Analytical solutions of nonlinear system of fractional-order Van der Pol equations

The double-well, in-phase and out-of-phase periodic solutions of the system of fractional-order Van der Pol equations and the exact solution of the nonlinear fractional-order Van der Pol equations with independent initial profiles are investigated in this paper. The influence of two main physical parameters such as angular frequency and the amplitude are included for the study. In addition, the difference between autonomous (i.e., f=1,g=M=0) and the non-autonomous (i.e., f=1,g0,M=0) nonlinear oscillators as well as the double-well VDPDO (i.e., f<0,g>0) cases is analysed. It is found that the variations in in-phase and out-of-phase periodic solutions and convergence rate strongly depend on the initial conditions with fractional orders. The effect of the physical parameters on phase portrait and the time history curves for various values of fractional orders are plotted and discussed.