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 \(n=1\).


34C25 Periodic solutions to ordinary differential equations
34C29 Averaging method for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI


[1] Hakl, R.; Torres, J. P.; Zamora, M., Periodic solutions of singular second order differential equations: upper and lower functions, Nonlinear Anal., 74, 18, 7078-7093 (2011) · Zbl 1232.34068
[2] Kadeřábek, Z., The autonomous system derived from Van der Pol-Mathieu equation, Aplimat - J. Appl. Math., Slovak Univ. Tech., 5, 2, 85-96 (2012)
[3] Kiguradze, I.; Lomtatidze, A., Periodic solutions of nonautonomous ordinary differential equations, Monatsh. Math., 159, 3, 235-252 (2010) · Zbl 1194.34076
[4] Momeni, M.; Kourakis, I.; Moslehi-Frad, M.; Shukla, P. K., A Van der Pol-Mathieu equation for the dynamics of dust grain charge in dusty plasmas, J. Phys. A: Math. Theor., 40, F473-F481 (2007) · Zbl 1119.76073
[5] Mawhin, J., Resonance and nonlinearity: a survey, Ukraïn. Mat. Zh., 59, 2, 190-205 (2007), Translation in Ukrainian Math. J. 59 (2) (2007) 197-214 · Zbl 1150.34424
[6] Perko, L., Differential Equations and Dynamical Systems (1996), Springer · Zbl 0854.34001
[7] Rachůnková, I.; Tvrdý, M.; Vrkoč, I., Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems, J. Differ. Equ., 176, 2, 445-469 (2001) · Zbl 1004.34008
[8] Szabelski, K.; Warmiński, J., Parametric self-excited non-linear system vibrations analysis with inertial excitation, Int. J. Non Linear Mech., 30, 2, 179-189 (1995) · Zbl 0821.70017
[9] Tondl, A., On The Interaction Between Self-Excited and Parametric Vibrations, Monographs and Memoranda (1978), National Research Institute for Machine Design: National Research Institute for Machine Design Běchovice, Prague
[10] Veerman, F.; Verhulst, F., Quasiperiodic phenomena in the Van der Pol-Mathieu equation, J. Sound Vib., 326, 1-2, 314-320 (2009)
[11] Verhulst, F., Nonlinear Differential Equations and Dynamical Systems (1996), Springer · Zbl 0854.34002
[12] Villari, G.; Zanolin, F., A geometric approach to periodically forced dynamical systems in presence of a separatrix, J. Differ. Equ., 208, 2, 292-311 (2005) · Zbl 1080.34023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.