Synthesis of a universal second-order limit cycle oscillator for prescribed phase-plane trajectories – A data-driven approach

This paper presents a novel data-driven framework for synthesizing structurally robust, second-order limit cycle oscillators capable of reproducing user-specified phase-plane trajectories. The proposed method builds upon Lyapunov-based design principles but crucially eliminates singularities and numerical instabilities that typically arise in gradient-based constructions. A singularity-free formulation is developed using conservative-dissipative decomposition and smoothed switching functions, ensuring global stability and computational efficiency. Filippov analysis is employed to characterize and mitigate sticking and sliding behaviors on discontinuity boundaries. Additionally, two data-driven approaches are proposed for constructing implicit limit cycle equations from discrete trajectory data—one using angular scaling and another via regression-based curve fitting—enabling flexibility in handling non-algebraic, non-smooth, or noisy data. The synthesized oscillators exhibit strong robustness to amplitude and phase perturbations, quantified using Phase Response Functions (PRFs). Comparative studies with existing models demonstrate the proposed oscillator's universality, precision, and resilience under various perturbation scenarios. Furthermore, the framework supports smooth and non-smooth geometries and accommodates both symbolic and data-defined limit cycles, making it suitable for a wide range of applications in robotics, bio-inspired control, and neural modeling. Numerical examples validate the theoretical claims and highlight the practical applicability of the proposed synthesis methodology.