Arnold Tongue Structure in a Phase-Coherent Chaotic Oscillator: A Validated Synchronization Pipeline with Diagnostic Transfer to a Reduced-Order Solar Dynamo Model

The physical origin of the approximately 11-year solar magnetic cycle and its irregular amplitude modulation remain open problems in solar physics. While mean-field magnetohydrodynamic (MHD) dynamo theory provides the accepted internal mechanism for large-scale magnetic field generation, the possibility that weak external perturbations may influence the cycle phase remains controversial and theoretically underconstrained. In this study, we pursue two complementary objectives. First, we derive a reduced-order nonlinear dynamo model via Galerkin projection of the mean-field MHD equations, yielding a three-variable autonomous system describing coupled poloidal field, toroidal field, and torsional oscillation amplitudes (Eq. 3a-3c). Second-constituting the primary numerical contribution-we develop and validate a complete Arnold tongue synchronization pipeline on the Rössler oscillator (a = 0.15, b = 0.2, c = 4.0), a well-characterized phase-coherent chaotic system that preserves the essential dynamical properties required: self-sustained oscillations, bounded phase diffusion, and a well-defined intrinsic frequency. The pipeline employs Hilbert-transform phase extraction, dual synchronization metrics (Phase Locking Value and rotation number), and systematic two-dimensional scanning over forcing frequency-amplitude space. The measured natural frequency is ωnat = 1.059188 rad/unit (Tnat ≈ 5.93 units). The primary 1:1 Arnold tongue is centered at ω/ωnat = 1.000 after normalization, with minimum detectable locking amplitude εcrit ≤ 10⁻⁴ (scan-resolution limited) and tongue width scaling Δω/ωnat ≈ 0.232 ε in the weak-forcing regime, consistent with classical Arnold resonance theory. The dual-metric consistency and correct apex alignment confirm that the pipeline is free of numerical artefacts. This validated methodology provides a rigorous foundation for future application to system (3a-3c); a preliminary diagnostic scan of that system confirms the pipeline's gate functionality and precisely characterizes the parameter-space conditions required before Arnold tongue analysis can proceed.