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Duffing Differential Equation


The most general forced form of the Duffing equation is

 x^..+deltax^.+(betax^3+/-omega_0^2x)=gammacos(omegat+phi).
(1)

Depending on the parameters chosen, the equation can take a number of special forms. For example, with no damping and no forcing, delta=gamma=0 and taking the plus sign, the equation becomes

 x^..+omega_0^2x+betax^3=0
(2)

(Bender and Orszag 1978, p. 547; Zwillinger 1997, p. 122). This equation can display chaotic behavior. For beta>0, the equation represents a "hard spring," and for beta<0, it represents a "soft spring." If beta<0, the phase portrait curves are closed.

If instead we take beta=1, omega_0=1, reset the clock so that phi=0, and use the minus sign, the equation is then

 x^..+deltax^.+(x^3-x)=gammacos(omegat).
(3)

This can be written as a system of first-order ordinary differential equations as

x^.=y,
(4)
y^.=x-x^3-deltay+gammacos(omegat)
(5)

(Wiggins 1990, p. 5) which, in the unforced case, reduces to

x^.=y
(6)
y^.=x-x^3-deltay
(7)

(Wiggins 1990, p. 6; Ott 1993, p. 3).

The fixed points of this set of coupled differential equations are given by

 x^.=y=0,
(8)

so y=0, and

y^.=x-x^3-deltay
(9)
=x(1-x^2)-0
(10)

giving x=0,+/-1. The fixed points are therefore (-1,0), (0,0), and (1,0).

Analysis of the stability of the fixed points can be point by linearizing the equations. Differentiating gives

x^..=y^.
(11)
=x-x^3-deltay
(12)
y^..=(1-3x^2)x^.-deltay^.,
(13)

which can be written as the matrix equation

 [x^..; y^..]=[0 1; 1-3x^2 -delta][x^.; y^.].
(14)

Examining the stability of the point (0,0):

 |0-lambda 1; 1 -delta-lambda|=lambda(lambda+delta)-1=lambda^2+lambdadelta-1=0
(15)
 lambda_+/-^((0,0))=1/2(-delta+/-sqrt(delta^2+4)).
(16)

But delta^2>=0, so lambda_+/-^((0,0)) is real. Since sqrt(delta^2+4)>|delta|, there will always be one positive root, so this fixed point is unstable. Now look at (+/-1, 0). The characteristic equation is

 |0-lambda 1; -2 -delta-lambda|=lambda(lambda+delta)+2=lambda^2+lambdadelta+2=0,
(17)

which has roots

 lambda_+/-^((+/-1,0))=1/2(-delta+/-sqrt(delta^2-8)).
(18)

For delta>0, R[lambda_+/-^((+/-1,0))]<0, so the point is asymptotically stable. If delta=0, lambda_+/-^((+/-1,0))=+/-isqrt(2), so the point is linearly stable (Wiggins 1990, p. 10). However, if delta in (-2sqrt(2),0), the radical gives an imaginary part and the real part is >0, so the point is unstable. If delta=-2sqrt(2), lambda_+/-^((+/-1,0))=sqrt(2), which has a positive real root, so the point is unstable. If delta<-2sqrt(2), then |delta|<sqrt(delta^2-8), so both roots are positive and the point is unstable.

DuffingOscillatorPhasePortrait

Interestingly, the special case delta=0 with no forcing,

x^.=y
(19)
y^.=x-x^3,
(20)

can be integrated by quadratures. Differentiating (19) and plugging in (20) gives

 x^..=y^.=x-x^3.
(21)

Multiplying both sides by x^. gives

 x^..x^.-x^.x+x^.x^3=0.
(22)

But this can be written

 d/(dt)(1/2x^.^2-1/2x^2+1/4x^4)=0,
(23)

so we have an invariant of motion h,

 h=1/2x^.^2-1/2x^2+1/4x^4.
(24)

Solving for x^.^2 gives

 x^.^2=((dx)/(dt))^2=2h+x^2-1/2x^4
(25)
 (dx)/(dt)=sqrt(2h+x^2-1/2x^4),
(26)

so

 t=intdt=int(dx)/(sqrt(2h+x^2-1/2x^4))
(27)

(Wiggins 1990, p. 29).

Note that the invariant of motion h satisfies

 x^.=(partialh)/(partialx^.)=(partialh)/(partialy)
(28)
 (partialh)/(partialx)=-x+x^3=-y^.,
(29)

so the equations of the Duffing oscillator are given by the Hamiltonian system

x^.=(partialh)/(partialy)
(30)
y^.=-(partialh)/(partialx)
(31)

(Wiggins 1990, p. 31).


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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.

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Duffing Differential Equation

Cite this as:

Weisstein, Eric W. "Duffing Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DuffingDifferentialEquation.html

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