# Talk:Duffing oscillator

Comments on The Duffing oscillator by T. Kanamaru:

The article generally appears correct and is at an appropriate level.

Suggestions:

"It is known that chaotic motions are observed in this case" To be correct, this should be revised to "It is known that chaotic motions can be observed in this case" For example, if damping delta is large enough, only stable periodic motions are observed.

Add a brief explanation of why dE/dt = -delta \dot{x}^2 implies approach of almost all trajectories to a sink in the unforced system (except for orbits in stable manifolds of the saddle when alpha < 0). Add link to articles on Liapunov functions and LaSalle's invariance principle, if they exist. This is a global stability claim, so needs some justification and discussion.

Add brief explanations of what Figs 1 and 8 actually show in the figure captions. The text explains that they are animations of points on the Poincaré cross section computed for single trajectories, as the phase at which the cross section is taken relative to forcing phase is changed. This is fine, but the reader would find it easier if it were also described in the caption.

Fixed points of Eqns (7-8) can be found analytically (they are simply solutions of a cubic equation. Explicit expressions for bifurcation sets are derived in [1] for example, a reference which the article already cites.

Add reference(s) to Y. Ueda's paper(s) on Duffing's equation (cited without reference in Fig 8 caption). Also, for historical accuracy, a reference to Duffing's 1918 book should be added; it may be found in Guckenheimer and Holmes 1983.

[1] P.J. Holmes an D.A. Rand, The bifurcations of Duffing's equation: An application of catastrophe theory, Journal of Sound and Vibration, 44, 237-253, 1976.

Philip Holmes, Princeton University.