Progress in Low-dimensional Chaos: on Phase Synchronization and Indeterminate Bifurcations

The field of chaotic dynamics has grown highly nonlinearly in the past few decades. Major progress occurred in traditional areas such as bifurcations, crises, basin boundaries, strengthening the mathematical foundations. New topics such as control and synchronization of chaos have emerged, addressing more practical questions. This book has a bit of both, addressing the initiates in low-dimensional chaos, from graduate students to researchers. First, we discuss the topic of phase synchronization of chaos. This phenomenon results from weak interactions between dynamical systems and found applications to neuroscience and communications. Here we focus on a competition phenomenon that occurs between signals in phase synchronization of a chaotic attractor. Second, we discuss the topic of indeterminate bifurcations. We study systems undergoing adiabatic drift that destroys an attracting periodic orbit through a saddle-node bifurcation placed on a fractal basin boundary. The fate of the system following the pre-bifurcation orbit is indeterminate; it is impossible to predict its final state past the bifurcation. We address this indeterminacy numerically and analytically.