Complex dynamics in two-dimensional coupling of quadratic maps

TL;DR

This work extends the classical Mandelbrot framework to two-dimensional Complex Quadratic Networks by introducing and analyzing the equi-M set M_2, a postcritically bounded locus for coupled quadratic maps. The authors develop a rigorous 2D-CQN model, derive the escape radius and stable fixed-point conditions, and characterize the main equi-cardioid via Schur–Cohn stability criteria, complemented by explicit analyses of feed-forward and equal-row-sum subfamilies. Key findings show that, unlike the univariate case, higher-period attractors and coexisting dynamics decouple the critical orbit from the full combinatorial dynamics, causing the equi-M set to be only a partial atlas of behavior; synchronization phenomena are also clarified, with coupling often forcing M-synchronization under broad conditions. Overall, the results reveal richer, more intricate dynamics in 2D CQNs and establish a bottom-up foundation for understanding how coupling shapes the dynamics of larger, high-dimensional networks.

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.

Figures (7)

• Figure 1: Main equi-cardioids for the feed-forward case with the coupling matrix $A=101f$ for increasing values of $f$, as labeled.
• Figure 2: Main equi-cardioids for the feed-forward case with the coupling matrix $A=\omega02\omega-\tau\tau-\omega$ (with equal row sum $\omega$) for increasing values of $\omega$, as labeled.
• Figure 3: Main equi-cardioids for coupled quadratic maps with the coupling matrix $A=\omega/2\omega/21\omega-1$ for increasing positive values of $\omega$, as labeled.
• Figure 4: Post-critically periodic regions ${\cal C}_2^k$, for periods up to k=20, are shown as subsets of the equi-M set for two example networks. A.$A=1011$ (Example 1 for $a=d=f=1$,$b=0$); B.$A=0.40.41-0.2$ (Example 2 for $\omega=0.8$). The colors correspond to the critical orbit being attracted respectively to a fixed point (purple region), period two orbit (cyan region), period three orbit (green regions), period four orbit (orange regions), and so on, up to period $k=20$.
• Figure 5: Regions with higher order periods for two example systems:A. The feed-forward network $z_1 = z_1^2+c$; $z_2=(z_1+z_2)^2+c$ and B. The system in Example 2(b), for $\omega = 0.4$. In each panel, the shaded regions correspond to the system having an attracting fixed point (blue), an attracting period 2 orbit (yellow) and a period 3 orbit (green). In the right panel, the brown region represents coexistence of an attracting fixed point and an attracting period two orbit. In both cases, the boundary of the equi-M set is shown as a black contour, for comparison.

Theorems & Definitions (19)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• Lemma 2.5
• Proposition 2.6
• Definition 3.1
• Lemma 3.2
• proof