View source for Duffing equation

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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 [[#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 \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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Duffing,   "Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung" , Vieweg  (1918)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.J. Stoker,   "Nonlinear vibrations in mechanical and electrical systems" , Interscience  (1950)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Hayashi,   "Nonlinear oscillations in physical systems" , McGraw-Hill  (1964)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Ph. Holmes,   "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer  (1983)</TD></TR></table>