A note on the periodic orbits of a kind of Duffing equations. (English) Zbl 1417.34095

Summary: We study the periodic orbits of the modified Duffing differential equation \(\ddot{y} + ay - \varepsilon y^3 = \varepsilon h(y, \dot{y})\) with \(a > 0, \varepsilon\) a small parameter and \(h\) a \(C^2\) function in its variables.


34C25 Periodic solutions to ordinary differential equations
34C29 Averaging method for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
Full Text: DOI Link


[1] Chen, H.; Li, Y., Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities, Proc. Amer. Math. Soc., 135, 3925-3932, (2007) · Zbl 1166.34313
[2] Chen, H.; Li, Y., Bifurcation and stability of periodic solutions of Duffing equations, Nonlinearity, 21, 2485-2503, (2008) · Zbl 1159.34033
[3] Duffing, G., Erzwungen schwingungen bei veränderlicher eigenfrequenz und ihre technisch bedeutung, Sammlung Vieweg Heft 41/42, (1918), Vieweg Braunschweig · JFM 46.1168.01
[4] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcation of vector fields, (1983), Springer · Zbl 0515.34001
[5] Hale, J. K.; Taboas, P. Z., Interaction of damping and forcing in a second order equation, Nonlinear Anal., 2, 77-84, (1978) · Zbl 0369.34014
[6] Hamel, G., Ueber erzwungene schingungen bei endlischen amplituden, Math. Ann., 86, 1-13, (1922) · JFM 48.0519.03
[7] J. Mawhin, Seventy-five years of global analysis around the forced pendulum equation, in: Proceedings of Equadiff 9 CD rom, Brno, Masaryk University, (1997), pp. 115-145.
[8] Sanders, J. A.; Verhulst, F., Averaging methods in nonlinear dynamical systems, Applied Mathematical Sciences, 59, (1985), Springer-Verlag New York · Zbl 0586.34040
[9] Verhulst, F., Nonlinear differential equations and dynamical systems, Universitext, (1996), Springer-Verlag Berlin · Zbl 0854.34002
[10] Wu, X.; Li, J.; Zhou, Y., A priori bounds for periodic solutions of a Duffing equation, J. Appl. Math. Comput., 26, 535-543, (2008) · Zbl 1162.34031
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.