Validation Evidence

The reporting contract required for publishing new hidden attractors.

When reporting a hidden-attractor candidate using this library, whether integer-order or fractional-order, the result must be accompanied by a validation package. A finite numerical package supports a candidate classification; it is not a global mathematical proof of hiddenness.

Because numerical integration of fractional systems is susceptible to artifacts (due to truncation errors and memory window length), a simple plot is insufficient evidence for publication.

The Validation Contract

A complete validation report must include:

The System Definition: Exact equations, parameter values, and fractional order $q$ .
Equilibria Proof: Analytical or high-precision numerical proof of the location of all equilibria.
Local Stability Analysis: Eigenvalue analysis for $q=1$ , or a Matignon-based analysis for fractional order.
Integration Setup: The solver, time step dt, total simulation time t_sim, and memory length only where a fractional computation uses one.
Initial Condition: The exact $(x_0, y_0, z_0)$ seed that produced the trajectory.
Trajectory Metrics: The bounding box and centroid.
Robustness Distance: The minimum Euclidean distance from the trajectory to the nearest equilibrium point over the final 50% of the simulation.

Integer Chua Baseline

The first promoted reference case is the non-smooth Chua system, whose characteristic is linear by pieces, at $q=1$ . Its theoretical derivation is recorded in the report dated 17 May 2026 and its generated evidence includes:

Check	Recorded result
Parameters	$\alpha=8.4562$ , $\beta=12.0732$ , $\gamma=0.0052$ , $m_0=-0.1768$ , $m_1=-1.1468$
Nyquist/DF seed	$\omega_0=2.03918694$ , $k=0.20986735$ , $a_0=5.85614509$
Closure residual	$9.93\times10^{-16}$
Lyapunov exponents after corrected EFORK-3 rerun	$(0.20825, 0.01373, -1.36693)$
Hiddenness probes	504 trajectories from equilibrium neighborhoods; `TARGET=0`

The library artifacts, the attached theoretical report, the locally executed MATLAB reproduction, the supplied Wolfram Language symbolic derivation, and Guan and Xie’s 2025 review are registered as separate evidence sources. The Wolfram file intentionally performs no numerical parameter evaluation, so it supports the algebra rather than replacing numerical reproduction.

Guan and Xie’s Example 6 reports the displayed values $\omega_0=2.0392$ , $k=0.2098$ , and $a_0=5.8576$ for this same Chua parameter set. Against those rounded paper values, the Python results differ by 0.000640%, 0.032104%, and 0.024838%, respectively. The full table is reported in Chua Integer q=1 Reference.

See Chua Integer q=1 Reference.

Fractional Non-Smooth Chua Algebra

For Danca’s non-smooth fractional Chua case at $q=0.9998$ , the algebra and Lur’e/DF stages have now been checked across Python and MATLAB, with the symbolic Wolfram source executing cleanly after its protected variable Tr was renamed.

Check	Recorded result
Equilibria	$E_0=(0,0,0)$ and $E_\pm=\pm(6.588307886539,\ 0.002836402256,\ -6.585471484283)$
Matignon classification	Stable `E0`; unstable external equilibria
Branch 1	$\omega_0=2.040286051079$ , $k=0.210022792962$ , $a_0=5.851767785486$
Branch 2	$\omega_0=3.244926730975$ , $k=0.956945404928$ , $a_0=1.053016610257$
Transfer convention	$W_{\rm code}=-W_{\rm report}$ , so $1+kW_{\rm code}=0$ equals $1-kW_{\rm report}=0$

Danca (2017) is the direct reference for the exact case. Petras (2008) supports the fractional PWL Chua circuit family but not these numeric parameters. Sene (2021) is a related Caputo-Liouville Chua analysis with a different third equation and cannot serve as direct numerical confirmation.

See Chua Non-Smooth.

EFORK-3 Method Validation

The numerical method is validated separately from any Chua candidate. A reference implementation reproduces the terminal errors in Tables 3, 4, 9, and 10 of Ghoreishi, Ghaffari, and Saad (2023) for their two manufactured Caputo examples. The maximum absolute difference from the displayed table values is 5.5489224e-9, under the 6e-9 display-rounding threshold.

The archived validation scripts were supplied by Dr. Luis Gerardo de la Fraga of CINVESTAV Unidad Zacatenco and register the third stage as K3 = a31*K1 + a32*K2. See EFORK-3 Published Validation.

After identifying the previous reversed third stage, all integration-dependent integer Chua outputs were deleted and regenerated. The theoretical report remains a source for algebra and seed construction; current continuation, Lyapunov, basin, spectrum, and hiddenness values are taken from the corrected run.

Automated Verification

The hidden-attractors-check-validation CLI tool can automatically verify if a results directory contains the necessary data to satisfy this contract.

hidden-attractors-check-validation --dir results/chua_hidden_01/ --strict

If it passes, the tool will generate a validation_report.tex file containing a formatted LaTeX table suitable for inclusion in an academic manuscript.

Independent Reintegration

If a hidden candidate is found using EFORKSolver, re-integrating the exact same initial condition with an independent solver such as ABMSolver is a useful numerical cross-check, especially for fractional systems.

Persistence under that check strengthens the evidence package; it does not by itself prove hiddenness. The Strict Refinement workflow is intended to automate this follow-up.