Duffing equation

The Duffing equation (or Duffing oscillator), named after <a href="/facts/Georg_Duffing/jP556VC5">Georg Duffing</a> (1861–1944), is a <a href="/facts/Non-linear/4aYItALC">non-linear</a> second-order <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a> used to model certain <a href="/facts/Harmonic_oscillator/7IQNToGt">damped and driven oscillators</a>. 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 <a href="/facts/Derivative/7Ito70Dh">derivative</a> of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 with respect to time, i.e. <a href="/facts/Velocity/Q3o1LVDe">velocity</a>, and 
 
 
 
 
 
 
 x
 ¨
 
 
 
 
 
 {\displaystyle {\ddot {x}}}
 
 is the second time-derivative of 
 
 
 
 x
 ,
 
 
 {\displaystyle x,}
 
 i.e. <a href="/facts/Acceleration/UVDmjYvR">acceleration</a>. 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 <a href="/facts/Potential/8Fcbtnt9">potential</a> than in <a href="/facts/Simple_harmonic_motion/u2cGAd91">simple harmonic motion</a> (which corresponds to the case 
 
 
 
 β
 =
 δ
 =
 0
 
 
 {\displaystyle \beta =\delta =0}
 
); in physical terms, it models, for example, an <a href="/facts/Elastic_pendulum/5SfuPxzt">elastic pendulum</a> whose spring's <a href="/facts/Stiffness/xlabLkfT">stiffness</a> does not exactly obey <a href="/facts/Hooke%2527s_law/7hTUcKUm">Hooke's law</a>.
The Duffing equation is an example of a dynamical system that exhibits <a href="/facts/Chaos_theory/LTC17vsF">chaotic behavior</a>. Moreover, the Duffing system presents in the <a href="/facts/Frequency_response/zsHmnCJ4">frequency response</a> the jump resonance phenomenon that is a sort of frequency <a href="/facts/Hysteresis/9M2pwR62">hysteresis</a> behaviour.

Duffing equation open-in-new

Duffing equation