Entropy-Based Measurements of Self-Organized Complexity: A View on the Dynamics of the Dissipative Standard Map

Entropic measures play a critical role in understanding self-organized complexity, particularly in dynamic systems. The dissipative Standard Map, originating from a one-dimensional atomic chain model, serves as an ideal framework for studying these phenomena. In this work, we employed the q-Renormalized entropy, a recently developed entropy-based complexity measure, to investigate the transitions between equilibrium and non-equilibrium states within the system. This measure has demonstrated its effectiveness in distinguishing the distinct phases of the system and identifying key transition points. Our study highlights that atomic chains, initiated with randomly selected conditions, exhibit a pathway to chaos characterized by periodic bifurcations. We also observed a correlation between entropy and organizational order: as entropy decreases, organizational order increases, peaking near the edge of chaos. Beyond this point, as the system enters a more chaotic regime, increasing entropy corresponds with a decline in organizational order.