An Approximate Solution for Period-1 Motions in a Periodically Forced Van Der Pol Oscillator

In this paper the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balance (HB) method. Such an approximate solution of periodic motion is given by the Fourier series expression, and the convergence of such an expression is guaranteed by the Fourier series theory of periodic functions. The approximate solution is different from traditional, approximate solution because the number of total harmonic terms (⁠$N$⁠) is determined by the precision of harmonic amplitude quantity level, set by the investigator (e.g., $AN≤ɛ$ and $ɛ=10-8$⁠). The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis of the coefficient dynamical systems of the Fourier series expressions of periodic solutions, and numerical illustrations of period-1 motions are compared to verify the analytical solutions of periodic motions. The trajectories and analytical harmonic amplitude spectrum for stable and unstable periodic motions are presented. The harmonic amplitude spectrum shows the harmonic term effects on periodic motions, and one can directly know which harmonic terms contribute on periodic motions and the convergence of the Fourier series expression is clearly illustrated.