Modeling Odd Nonlinear Oscillators with Fifth–Order Truncated Chebyshev Series

The aim of this work is to model the nonlinear dynamics of conservative oscillators, with restoring force originating from even-order potentials. In particular, we extend our previous findings on inverting the time-integral equation that arises in the solution of such dynamical systems, a task that is almost always intractable in exact form. This is faced and solved by approximating the restoring force with its Chebyshev series truncated to order five; such a quintication approach yields a quinticate oscillator, whose associated time-integral can be inverted in closed form. Our solution procedure is based on the quinticate oscillator coefficients, upon which a second-order polynomial is constructed, which appears in the time-integrand of the quinticate problem, and whose roots determine the expression of the closed-form solution, as well as that of its period. The presented algorithm is implemented in the Mathematica software and validated on some conservative nonlinear oscillators taken from the relevant literature.