Repository LaTeX Notes

A comprehensive guide to the mathematical LaTeX publications, user manuals, and validation reports available in the repository's docs folder.

The repository’s docs/ folder contains a rich library of formal LaTeX publications, didactic user manuals, and markdown technical audits. These documents establish the mathematical rigor of the fractional chaotic attractor localization project, bridging the gap between theoretical physics and active Python/C implementations.

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Get the compiled, print-ready PDF versions of these manuscripts directly from the portal server or view their LaTeX sources on GitHub:

1. The LaTeX Scientific Publications

Below is the directory of compiled LaTeX documents that serve as the scientific core of the project. They can be found directly under version_2/docs/ in the repository.

📄 reporte_unificado_chua_fraccionario.tex

• Title: Reporte Unificado de Sistemas Fraccionarios de Chua
• Purpose: The main theoretical reference for the fractional Chua research.
• Key Topics:
  • Commensurate Caputo fractional Chua equations.
  • Stability mapping on the complex Matignon plane.
  • Lur’e split decomposition and Describing Function harmonic analysis.
  • Analytical search protocols for hidden attractors.
  • Visual index of reference attractors and target-hit basin grids.

📄 reporte_matematico_hidden_attractors.tex

• Title: Complemento matemático y operativo del reporte de la librería hidden-attractors-fo, version_2
• Purpose: A comprehensive technical audit and mathematical expansion.
• Key Topics:
  • Complete derivation of coordinate relations for equilibria $E_0, E_+, E_-$.
  • Closing slope $s_c = -\frac{\beta}{\beta + \gamma}$ formulation.
  • Explicit regional Jacobian matrices $J(a)$ for slopes $a \in \{m_0, m_1\}$.
  • Detailed rational transfer function $\widehat W_q(\lambda)$ and complex minor determinant $N(\lambda)$.
  • Mathematical definitions for trajectory diagnostics (variance, spectral entropy, cloud distance).
  • Comprehensive table of basin labels (CLASS_LABELS) and sphere-control protocols.

📄 manual_usuario_hidden_attractors_fo_didactico.tex

• Title: Manual didáctico de usuario de la librería hidden-attractors-fo
• Purpose: A step-by-step user guide explaining the mental model, environment setup, and execution commands.
• Key Topics:
  • Mental model mapping: Model $\rightarrow$ Equilibria $\rightarrow$ Candidates $\rightarrow$ Trajectories $\rightarrow$ Robustness.
  • Complete installation guidelines using virtual environments and dependency management.
  • Step-by-step execution walkthroughs for robustness_overlay, sphere_controls, and refined_basin workflows.
  • Clear operational advice on how to interpret numerical results to defend or reject candidate hiddenness.

📄 protocolo_pruebas_chua_fraccionario_no_suave.tex

• Title: Protocolo de Pruebas de Sistemas de Chua Fraccionarios No Suaves
• Purpose: Systematic testing protocol mapping the non-smooth piecewise characteristic and numerical constraints.
• Key Topics:
  • Setup matrices for the Lur’e split configuration.
  • Tolerances and convergence criteria for Caputo integrations.
  • Robustness checklists under variable step sizes.

📄 protocolo_chua_no_suave_flujo_secuencial.tex

• Title: Protocolo Chua No Suave Flujo Secuencial
• Purpose: Defines the sequential processing flow for numerical trials.
• Key Topics:
  • Trajectory parsing pipeline stages.
  • Post-processing criteria for Poincaré cross sections.
  • Spectrum-based chaotic signature validation.

2. The Markdown Technical Audits

In addition to LaTeX sources, the docs/ folder contains active markdown documentation tracking specific verification checks and mathematical conversions.

3. How to Compile the LaTeX Documentation

The documents use standard LaTeX packages (amsmath, amssymb, booktabs, tcolorbox, listings, fancyhdr, hyperref).

To compile any of these documents into high-quality PDFs, navigate to the docs/ directory of the repository in your terminal and run:

Command Option A: Latexmk (Automatic, Recommended)

latexmk -pdf -interaction=nonstopmode -halt-on-error <document_name>.tex

Command Option B: PdfLaTeX (Standard Compiler)

pdflatex -interaction=nonstopmode -halt-on-error <document_name>.tex

Note: Running latexmk will automatically resolve bibliography references and execute multiple passes to correctly compile the table of contents and internal page links.

4. Key Mathematical Formulations Formalized

To maintain perfect alignment between the LaTeX manuscripts and the active Python code, the following formulas are explicitly audited:

The Closing Slope & Equilibria Coordinated Relations

Equilibria points are solved from the vector field $F(X) = 0$. By substituting coordinate equations, all equilibria coordinates are roots of $f(x) = s_c x$, where the closing slope $s_c$ is:

s_c = -beta / (beta + gamma)

This yields the exact coordinate coordinates of the symmetric external equilibria:

x_plus  = (m0 - m1) / (s_c - m1)
x_minus = -(m0 - m1) / (s_c - m1)

The Matignon Sector Criterion

A commensurate fractional system of order $q$ is locally asymptotically stable under regional Jacobian $J(a)$ if all its complex eigenvalues $\lambda_i$ satisfy:

|arg(eigenvalue_i)| > q * pi / 2

The EFORK-3 Stage Order Correction

The third stage of the 3-stage Fractional Runge-Kutta method is implemented and validated in the solver as:

K3 = h^q * F_n(t_n + c3 * h,  y_n + a31 * K1 + a32 * K2)

This matches the ordering validated against CINVESTAV reference scripts, ensuring high-order convergence.