Universal spectral structure in pendulum-like systems

Pendulum-like dynamics is a universal motif across many areas of physics, underlying systems ranging from classical nonlinear oscillators to superconducting qubits and cold-atom tunneling platforms. Here we present an exact frequency-domain formulation of the pendulum equation that applies uniformly across oscillatory, separatrix, and rotational regimes. The resulting spectral representation reveals a previously hidden unification: all regimes share the same analytic spectral structure and characteristic frequency scale. We discover that all regimes arise from a single universal spectral kernel, with parity selection distinguishing the periodic motions and the separatrix representing their discrete-to-continuum limit. Regime changes thus correspond to symmetry-driven reorganizations in frequency space rather than changes in the underlying spectral structure, with the stopping trajectory representing the continuum limit reached without system-size scaling. The spectral structure can be derived via a spectral discretization approach starting from the separatrix solution, without relying on the classical Jacobi elliptic formulation. Beyond providing closed-form solutions, the framework reveals a transparent spectral structure underlying a broad class of classical and quantum pendulum-like systems.