Equivalent representation form in the sense of Lyapunov, of nonlinear forced, damped second-order differential equations

This paper focuses on finding in the sense of Lyapunov, the equivalent forced, damped cubic–quintic Duffing equation of nonlinear forced damped, second-order ordinary differential equations that could contain rational or irrational restoring elastic terms. The accuracy obtained from the equivalent expressions of dynamical systems such as the generalized pendulum equation, the power-form elastic term oscillator, oscillatory systems with irrational elastic restoring forces that modeled the motion of a mass attached to two stretched elastic springs, and the motion of the mechanism of dipteran flight motor, is numerically evaluated by computing their corresponding Lyapunov characteristic exponents, the amplitude–frequency, the amplitude–time, phase portraits, Poincare’s maps, and their Kaplan–Yorke dimension plots.