Amplitude decay of damped non-linear oscillators studied with Jacobian elliptic functions

Approximate bounded solutions of the equation x " +bx ' +c 1 x+c 3 x 3 =0 with b >0, c 1 ≷0 and c 3 ≷0 are developed in terms of the Jacobian elliptic functions cn, cd and dn. The solutions are found by following the method developed by Christopher in 1973 for the case with c 1 >0 and c 3 >0. Formulas for the amplitude decay are given in two different approximations. The solutions are compared with Runge-Kutta numerical integration results and shown to be accurate for a wide range of b, c 1 , c 3 , and initial conditions.