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

Abstract

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 Čebyšëv 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.