Namespaces
Variants
Actions

Duffing equation

From Encyclopedia of Mathematics
Revision as of 16:58, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

A second-order ordinary differential equation

(*)

where , , , , are constants. This equation is an important example of a system with one degree of freedom with a non-linear restoring force and damping, which executes forced oscillations under the effect of a harmonic external force . If one speaks of a rigid elastic force while if one speaks of a soft force. G. Duffing [1] was the first to study the solutions of equation (*).

Solutions of Duffing's equation cannot be obtained in closed form. It has been proved that the equation has a large number of distinct periodic solutions. In equation (*), possible harmonic oscillations are with an amplitude which is a function of the frequency (an amplitude curve); to certain values of the frequency there may correspond several types of oscillations with different amplitudes. Under certain conditions Duffing's equation yields subharmonic oscillations with frequencies , where is an integer. The solutions of equation (*) are often studied by the method of the small parameter.

References

[1] G. Duffing, "Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung" , Vieweg (1918)
[2] J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950)
[3] C. Hayashi, "Nonlinear oscillations in physical systems" , McGraw-Hill (1964)


Comments

References

[a1] Ph. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)
How to Cite This Entry:
Duffing equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Duffing_equation&oldid=33326
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article