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Difference between revisions of "Duffing equation"

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A second-order ordinary differential equation
 
A second-order ordinary differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d0341501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$x''+kx'+\omega_0^2x+\alpha x^3=F\cos\omega t,\tag{*}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d0341502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d0341503.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d0341504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d0341505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d0341506.png" /> are constants. This equation is an important example of a system with one degree of freedom with a non-linear restoring force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d0341507.png" /> and damping, which executes forced oscillations under the effect of a harmonic external force <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d0341508.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d0341509.png" /> one speaks of a rigid elastic force while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d03415010.png" /> one speaks of a soft force. G. Duffing [[#References|[1]]] was the first to study the solutions of equation (*).
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where $k>0$, $\omega_0$, $\alpha$, $F$, $\omega$ are constants. This equation is an important example of a system with one degree of freedom with a non-linear restoring force $f(x)=-\omega_0^2x-\alpha x^3$ and damping, which executes forced oscillations under the effect of a harmonic external force $F(t)=F\cos\omega t$. If $\alpha>0$ one speaks of a rigid elastic force while if $\alpha<0$ one speaks of a soft force. G. Duffing [[#References|[1]]] was the first to study the solutions of equation \ref{*}.
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d03415011.png" /> with an amplitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d03415012.png" /> which is a function of the frequency (an amplitude curve); to certain values of the frequency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d03415013.png" /> there may correspond several types of oscillations with different amplitudes. Under certain conditions Duffing's equation yields subharmonic oscillations with frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d03415014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034150/d03415015.png" /> is an integer. The solutions of equation (*) are often studied by the method of the small parameter.
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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 \ref{*}, possible harmonic oscillations are $x=A\cos\omega t$ with an amplitude $A=A(\omega)$ which is a function of the frequency (an amplitude curve); to certain values of the frequency $\omega$ there may correspond several types of oscillations with different amplitudes. Under certain conditions Duffing's equation yields subharmonic oscillations with frequencies $\omega/n$, where $n$ is an integer. The solutions of equation \ref{*} are often studied by the method of the small parameter.
  
 
====References====
 
====References====

Revision as of 17:02, 19 September 2014

A second-order ordinary differential equation

$$x''+kx'+\omega_0^2x+\alpha x^3=F\cos\omega t,\tag{*}$$

where $k>0$, $\omega_0$, $\alpha$, $F$, $\omega$ are constants. This equation is an important example of a system with one degree of freedom with a non-linear restoring force $f(x)=-\omega_0^2x-\alpha x^3$ and damping, which executes forced oscillations under the effect of a harmonic external force $F(t)=F\cos\omega t$. If $\alpha>0$ one speaks of a rigid elastic force while if $\alpha<0$ one speaks of a soft force. G. Duffing [1] was the first to study the solutions of equation \ref{*}.

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 \ref{*}, possible harmonic oscillations are $x=A\cos\omega t$ with an amplitude $A=A(\omega)$ which is a function of the frequency (an amplitude curve); to certain values of the frequency $\omega$ there may correspond several types of oscillations with different amplitudes. Under certain conditions Duffing's equation yields subharmonic oscillations with frequencies $\omega/n$, where $n$ is an integer. The solutions of equation \ref{*} 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