  Emergence, the phenomena where a system's micro-scale dynamics facilitate the
development of non-trivial, informative higher scales, has become a
foundational concept in modern sciences, tying together fields as diverse as
physics, biology, economics, and ecology. Despite it's apparent universality
and the considerable interest, historically researchers have struggled to
provide a rigorous, formal definition of emergence that is applicable across
fields. Recent theoretical work using information theory and network science to
formalize emergence in state-transition networks (causal emergence) has
provided a promising way forward, however the relationship between this new
framework and other well-studied system dynamics is unknown. In this study, we
apply causal emergence analysis to two well-described dynamical systems: the 88
unique elementary cellular automata and the continuous Rossler system in
periodic, critical, and chaotic regimes. We find that emergence, as well as its
component elements (determinism, degeneracy, and effectiveness) vary
dramatically in different dynamical regimes in sometimes unexpected ways. We
conclude that the causal emergence framework provides a rich new area of
research to explore both to theoreticians and natural scientists in many
fields.
