A Parameter-Free Closed-Form Approximation for the Asymmetric Cubic Oscillator Period

A closed-form approximation for the period of the asymmetric cubic oscillator is derived from first principles. The key steps are: a rescaling that eliminates the nonlinearity parameter entirely, leaving a single dimensionless energy ratio μ ∈ (0,1); an exact reduction of the period integral to T = (4/√(2(u₊−u₃)))K(k²), where k² is determined by the three roots of the cubic; and a two-stage normalization producing a smooth bounded function on [0,1]. Four anchor conditions from the asymptotic expansions of K at both endpoints determine the approximation with no free parameters. The result achieves a maximum error of −0.013041% near μ ≈ 0.597, matching the companion pendulum paper exactly — a necessary consequence of the two systems being identical as normalized objects. All constants are composed exclusively of π and ln 4, with no free parameters. Part of a series: AGM · Ellipse perimeter · Nonlinear pendulum · Quartic anharmonic oscillator · Relativistic harmonic oscillator