The most general forced form of the Duffing equation is
![\begin{displaymath}
\ddot x+\delta\dot x+(\beta x^3\pm{\omega_0}^2 x)=A\sin(\omega t+\phi).
\end{displaymath}](d2_1647.gif) |
(1) |
If there is no forcing, the right side vanishes, leaving
![\begin{displaymath}
\ddot x+\delta\dot x+(\beta x^3\pm{\omega_0}^2 x)=0.
\end{displaymath}](d2_1648.gif) |
(2) |
If
and we take the plus sign,
![\begin{displaymath}
\ddot x+{\omega_0}^2x+\beta x^3=0.
\end{displaymath}](d2_1650.gif) |
(3) |
This equation can display chaotic behavior. For
, the equation represents a ``hard spring,'' and for
, it represents a ``soft spring.'' If
, the phase portrait curves are closed.
Returning to (1),
take
,
,
, and use the minus sign. Then the equation is
![\begin{displaymath}
\ddot x+\delta\dot x+(x^3-x) = 0
\end{displaymath}](d2_1656.gif) |
(4) |
(Ott 1993, p. 3). This can be written as a system of first-order
ordinary differential equations by writing
The fixed points of these differential equations
so
, and
giving
. Differentiating,
![\begin{displaymath}
\left[{\matrix{\ddot x\cr \ddot y\cr}}\right] = \left[{\matr...
...delta\cr}}\right] \left[{\matrix{\dot x\cr \dot y\cr}}\right].
\end{displaymath}](d2_1669.gif) |
(11) |
Examine the stability of the point (0,0):
![\begin{displaymath}
\left\vert\matrix{0-\lambda & 1\cr 1 & -\delta-\lambda \cr}\...
...rt = \lambda(\lambda+\delta)-1 = \lambda^2+\lambda\delta-1 = 0
\end{displaymath}](d2_1670.gif) |
(12) |
![\begin{displaymath}
\lambda^{(0,0)}_\pm = {\textstyle{1\over 2}}(-\delta\pm\sqrt{\delta^2+4}\,).
\end{displaymath}](d2_1671.gif) |
(13) |
But
, so
is real. Since
, there will always be one
Positive Root, so this fixed point is unstable. Now look at (
, 0).
![\begin{displaymath}
\left\vert\matrix{0-\lambda & 1\cr -2 & -\delta-\lambda\cr}\...
...rt = \lambda(\lambda+\delta)+2 = \lambda^2+\lambda\delta+2 = 0
\end{displaymath}](d2_1675.gif) |
(14) |
![\begin{displaymath}
\lambda^{(\pm 1,0)}_\pm = {\textstyle{1\over 2}}(-\delta\pm\sqrt{\delta^2-8}\,).
\end{displaymath}](d2_1676.gif) |
(15) |
For
,
, so the point is asymptotically stable. If
,
, so the point is linearly stable. If
, the radical gives an
Imaginary Part and the Real Part is
, so the point is unstable. If
,
, which has a Positive Real Root, so the point is unstable. If
, then
, so both Roots are Positive and the point is unstable.
The following table summarizes these results.
![$\delta>0$](d2_1677.gif) |
asymptotically stable |
![$\delta=0$](d2_1649.gif) |
linearly stable (superstable) |
![$\delta<0$](d2_1686.gif) |
unstable |
Now specialize to the case
, which can be integrated by quadratures.
In this case, the equations become
Differentiating (16) and plugging in (17) gives
![\begin{displaymath}
\ddot x=\dot y=x-x^3.
\end{displaymath}](d2_1689.gif) |
(18) |
Multiplying both sides by
gives
![\begin{displaymath}
\ddot x \dot x -\dot x x+\dot x x^3=0
\end{displaymath}](d2_1691.gif) |
(19) |
![\begin{displaymath}
{d\over dt} ({\textstyle{1\over 2}}\dot x^2-{\textstyle{1\over 2}}x^2+{\textstyle{1\over 4}}x^4)=0,
\end{displaymath}](d2_1692.gif) |
(20) |
so we have an invariant of motion
,
![\begin{displaymath}
h\equiv {\textstyle{1\over 2}}\dot x^2-{\textstyle{1\over 2}}x^2+{\textstyle{1\over 4}}x^4.
\end{displaymath}](d2_1693.gif) |
(21) |
Solving for
gives
![\begin{displaymath}
\dot x^2=\left({dx\over dt}\right)^2=2h+x^2-{\textstyle{1\over 2}}x^4
\end{displaymath}](d2_1695.gif) |
(22) |
![\begin{displaymath}
{dx\over dt}=\sqrt{2h+x^2+{\textstyle{1\over 2}}x^2},
\end{displaymath}](d2_1696.gif) |
(23) |
so
![\begin{displaymath}
t=\int dt=\int{dx\over\sqrt{2h+x^2+{\textstyle{1\over 2}}x^2}}.
\end{displaymath}](d2_1697.gif) |
(24) |
Note that the invariant of motion
satisfies
![\begin{displaymath}
\dot x={\partial h\over\partial\dot x} ={\partial h\over\partial y}
\end{displaymath}](d2_1698.gif) |
(25) |
![\begin{displaymath}
{\partial h\over \partial x}=-x+x^3=-\dot y,
\end{displaymath}](d2_1699.gif) |
(26) |
so the equations of the Duffing oscillator are given by the Hamiltonian System
![\begin{displaymath}
\cases{
\dot x = {\partial h\over\partial y}\cr
\dot y =-{\partial h\over\partial x}.\cr}
\end{displaymath}](d2_1700.gif) |
(27) |
References
Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993.
© 1996-9 Eric W. Weisstein
1999-05-24