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.