles. To examine the universality of this quasiperiodic transition, we studied two-dimensional standardlike map that is a more real model system. Unlike the case of the circle map, as the control parameter is changed, reappearance of the invariant circle after its breakup was observed. This Recurrence of Invariant Circle occurs because the nearby mode-locked resonances separate after they overlapped. However, as the dissipation is increased, the number of recurrences gradually decreases, and ultimately reappearance ceases at some threshold value of the dissipation parameter. Note also that the scaling behaviors at the disappearance and reappearance points are the same as those in the circle map. Furthermore, we have also found this kind of recurrence of invariant circles in area-preserving standardlike maps. For more details, see the following publications:

[1] B. Hu, J. Shi, and S.-Y. Kim, "Recurrence of Kolmogorov-Arnold-Moser tori in nonanalytic twist maps," J. Stat. Phys. 62, 631-649 (1991).
[2] B. Hu, J. Shi, and S.-Y. Kim, "Critical phenomena of invariant circles," Phys. Rev. A 43, 4249-4253 (1991).
[3] S.-Y. Kim and B. Hu, "Recurrence of invariant circles in a dissipative standardlike map," Phys. Rev. A 44, 934-939 (1991).
[4] S.-Y. Kim and D.-S. Lee, "Transition to chaos in a dissipative standardlike map," Phys. Rev. A 45, 5480-5487 (1992).

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