where the (unknown) function is the displacement at time is the first derivative of with respect to time, i.e. velocity, and is the second time-derivative of i.e. acceleration. The numbers and are given constants.
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.
The Duffing equation can be seen as describing the oscillations of a mass attached to a nonlinear spring and a linear damper. The restoring force provided by the nonlinear spring is then
When and the spring is called a hardening spring. Conversely, for it is a softening spring (still with ). Consequently, the adjectives hardening and softening are used with respect to the Duffing equation in general, dependent on the values of (and ).
The number of parameters in the Duffing equation can be reduced by two through scaling, e.g. the excursion and time can be scaled as: and assuming is positive (other scalings are possible for different ranges of the parameters, or for different emphasis in the problem studied). Then:
The dots denote differentiation of with respect to This shows that the solutions to the forced and damped Duffing equation can be described in terms of the three parameters ( and ) and two initial conditions (i.e. for and ).
since for damping. Without forcing the damped Duffing oscillator will end up at (one of) its stableequilibrium point(s). The equilibrium points, stable and unstable, are at If the stable equilibrium is at If and the stable equilibria are at and
Frequency response as a function of for the Duffing equation, with and damping The dashed parts of the frequency response are unstable.
The forced Duffing oscillator with cubic nonlinearity is described by the following ordinary differential equation:
The frequency response of this oscillator describes the amplitude of steady state response of the equation (i.e. ) at a given frequency of excitation For a linear oscillator with the frequency response is also linear. However, for a nonzero cubic coefficient, the frequency response becomes nonlinear. Depending on the type of nonlinearity, the Duffing oscillator can show hardening, softening or mixed hardening–softening frequency response. Anyway, using the homotopy analysis method or harmonic balance, one can derive a frequency response equation in the following form:
For the parameters of the Duffing equation, the above algebraic equation gives the steady state oscillation amplitude at a given excitation frequency.
Derivation of the frequency response
Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form:
Application in the Duffing equation leads to:
Neglecting the superharmonics at the two terms preceding and have to be zero. As a result,
Squaring both equations and adding leads to the amplitude frequency response:
Jumps in the frequency response. The parameters are: , and 
For certain ranges of the parameters in the Duffing equation, the frequency response may no longer be a single-valued function of forcing frequency For a hardening spring oscillator ( and large enough positive ) the frequency response overhangs to the high-frequency side, and to the low-frequency side for the softening spring oscillator ( and ). The lower overhanging side is unstable – i.e. the dashed-line parts in the figures of the frequency response – and cannot be realized for a sustained time. Consequently, the jump phenomenon shows up:
when the angular frequency is slowly increased (with other parameters fixed), the response amplitude drops at A suddenly to B,
if the frequency is slowly decreased, then at C the amplitude jumps up to D, thereafter following the upper branch of the frequency response.
The jumps A–B and C–D do not coincide, so the system shows hysteresis depending on the frequency sweep direction.
^Thompson, J.M.T.; Stewart, H.B. (2002). Nonlinear Dynamics and Chaos. John Wiley & Sons. p. 66. ISBN9780471876847.
^Lifshitz, R.; Cross, M.C. (2008). "Nonlinear mechanics of nanomechanical and micromechanical resonators". In Schuster, H.G. (ed.). Reviews of Nonlinear Dynamics and Complexity. Wiley. pp. 8–9. ISBN9783527407293. LCCN2008459659.
^ abBrennan, M.J.; Kovacic, I.; Carrella, A.; Waters, T.P. (2008). "On the jump-up and jump-down frequencies of the Duffing oscillator". Journal of Sound and Vibration. 318 (4–5): 1250–1261. doi:10.1016/j.jsv.2008.04.032.
Duffing, G. (1918), Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung [Forced oscillations with variable natural frequency and their technical relevance] (in German), Heft 41/42, Braunschweig: Vieweg, vi+134 , pp., OCLC12003652