Period-m motions and bifurcation trees in a periodically forced, van der Pol-Duffing oscillator

Analytical period-m motions and bifurcation trees in a periodically forced, van der Pol-Duffing oscillator are obtained through the Fourier series, and the corresponding stability and bifurcation of such period-m motions are discussed. To verify the approximate, analytical solutions of period-m motions on the bifurcation trees, numerical simulations are carried out, and the numerical results are compared with analytical solutions. The harmonic amplitude distributions are presented to show the significance of harmonic terms in the finite Fourier series of the analytical periodic solutions. The bifurcation trees of period-m motion to chaos via period-doubling are individually embedded in the quasi-periodic and chaotic motions without period-doubling.