Discontinuous generalized synchronization of chaos

We study synchronization functions in basic examples of discontinuous forced systems with contractive response and chaotic driving. The forcing is given by baker-type maps and the response is assumed to depend monotonically on the drive. The resulting synchronization functions have dense sets of discontinuities and their graphs appear to be extremely choppy. We show that these functions have bounded variation when the contraction is strong, and conversely, that their total variation is infinite when the contraction becomes weak. In the first case, we also analyze in detail smoothness properties of the corresponding continuous component. 2010 Mathematical Subject Classification. 37A99, 26A45