Bubbling, riddling, blowout and critical curves

This paper provides a survey of some recent results and examples concerning the use of the method of critical curves in the study of chaos synchronization in discrete dynamical systems with an invariant one-dimensional submanifold. Some examples of two-dimensional discrete dynamical systems, which exhibit synchronization of chaoti1c trajectories with the related phenomena of bubbling, on–off intermittency, blowout and riddles basins, are examined by the usual local analysis in terms of transverse Lyapunov exponents, whereas segments of critical curves are used to obtain the boundary of a two-dimensional compact trapping region containing the one-dimensional Milnor chaotic attractor on which synchronized dynamics occur. Thanks to the folding action of critical curves, the existence of such a compact region may strongly influence the effects of bubbling and blowout bifurcations, as it acts like a ‘trapping vessel’ inside which bubbling and blowout phenomena are bounded by the global dynamical forces of the dynamical system.