Duffing equation

Duffing the oscillator is a non-linear oscillator. It may as an extension of the harmonic oscillator, the potential of the linear Hooke's law underlies to be regarded as a cubic restoring force. Its behavior is described by the following differential equation with the time derivatives of x:

Is the attenuation, the amplitude and frequency of the excitation, are system- specific parameters which characterize the non-linear restoring force.

Duffing oscillator without excitation

The state space representation of the undriven Duffing oscillator

For the stationary case

And thus

The equation provides for three stationary solutions

These are only real if is. To assess which of these stationary solutions is stable, the system of differential equations is linearized around these points. The Jacobi matrix of the system

Has the eigenvalues

And for the eigenvalues

The condition provides two cases.

Case 1: and

Case 2: and

The differential equation

With describing the stable Duffing oscillator.

Duffing equation

Duffing oscillator without excitation

See also