Duffing equation

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 x ¨ + δ x ˙ + α x + β x 3 = γ cos ⁡ ( ω t ) , {\displaystyle {\ddot {x}}+\delta {\dot {x}}+\alpha x+\beta x^{3}=\gamma \cos(\omega t),} where the (unknown) function x = x ( t ) {\displaystyle x=x(t)} is the displacement at time t, x ˙ {\displaystyle {\dot {x}}} is the first derivative of x {\displaystyle x} with respect to time, i.e. velocity, and x ¨ {\displaystyle {\ddot {x}}} is the second time-derivative of x , {\displaystyle x,} i.e. acceleration. The numbers δ , {\displaystyle \delta ,} α , {\displaystyle \alpha ,} β , {\displaystyle \beta ,} γ {\displaystyle \gamma } and ω {\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 β = δ = γ = 0 {\displaystyle \beta =\delta =\gamma =0} and α > 0 {\displaystyle \alpha >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.