Periodic solutions of a generalized Van der Pol-Mathieu differential equation
The paper investigates the generalized Van der Pol-Mathieu equation NEWLINE\[NEWLINE \frac{d^2x}{dt^2}-\varepsilon(\alpha_0-\beta_0x^{2n})\frac{dx}{dt}+\omega_0^2(1+\varepsilon h_0 \cos \gamma t)x=0, \eqno(1) NEWLINE\]NEWLINE where \(n\in N\), \(\gamma=2\omega_0+2d_0\varepsilon\), \(\alpha_0>0\), \(\beta_0>0\), \(h_0>0\), \(\omega_0>0\), \(d_0\in R\), and \(\varepsilon>0\) is a small parameter. The authors prove the existence of nontrivial oscillatory periodic solutions of (1). Their proofs are based on the averaging method and the Bogoliubov theorem about the existence and stability of periodic solutions. They also use the method of complexification and the phase space analysis of a derived autonomous equation. In addition, the existence of oscillatory quasiperiodic solutions is discussed. It was shown that equation (1) has similar behaviour for \(n>1\) as for \(n=1\).
scientific article; zbMATH DE number 6369140