Solution of the relativistic (an)harmonic oscillator using the harmonic balance method

The harmonic balance method is used to construct approximate frequency–amplitude relations and periodic solutions to the relativistic oscillator. By combining linearization of the governing equation with the harmonic balance method, we construct analytical approximations to the oscillation frequencies and periodic solutions for the oscillator. To solve the nonlinear differential equation, firstly we make a change of variable and secondly the differential equation is rewritten in a form that does not contain the square-root expression. The approximate frequencies obtained are valid for the complete range of oscillation amplitudes, A, while the discrepancy between the second approximate frequency and the exact one never exceeds 0.82% and tends to 0.52% when A tends to infinity. Excellent agreement of the approximate frequencies and periodic solutions with the exact ones are demonstrated and discussed.