Duffing oscillator

Poincaré image of a driven Duffing oscillator

The Duffing oscillator , named after Georg Duffing , is a non-linear oscillator . It can be viewed as an extension of the harmonic oscillator , whose potential is based on the linear Hooke's law , by a cubic restoring force. Its behavior is described by the following differential equation with the time derivatives of x:

${\displaystyle {\ddot {x}}+\delta {\dot {x}}+\alpha x+\beta x^{3}=\gamma \cos(\omega _{0}t)}$

${\displaystyle \delta }$is the damping, are the amplitude and frequency of the excitation, are system-specific parameters that characterize the non-linear, restoring force. ${\displaystyle \gamma ,\omega _{0}}$ ${\displaystyle \alpha ,\beta }$

Duffing oscillator without excitation

The state space representation of the homogeneous Duffing oscillator is ${\displaystyle {\ddot {x}}+\delta {\dot {x}}+\alpha x+\beta x^{3}=0}$

${\displaystyle {\begin{bmatrix}{\dot {x}}_{1}\\{\dot {x}}_{2}\\\end{bmatrix}}={\begin{bmatrix}x_{2}\\-\delta x_{2}-\alpha x_{1}-\beta x_{1}^{3}\\\end{bmatrix}}}$

The following applies to the inpatient case

${\displaystyle {\begin{bmatrix}0\\0\\\end{bmatrix}}={\begin{bmatrix}x_{2}\\-\delta x_{2}-\alpha x_{1}-\beta x_{1}^{3}\\\end{bmatrix}}}$

and thus

${\displaystyle x_{2}=0\,}$and . ${\displaystyle \alpha x_{1}+\beta x_{1}^{3}=0}$

The equation yields stationary solutions for three ${\displaystyle x_{1}\;}$

${\displaystyle x_{1_{0}}=0,x_{1_{1,2}}=\pm {\sqrt {-{\frac {\alpha }{\beta }}}}}$

These are only real if is. To assess which of these stationary solutions are stable, the system of differential equations is linearized around these points. The Jacobian matrix of the system ${\displaystyle {\frac {\alpha }{\beta }}<0}$

${\displaystyle {\textbf {J}}={\begin{bmatrix}0&1\\-\alpha -3\beta x_{1}^{2}&-\delta \\\end{bmatrix}}}$

has for the eigenvalues ${\displaystyle x_{1_{0}}\;}$

${\displaystyle \lambda _{0}={\frac {-\delta \pm {\sqrt {\delta ^{2}-4\alpha }}}{2}}}$

and for the eigenvalues ${\displaystyle x_{1_{1,2}}\;}$

${\displaystyle \lambda _{1}={\frac {-\delta \pm {\sqrt {\delta ^{2}+8\alpha }}}{2}}}$.

The condition gives two cases. ${\displaystyle {\frac {\alpha }{\beta }}<0}$

Case 1: and ${\displaystyle \alpha >0\;}$ ${\displaystyle \beta <0\;}$

${\displaystyle \lambda _{0}\;}$has negative real parts, i.e. H. this point is stable.

${\displaystyle \lambda _{1}\;}$has a positive real part, i.e. H. these points are unstable.

Case 2: and ${\displaystyle \alpha <0\;}$ ${\displaystyle \beta >0\;}$

${\displaystyle \lambda _{0}\;}$has a positive real part, i.e. H. this point is unstable.

${\displaystyle \lambda _{1}\;}$has negative real parts, i.e. H. these points are stable.

The differential equation

${\displaystyle {\ddot {x}}+\delta {\dot {x}}-ax+bx^{3}=0}$

with describes the stable duffing oscillator. ${\displaystyle \delta >0,a>0,b>0\;}$

Web links

Commons : Duffing Oscillator  - collection of images, videos and audio files

• Duffing oscillator at Scholarpedia