An intelligent hybrid temporal-scale framework for discovering governing equations of limit-cycle vibrations

Discovering sparse and interpretable governing equations for weakly nonlinear limit-cycle vibration remains challenging because the nonlinear terms governing amplitude saturation and phase modulation are often much weaker than the dominant linear oscillatory components. To address this issue, this study proposes a Hybrid Multi-Temporal Dynamic Equation Learning framework, termed HyMTD-EQL, for data-driven discovery of governing equations in weakly nonlinear limit-cycle dynamical systems. The proposed method reorganizes measured trajectories into phase-space-structured hybrid temporal data, thereby encoding both fast-time-scale local dynamics and slow-time-scale global evolution toward stable limit-cycle attractors. An Equation Learner network is further embedded into a recurrent numerical-integration framework to learn sparse explicit equations under trajectory-level dynamical consistency, while a nondimensionalized learning strategy is introduced to improve robustness for variables with different physical scales. The framework is first validated on five representative numerical limit-cycle systems, including the Van der Pol oscillator, Duffing–Van der Pol oscillator, Hopf normal form, Brusselator model, and FitzHugh–Nagumo model. It is then applied to wind-tunnel-measured vortex-induced vibration and limit-cycle flutter of bluff-body sectional models. The discovered equations accurately reproduce time-domain responses and phase-space structures, and reveal physically interpretable stiffness, damping, coupling, and nonlinear saturation mechanisms. Ablation and robustness analyses further confirm the necessity of phase-space clustering and the stability of the identified equation structures under noise, sparse sampling, and random initialization. These results demonstrate that HyMTD-EQL provides an effective and interpretable framework for discovering governing equations of weakly nonlinear limit-cycle vibration systems.