Character sum

A Poincaré section of the forced Duffing equation suggesting chaotic behaviour

The Duffing equation, named after Georg Duffing, is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by

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

where the (unknown) function x=x(t) is the displacement at time t, $\dot{x}$ is the first derivative of x with respect to time, i.e. velocity, and $\ddot{x}$ is the second time-derivative of x, i.e. acceleration. The numbers $\delta$, $\alpha$, $\beta$, $\gamma$ and $\omega$ are given constants.

The equation describes the motion of a damped oscillator with a more complicated potential than in simple harmonic motion (which corresponds to the case β=δ=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.

Parameters

• $\delta$ controls the size of the damping.
• $\alpha$ controls the size of the stiffness.
• $\beta$ controls the amount of non-linearity in the restoring force. If $\beta =0$, the Duffing equation describes a damped and driven simple harmonic oscillator.
• $\gamma$ controls the amplitude of the periodic driving force. If $\gamma =0$ we have a system without driving force.
• $\omega$ controls the frequency of the periodic driving force.

Methods of solution

Duffing oscillator limit cycle γ>0

Duffing oscillator limit cycle phase animation γ>0

Duffing oscillator chaos oscillation γ<0

Duffing oscillator attractors animation γ<0

In general, the Duffing equation does not admit an exact symbolic solution. However, many approximate methods work well:

• Expansion in a Fourier series will provide an equation of motion to arbitrary precision.
• The $x^3$ term, also called the Duffing term, can be approximated as small and the system treated as a perturbed simple harmonic oscillator.
• The Frobenius method yields a complicated but workable solution.
• Any of the various numeric methods such as Euler's method and Runge-Kutta can be used.

In the special case of the undamped ($\delta =0$) and undriven ($\gamma =0$) Duffing equation, an exact solution can be obtained using Jacobi's elliptic functions.

References

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External links

• Duffing oscillator on Scholarpedia
• MathWorld page