Duffing equation
The equation describes the motion of a damped oscillator with a more complex potential than in simple harmonic motion (which corresponds to the case
); in physical terms, it models, for example, an elastic 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.
The parameters in the above equation are: 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
Consequently, the adjectives hardening and softening are used with respect to the Duffing equation in general, dependent on the values of
[1] The number of parameters in the Duffing equation can be reduced by two through scaling (in accord with the Buckingham π theorem), e.g. the excursion
is positive (other scalings are possible for different ranges of the parameters, or for different emphasis in the problem studied).
This shows that the solutions to the forced and damped Duffing equation can be described in terms of the three parameters (
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 (
) Duffing equation, an exact solution can be obtained using Jacobi's elliptic functions.
[6] Multiplication of the undamped and unforced Duffing equation,
Similarly, the damped oscillator converges globally, by Lyapunov function method[8]
Without forcing the damped Duffing oscillator will end up at (one of) its stable equilibrium point(s).
The forced Duffing oscillator with cubic nonlinearity is described by the following ordinary differential equation:
The frequency response of this oscillator describes the amplitude
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:[9][5]
Using the method of harmonic balance, an approximate solution to the Duffing equation is sought of the form:[9]
Squaring both equations and adding leads to the amplitude frequency response:
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: The jumps A–B and C–D do not coincide, so the system shows hysteresis depending on the frequency sweep direction.
[9] The above analysis assumed that the base frequency response dominates (necessary for performing harmonic balance), and higher frequency responses are negligible.
This assumption fails to hold when the forcing is sufficiently strong.
Higher order harmonics cannot be neglected, and the dynamics become chaotic.
There are different possible transitions to chaos, most commonly by successive period doubling.
[10] Some typical examples of the time series and phase portraits of the Duffing equation, showing the appearance of subharmonics through period-doubling bifurcation – as well chaotic behavior – are shown in the figures below.
The red dots in the phase portraits are at times
