Frequency-amplitude characteristics of periodic motions in a periodically forced van der Pol oscillator

In this paper, nonlinear frequency-amplitude characteristics of periodic motions in a periodically forced van der Pol oscillator are studied systematically. The periodic motions of the van der Pol oscillator are determined by the semi-analytical method, and the corresponding stability and bifurcation analysis is completed through the eigenvalue analysis. From the finite Fourier series analysis, the nonlinear frequency-amplitude characteristics of periodic motions are analyzed. From the frequency-amplitude analysis, the limit cycle of the van der Pol oscillator can be obtained analytically as excitation amplitude approach to zero, rather than numerically. For the van der Pol oscillator, most of periodic motions in the van der pol oscillator are symmetric. However, an asymmetric period-1 motion in the van der Pol oscillator is discovered. Thus a bifurcation tree of period-1 motion to chaos can be found.