The most general forced form of the Duffing equation is
|
(1) |
If there is no forcing, the right side vanishes, leaving
|
(2) |
If and we take the plus sign,
|
(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
|
(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,
|
(11) |
Examine the stability of the point (0,0):
|
(12) |
|
(13) |
But
, so
is real. Since
, there will always be one
Positive Root, so this fixed point is unstable. Now look at (, 0).
|
(14) |
|
(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.
|
asymptotically stable |
|
linearly stable (superstable) |
|
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
|
(18) |
Multiplying both sides by gives
|
(19) |
|
(20) |
so we have an invariant of motion ,
|
(21) |
Solving for gives
|
(22) |
|
(23) |
so
|
(24) |
Note that the invariant of motion satisfies
|
(25) |
|
(26) |
so the equations of the Duffing oscillator are given by the Hamiltonian System
|
(27) |
References
Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993.
© 1996-9 Eric W. Weisstein
1999-05-24