Accurate Closed‐Form First‐Harmonic Approximation for the Frequency‐Amplitude Relationship of the Relativistic Duffing Oscillator via Hypergeometric Functions

Source abstract

ABSTRACT The relativistic Duffing oscillator presents a significant mathematical challenge due to its irrational Lorentz factor singularity, which causes traditional series‐expansion approximation techniques to severely diverge at velocities approaching the speed of light. To overcome this limitation, this article introduces a novel, non‐perturbative first‐harmonic analytical framework that accurately captures the system's dynamics across the entire physical velocity domain. The adopted methodology systematically clears the singularity by absorbing relativistic effects entirely into the restoring force term. Based on an assumed single‐harmonic cosine trial solution, a first‐order harmonic balance method via a Galerkin projection is utilized, and the resulting integrals are evaluated exactly in closed form using the Beta function and Gaussian hypergeometric functions. To overcome the implicit nature of the resulting transcendental equation, a novel quadratic parametric inversion framework is applied, explicitly mapping the physical amplitude to the nonlinear frequency without relying on iterative numerical root‐finding. Validation against high‐precision numerical integrations across a range of strong cubic nonlinearity parameters demonstrates exceptional accuracy. The main results show that the proposed approximation maintains relative errors below 1% all tested cubic nonlinearity parameters in weakly‐to‐moderately relativistic regimes (velocities up to half the speed of light). At the ultra‐relativistic limit, the relative error remains tightly bounded, peaking at approximately 5.7% for the strongest evaluated nonlinearity, which is a natural residual consequence of the single‐harmonic truncation. Comparative benchmarking confirms that while traditional first‐order approximations specifically the Taylor‐expanded Harmonic Balance Method and He's Energy Balance Method suffer severe divergence near the speed of light, the proposed hypergeometric framework remains physically robust.