Recurrence of invariant circles in a dissipative standardlike map

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Abstract
We study a dissipative standardlike map that contains a parameter z that can be used to tune the map from a piecewise-linear map (z=0) to the dissipative standard map (z=∞). When both the tuning parameter z and dissipation are small, the reappearance of an invariant circle after its breakup is observed. This recurrence phenomenon takes place as 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 dissipation parameter value dependent on z. Scaling behavior at the disappearance and reappearance points is also discussed.