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Periodic solutions of a generalized Van der Pol-Mathieu differential equation. (English) Zbl 1309.34068

The paper investigates the generalized Van der Pol-Mathieu equation \[ \frac{d^2x}{dt^2}-\varepsilon(\alpha_0-\beta_0x^{2n})\frac{dx}{dt}+\omega_0^2(1+\varepsilon h_0 \cos \gamma t)x=0, \eqno(1) \] 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 \(n=1\).

MSC:

34C25 Periodic solutions to ordinary differential equations
34C29 Averaging method for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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