Exact time–integral inversion via Čebyšëv quintic approximations for nonlinear oscillators

The focus of this work is the solution of a fundamental problem that arises in non-dissipative nonlinear oscillators and related applications, namely the rare possibility of explicitly inverting the associated time-integral. Here, the inversion issue is treated by near-minimax approximation of the restoring force via fifth-order Cebishev polynomials on a normalised integration interval: this gives rise to a Duffing-type quintic oscillator, whose solutions effectively represent those of the original problem. Indeed, when an odd function describes the restoring force, the elliptic time-integral associated with the quinticate oscillator can be inverted in closed form. This is obtained here, by observing that the integrand involves a quadratic polynomial, built on the quinticate oscillator coefficients, and by studying its discriminant. Based on these findings, we provide a novel solution procedure, implemented within the Mathematica scientific environment, that exploits elliptic integrals of the first kind and whose effectiveness is tested on three well-known conservative nonlinear oscillator models.