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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 onedimensional circle map. As a control parameter passes through a critical
value, a quasiperiodic transition to chaos occurs through the breakup of the
invariant circles. To examine the universality of this quasiperiodic transition,
we studied twodimensional 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 modelocked
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 areapreserving standardlike maps. For
more details, see the following publications:
[1] B. Hu, J. Shi, and S.Y. Kim, "Recurrence of KolmogorovArnoldMoser tori in
nonanalytic twist maps," J. Stat. Phys. 62, 631649 (1991).
[2] B. Hu, J. Shi, and S.Y. Kim, "Critical phenomena of invariant circles,"
Phys. Rev. A 43, 42494253 (1991).
[3] S.Y. Kim and B. Hu, "Recurrence of invariant circles in a dissipative
standardlike map," Phys. Rev. A 44, 934939 (1991).
[4] S.Y. Kim and D.S. Lee, "Transition to chaos in a dissipative standardlike
map," Phys. Rev. A 45, 54805487 (1992).
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