On parameter size for existence of periodic solutions by the method of averaging (Q2784702)

It is shown that the Krylov-Bogoliubov-Mitropolskij averaging method applied to the ordinary differential equation NEWLINE\[NEWLINE\ddot x+ x= \varepsilon[f_1(x)+ f_2(x, \dot x^2)\dot x]NEWLINE\]NEWLINE can predict the existence of spurious limit cycles under certain conditions. One such limit cycle appears in the case of the modified van der Pol equation NEWLINE\[NEWLINE\ddot x+ x= \varepsilon[- \beta x^3+ (1- x^2)\dot x],\quad \beta> 0,NEWLINE\]NEWLINE in addition to the expected one. A possible resolution of this inconsistency in the perturbation method is suggested.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00016].