Complex dynamics in two-dimensional coupling of quadratic maps

TL;DR

This paper explores the complex behavior of two-dimensional coupled quadratic maps, analyzing how their parameter spaces and attractors relate, revealing limitations of existing sets in describing coupled dynamics and highlighting increased complexity with more nodes.

Contribution

It provides an analytical and numerical study of two-node quadratic networks, clarifying the relationship between equi-M sets and attractors, and identifying how coupling affects system dynamics.

Findings

Main cusp of equi-M set aligns with period-1 pseudo-bulb boundary

Discrepancies occur for higher periods and coexisting attractors

Coupled dynamics exhibit greater complexity than single-map systems

Abstract

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.