The structure of mode-locked regions in quasi-periodically forced circle maps

Using a mixture of analytic and numerical techniques we show that the mode- locked regions of quasi-periodically forced Arnold circle maps form complic ated sets in parameter space. These sets are characterized by 'pinched-off' regions, where the width of the mode-locked region becomes very small. By considering general quasi-periodically forced circle maps we show that this pinching occurs in a broad class of such maps having a simple symmetry. (C ) 2000 Elsevier Science B.V. All rights reserved.