The Duffing equation (or Duffing oscillator), named after Georg Duffing (1861–1944), 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 ${\displaystyle x=x(t)}$ is the displacement at time t, ${\displaystyle {\dot {x}}}$ is the first derivative of ${\displaystyle x}$ with respect to time, i.e. velocity, and ${\displaystyle {\ddot {x}}}$ is the second time-derivative of ${\displaystyle x,}$ i.e. acceleration. The numbers ${\displaystyle \delta ,}$ ${\displaystyle \alpha ,}$ ${\displaystyle \beta ,}$ ${\displaystyle \gamma }$ and ${\displaystyle \omega }$ are given constants.

The equation describes the motion of a damped oscillator with a more complex potential than in simple harmonic motion (which corresponds to the case ${\displaystyle \beta =\delta =0}$); 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.

Parameters

The parameters in the above equation are:

• ${\displaystyle \delta }$ controls the amount of damping,
• ${\displaystyle \alpha }$ controls the linear stiffness,
• ${\displaystyle \beta }$ controls the amount of non-linearity in the restoring force; if ${\displaystyle \beta =0,}$ the Duffing equation describes a damped and driven simple harmonic oscillator,
• ${\displaystyle \gamma }$ is the amplitude of the periodic driving force; if ${\displaystyle \gamma =0}$ the system is without a driving force, and
• ${\displaystyle \omega }$ is the angular frequency of the periodic driving force.

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 ${\displaystyle \alpha x+\beta x^{3}.}$

When ${\displaystyle \alpha >0}$ and ${\displaystyle \beta >0}$ the spring is called a hardening spring. Conversely, for ${\displaystyle \beta <0}$ it is a softening spring (still with ${\displaystyle \alpha >0}$). Consequently, the adjectives hardening and softening are used with respect to the Duffing equation in general, dependent on the values of ${\displaystyle \beta }$ (and ${\displaystyle \alpha }$).^[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 ${\displaystyle x}$ and time ${\displaystyle t}$ can be scaled as:^[2] ${\displaystyle \tau =t{\sqrt {\alpha }}}$ and ${\displaystyle y=x\alpha /\gamma ,}$ assuming ${\displaystyle \alpha }$ is positive (other scalings are possible for different ranges of the parameters, or for different emphasis in the problem studied). Then:^[3]

where

• ${\displaystyle \eta ={\frac {\delta }{2{\sqrt {\alpha }}}},}$
• ${\displaystyle \mathrm{\epsilon <!--\; \epsilon \; -->}=\frac{\mathrm{\beta <!--\; \beta \; -->}{\mathrm{\gamma <!--\; \gamma \; -->}}^2}{}}$Zdroj:https://en.wikipedia.org?pojem=Duffing_equation
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