Duffing equation

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A Poincaré section of the forced Duffing equation suggesting chaotic behaviour

The Duffing equation (or Duffing oscillator), named after Georg Duffing, is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by

where the (unknown) function x=x(t) is the displacement at time t, is the first derivative of x with respect to time, i.e. velocity, and is the second time-derivative of x, i.e. acceleration. The numbers , , , and are given constants.

The equation describes the motion of a damped oscillator with a more complicated potential than in simple harmonic motion (which corresponds to the case β=δ=0); in physical terms, it models, for example, a spring pendulum whose spring's stiffness does not exactly obey Hooke's law.

The Duffing equation is an example of a dynamical system that exhibits chaotic behavior. Moreover, the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour.


  • controls the size of the damping.
  • controls the size of the stiffness.
  • controls the amount of non-linearity in the restoring force. If , the Duffing equation describes a damped and driven simple harmonic oscillator.
  • controls the amplitude of the periodic driving force. If we have a system without driving force.
  • controls the frequency of the periodic driving force.

Methods of solution[edit]

In general, the Duffing equation does not admit an exact symbolic solution. However, many approximate methods work well:

In the special case of the undamped () and undriven () Duffing equation, an exact solution can be obtained using Jacobi's elliptic functions.

Boundedness of the solution for the undamped and unforced oscillator[edit]

Multiplication of the undamped and unforced Duffing equation, with gives:[2]

with H a constant. The value of H is determined by the initial conditions and

The substitution in H shows that the system is Hamiltonian:


When both and are positive, the solution is bounded:[2]


with the Hamiltonian H being positive.



  1. ^ Tajaddodianfar, Farid (2016). "Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method". Microsystem Technologies. doi:10.1007/s00542-016-2947-7. 
  2. ^ a b Bender & Orszag (1999, p. 546)


  • Bender, C.M.; Orszag, S.A. (1999), Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer, pp. 545–551, ISBN 9780387989310 
  • Addison, P.S. (1997), Fractals and Chaos: An illustrated course, CRC Press, pp. 147–148, ISBN 9780849384431 

External links[edit]