Smooth analog of standard mapping

Twenty years ago Bullett [1] published an article [1] where he found the invariant curves of standard mapping, having replaced the sinusoidal force by its smooth analog, a piecewise linear saw. His studies discovered an unexpected effect: at certain values of the perturbation parameter, unsplit separatrices of integer and fractional resonances arise among global invariant curves, while chaotic layers, which are usually attendant to these separatrices, are absent. Interestingly, the system remains nonintegrable in this case and the separatrices persist, confining momentum global diffusion under the condition of strong local chaos. For reasons not well understood, this important effect and its consequences had gone largely unnoticed until Ovsyannikov [2] independently proved a similar theorem for integer resonances in terms of the same model of symmetric piecewise linear 2D mapping. Since then, piecewise linear maps and their related continuous systems have become a subject of extensive research. Both Bullett and Ovsyannikov restricted analysis to the invariant curves of the new type, since the two branches of split separatrices form chaotic trajectories that are impossible to treat analytically. To the author's knowledge, examples of persisting separatrices other than those given in [1, 2] have not been reported. In this work, the author presents numerical and analytical results that directly or indirectly concern the effect of separatrix persistence in the absence of attendant dynamic chaos. Issues remaining to be understood are noted.