Chaotic dynamics has been a fertile area of mathematical research in recent years, and chaos has now been observed in many scientific fields. Much of the progress to date in the study of chaotic dynamics has focused on low-dimensional systems, and many phenomena are well understood in certain low-dimensional cases. Even when systems with higher- dimensional state spaces are considered, often the dynamics takes place on a low-dimensional attractor (by "low", we mean up to dimension 2 for maps and 3 for flows). This proposal seeks to explore how some of the phenomena associated with chaotic dynamical systems extend to cases that are fundamentally higher-dimensional -- that is, when the system is expanding in more than one dimension, or in other words has more than one positive Lyapunov exponent.
Higher-dimensional chaos can occur in models of many engineering devices and in meteorological models and climate models. This research will tell us, for example, if mathematical models can suddenly get trapped in artificial windows of regular behavior. These windows would be artificial in that any realistic level of noise would disrupt the regular behavior. Our research will also shed light on the reliability of computer simulations for higher dimensional models. This study is critical to realistic modeling of complex systems.
