Duffing Differential Equation
المؤلف:
Bender, C. M. and Orszag, S. A
المصدر:
dvanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill
الجزء والصفحة:
...
11-6-2018
1903
Duffing Differential Equation
The most general forced form of the Duffing equation is
 |
(1)
|
Depending on the parameters chosen, the equation can take a number of special forms. For example, with no damping and no forcing, 
and taking the plus sign, the equation becomes
 |
(2)
|
(Bender and Orszag 1978, p. 547; Zwillinger 1997, p. 122). 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.
If instead we take 
, 
, reset the clock so that 
, and use the minus sign, the equation is then
 |
(3)
|
This can be written as a system of first-order ordinary differential equations as
(Wiggins 1990, p. 5) which, in the unforced case, reduces to
(Wiggins 1990, p. 6; Ott 1993, p. 3).
The fixed points of this set of coupled differential equations are given by
 |
(8)
|
so 
, and
giving 
. The fixed points are therefore 
, 
, and 
.
Analysis of the stability of the fixed points can be point by linearizing the equations. Differentiating gives
which can be written as the matrix equation
![[x^..; y^..]=[0 1; 1-3x^2 -delta][x^.; y^.].](http://mathworld.wolfram.com/images/equations/DuffingDifferentialEquation/NumberedEquation5.gif) |
(14)
|
Examining the stability of the point (0,0):
 |
(15)
|
 |
(16)
|
But 
, so 
is real. Since 
, there will always be one positive root, so this fixed point is unstable. Now look at (
, 0). The characteristic equation is
 |
(17)
|
which has roots
 |
(18)
|
For 
, ![R[lambda_+/-^((+/-1,0))]<0](http://mathworld.wolfram.com/images/equations/DuffingDifferentialEquation/Inline45.gif)
, so the point is asymptotically stable. If 
, 
, so the point is linearly stable (Wiggins 1990, p. 10). However, 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.
Interestingly, the special case 
with no forcing,
can be integrated by quadratures. Differentiating (19) and plugging in (20) gives
 |
(21)
|
Multiplying both sides by 
gives
 |
(22)
|
But this can be written
 |
(23)
|
so we have an invariant of motion 
,
 |
(24)
|
Solving for 
gives
 |
(25)
|
 |
(26)
|
so
 |
(27)
|
(Wiggins 1990, p. 29).
Note that the invariant of motion 
satisfies
 |
(28)
|
 |
(29)
|
so the equations of the Duffing oscillator are given by the Hamiltonian system
(Wiggins 1990, p. 31).
REFERENCES:
Bender, C. M. and Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill, p. 547, 1978.
Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, 1993.
Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, p. 35, 1989.
Trott, M. "The Mathematica Guidebooks Additional Material: Wigner Function of a Duffing Oscillator." http://www.mathematicaguidebooks.org/additions.shtml#N_1_08.
Wiggins, S. "Application to the Dynamics of the Damped, Forced Duffing Oscillator." §1.2E in Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag, pp. 5-6, 10, 23, 26-32, 44-45, 50-51, and 153-175, 1990.
Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995.
Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 122, 1997.
الاكثر قراءة في معادلات تفاضلية
اخر الاخبار
اخبار العتبة العباسية المقدسة
