Research Alpha · Python 3.11+
hidden‑attractors‑fo
Numerical workflows for locating and verifying hidden attractor candidates in fractional-order and integer-order Chua/Lur'e systems.
$ pip install hidden-attractors-fo
From fractional-order integration to basin classification and publication-ready figures.
Register your own systems with register_system().
| System | Type | Dim | Description | Docs |
|---|---|---|---|---|
chua-nonsmooth | Integer & Fractional | 3 | Non-smooth Chua circuit with a linear-by-pieces characteristic | Docs → |
chua-arctan | Integer & Fractional | 3 | Chua variant with arctangent nonlinearity | Docs → |
lorenz | Integer | 3 | Classic Lorenz '63 — reference self-excited system | Docs → |
rossler | Integer | 3 | Rössler attractor — simple spiral chaos | Docs → |
custom | Any | Any | User-defined via register_system() — provide RHS, Jacobian, equilibria | Docs → |
To guarantee numerical validity and prevent computational artifacts, the library implements an automated 8-stage verification contract.
Regional vector field residuals, equilibria locations, analytic Regional Jacobians, and eigenvalues mapped onto the complex Matignon plane at q = 0.9998. Mapped across Python, MATLAB, and Wolfram.
python tools/validation/validate_chua_fractional_nonsmooth_algebra.py Analytical Lur'e decomposition vector splits, Machado describing-function limit cycles, transfer function sign conventions (W_code = -W_report), and harmonic continuation branch matching.
Verified via Stage 01 PackageExtended Fractional Runge-Kutta (EFORK-3) convergence. Verifies analytical Mittag-Leffler convergence to 6 × 10-9 tolerance, Runge-Kutta q = 1 stage orders, and ABM cross-comparisons.
python tools/validation/validate_efork_integrator.py Locating viable, non-collapsing initial attractor seeds outside the immediate neighborhood of regional stable/unstable manifolds.
Computing ROSENSTEIN Lyapunov exponent spectra, trajectory clouds, Poincaré sections, and sample/permutation entropy metrics.
Thorough spherical coordinate grids swept systematically from equilibria neighborhoods to confirm strict mathematical hiddenness.
Perturbing system parameters (alpha, beta) to guarantee structural stability and ensure attractors don't collapse into a stable limit cycle.
Formal alignment, cross-comparisons, and coordinate centroid matchings against original publications (e.g., Danca 2017).
The installable command line interface contains a validation contract checker that verifies your local files comply with the machine-readable schema definition in configs/validation_contract.json.
hidden-attractors-check-validation --dir validation/ --strict hidden-attractors-check-validation --contract configs/validation_chua_integer_q1.json --validation-root validation/reference_cases/chua_integer_q1 hidden-attractors-check-validation --contract configs/validation_efork3_ghoreishi_ghaffari.json --validation-root validation/reference_cases/efork3_ghoreishi_ghaffari Run any of the included user-facing examples from the root of the project to test systems, solvers, and seed generators.
Evaluates all three equilibria points and prints non-smooth vector field RHS residuals to machine precision.
python examples/quickstart_equilibria.py Runs a fast trajectory integration of the fractional Chua system using compiled C-accelerated EFORK-3 solver.
python examples/minimal_chua_protocol.py Queries the internal system candidate records and details all verified bounding boxes and centroids.
python examples/list_final_candidates.py Registers a custom fractional-order chaotic attractor, defines its vector equations, and initiates solver loops.
python examples/custom_system_definition.py Configures and coordinates the parameters for sweeping system parameter maps and plotting overlap maps.
python examples/create_robustness_overlay_config.py After editable installation, all workflows are available as terminal commands.
If you use hidden-attractors-fo in academic work, please cite the repository.