A Nonlinear Three-Mode Model of Solar Activity Based on Coupled Oscillators and Nonlinear Energy Transfer

Solar activity exhibits a dominant decadal timescale embedded in nonlinear multiscale dynamics. We present a minimal autonomous three-mode hybrid model to investigate whether such a timescale can arise from internal nonlinear energy exchange rather than external periodic forcing or direct inheritance of a prescribed modal period. The model couples a slow internal reservoir, a tachocline-associated transfer mode, and a nonlinear magnetic subsystem described by Duffing and Van der Pol-Duffing dynamics. Magnetic-energy release is implemented through a threshold-triggered reset, while a D-index-based structural energy-exchange index characterizes relative magnetic-energy changes. The system develops a persistent bounded hybrid regime with a dominant period of approximately 10 yr. Initial-condition ensembles and variations in the intrinsic modal periods demonstrate that this timescale does not simply track the prescribed 11 yr tachocline period. Parameter studies identify the release depth and magnetic recovery rate as stronger controls, supporting an accumulation-release mechanism of timescale selection. Release events exhibit organized recurrence and significant tachocline-phase preference, with localized event skipping separating spectral and release timescales. Saltation-based Lyapunov analysis reveals no robust positive largest exponent for the baseline regime over the investigated finite-time horizons. These results demonstrate that nonlinear energy exchange combined with threshold-mediated accumulation and release can autonomously generate a robust decadal collective timescale in a reduced nonlinear system.