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 where the (unknown) function x = x ( t ) {displaystyle x=x(t)} is the displacement at time t , {displaystyle 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 eta ,} 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 = 0 {displaystyle eta =delta =0} ); in physical terms, it models, for example, a spring 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.