Complex dynamics in two-dimensional coupling of quadratic maps

This paper examines the structure and limitations of equi-M sets in two-dimensional Complex Quadratic Networks (CQNs). In particular, we aim to describe the relationship between the equi-M set and the parameter domains where the critical orbit converges to periodic attractors (pseudo-bulbs). The two-node case serves as a foundational testbed: its analytical tractability enables the identification of critical phenomena and their dependence on coupling, while offering insight into more general principles. The two-node case is also simple enough to allow for explicit coupling conditions that govern phase transitions between synchronized and desynchronized behavior. Using a combination of analytical and numerical methods, the study reveals that while the period-1 pseudo-bulb closely tracks the boundary of the equi-M set near its main cusp, this correspondence breaks down for higher periods and in regions supporting coexisting attractors. These discrepancies highlight key differences between single-map and coupled dynamics, where equi-M sets no longer provide a full encoding of system combinatorics. These findings clarify the topological and dynamical behavior of low-dimensional CQNs and point toward a sharp increase in complexity as the number of nodes grows, laying the groundwork for future studies of high-dimensional dynamics.