Superstable Cycles in Triadic Percolation

• Superstable cycles are defined as cycles that capture the map’s maximum with a vanishing derivative, serving as a diagnostic for critical transitions.
• The superstable geometry scaling law |δ| ∝ |p – pₙ|^(1/z) quantifies local nonlinearity and identifies universality classes in unimodal maps.
• This diagnostic protocol has practical applications in triadic percolation and higher-order network models, revealing the route to chaos through precise scaling relations.

Superstable Cycles

Superstable cycles are central invariant objects in the dynamics of one-dimensional unimodal maps, with direct relevance for the nonlinear criticality and universality classes of percolation processes governed by higher-order (specifically, triadic) regulatory interactions. In the context of triadic percolation, the geometry and scaling behavior of superstable cycles provides a parameter-free, map-agnostic diagnostic for the local nonlinearity at the map’s maximum, characterizing the universality class (e.g., $z$-logistic) that governs the route to chaos in such systems (Aghaei et al., 1 Feb 2026). The superstable geometry is not merely a technical tool, but a direct probe of the underlying bifurcation scenario and scaling exponents, with immediate applicability to dynamical higher-order network models including synthetic, multilayer, and hypergraph-based systems.

1. Formal Definition of Superstable Cycles in Unimodal Maps

Let $H_p:[0,1]\to[0,1]$ be a family of smooth, unimodal maps parameterized by $p$ (e.g., the effective dynamical map for the regulator-GCC density in triadic percolation (Aghaei et al., 1 Feb 2026)). A period-$2^n$ cycle $\{R_1,\dots,R_{2^n}\}$ is called superstable if it contains the map’s interior maximum $R_m=\arg\max H_p(R)$, i.e., if $H_p^{(k)}(R_m)=R_{k+m}$ for some $k\in\{0,\dots,2^n-1\}$ and $H_p^{(2^n)}(R_m)=R_m$. The superstable parameter value $p_n$ is defined by the requirement that $F_{p_n}(R_m)=R_m$ and $F_{p_n}'(R_m)=0$, where $F_p=H_p^{(2^n)}$ is the $2^n$-iterate return map. At these parameter values the cycle multiplier vanishes and the maximal Lyapunov exponent diverges to $-\infty$.

2. Superstable Geometry and Scaling Law

Close to the superstable point $p_n$, the $2^n$-return map $F_p$ near $R_m$ admits the expansion

$$F_p(R_m+\delta)=R_m+\mu_n(p)-A_n|\delta|^z+O(\delta^{z+1},(p-p_n)^2)$$

where $\mu_n(p)=F_p(R_m)-R_m\to0$ as $p\to p_n$, $A_n>0$ is the leading nonvanishing coefficient, and $z$ is the nonflat order of the maximum, i.e., the smallest even integer such that $F^{(j)}_p(R_m)=0$ for $j=1,\dots,z-1$ but $F^{(z)}_p(R_m)\neq 0$.

The distinguished “next-to-maximum” branch $R_n(p)$, defined by $F_p(R_n(p))=R_m$ with $R_n(p_n)=R_m$, exhibits

$|\delta_n(p)|=|R_n(p)-R_m|\;\propto\;|p-p_n|^{1/z}$

This is the superstable-geometry scaling law. The exponent $\gamma=1/z$ provides a direct geometric probe of the local nonlinearity order $z$ at the maximum of $H_p$.

3. Determination of Universality Class via $z$-Nonflatness

The local order $z$ is controlled by the analytic structure of the “activation kernel” $F(x)=f(x)[1-g(x)]$ at its interior maximum $x_m$: $$F'(x_m)=\cdots=F^{(z-1)}(x_m)=0,\quad F^{(z)}(x_m)\neq 0$$ Generic Poisson or Hill-type kernels yield $z=2$ (logistic universality class, with quadratic maximum), but it is possible to construct models (e.g., using cap-type rules) where $z=d>2$ is realized, leading to $z$-logistic universality. This local nonflatness is significant: the universality class (and scaling exponents for bifurcations, Feigenbaum constants, etc.) is determined entirely by $z$—not by the global functional form.

4. Numerical and Analytical Verification

The scaling law $|\delta_n(p)|\propto|p-p_n|^{1/z}$ is verified numerically for both canonical unimodal families (e.g., quadratic map: $z=2\Rightarrow \gamma=1/2$, quartic map: $z=4\Rightarrow \gamma=1/4$), and for heterogeneous triadic percolation ensembles (Poisson structural degrees and regulators). Fits of $\log|\delta_n|$ vs. $\log|p-p_n|$ yield slopes closely matching the theoretical $1/z$ values, demonstrating the robustness of the diagnostic (Aghaei et al., 1 Feb 2026).

In synthetic triadic percolation, orbit diagrams $R$ vs. $p$ display period-doubling cascades and chaos, with Lyapunov spectra confirming effective one-dimensionality. The one-dimensional Lyapunov exponent $\lambda_1$ dominates; computing it across $p$ exposes sharp minima at superstable parameters $p_n$, serving as excellent detection points for the geometry-based scaling procedure. Superstable scaling fits yield $\gamma\approx0.5$, confirming $z=2$ in typical triadic percolation.

5. Diagnostic Procedure and Practical Methodology

A highly explicit, map-agnostic recipe is available for extracting $z$ from raw orbit data:

1. Compute the Lyapunov profile $\lambda(p)$ over $p$ and identify superstable minima $p_n$.
2. Refine $p_n$ to high accuracy via root-finding: enforce $H_p^{(2^n)}(R_m)=R_m$ locally in $p$.
3. Identify $R_m$ as the unique interior maximum of the orbit diagram.
4. Solve $H_p^{(2^n)}(R)=R_m$ near $p_n$ to trace the “next-to-maximum” branch $R_n(p)$.
5. Collect data pairs $$(\Delta p_{ni}=|p_n-p_i|,\ \delta_{ni}=|R_n(p_i)-R_m|)$$ for small $|\Delta p_{ni}|$.
6. Fit $\log \delta_{ni}$ vs. $\log \Delta p_{ni}$; the slope converges to $\gamma=1/z$ in the accumulation limit.
7. Deduce $z=1/\gamma$ as the nonflat order.

This protocol yields a geometric classifier for universality class, without requiring explicit knowledge or reconstruction of the map $H_p$.

6. Significance in Triadic Percolation and Higher-Order Network Dynamics

In higher-order percolation models (including single-layer (Sun et al., 2022, Sun et al., 10 Oct 2025), multilayer (Sun et al., 10 Oct 2025), and higher-order hypergraphs (Sun et al., 2024)) where the macroscopic order parameter is governed by a one-dimensional unimodal map, the superstable-geometry diagnostic provides a universal tool for determining the criticality class. In all these examples, the observed period-doubling cascade, onset of chaos, and corresponding bifurcation sequence are governed by the $z$-logistic universality, with $z$ determined directly by the nonflatness of the activation kernel at its maximum.

The ability to tune $z$ (and thus the critical scaling) through regulatory statistics or functional forms enables modelers to construct network dynamics with arbitrary dynamical complexity and targeted route-to-chaos properties. A plausible implication is that heterogeneous real-world networks with higher-order regulations may be classified empirically via superstable geometry into universal classes, providing a dynamical analogue of structural or spectral universality in random networks.

7. Extensions and Future Directions

The superstable-geometry paradigm extends naturally to noncanonical maps (multi-peak, nonunimodal), multilayer or hierarchical regulatory structures (Sun et al., 10 Oct 2025, Sun et al., 2024), and spatially embedded networks (Millán et al., 2023, Millán et al., 2024). For maps with $z>2$, nonstandard critical exponents and scaling laws (non-Feigenbaum universality) can emerge, opening new territory for theoretical analysis. The technique is robust even for empirical data, provided basic regularity and unimodality assumptions are met. Explicit extension to maps arising in hierarchical triadic percolation, interdependent hyperedge regulation, or spatially organized blinking phases remains an active direction for research (Sun et al., 2024, Araujo et al., 2023, Millán et al., 2023).

In summary, superstable cycles, through their geometric scaling structure, serve as both a rigorous classification tool for dynamical phase transitions in higher-order percolation systems and a bridge between network regulatory details and universal nonlinear dynamics (Aghaei et al., 1 Feb 2026).