Duffing equation

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The Duffing equation is a non-linear second-order differential equation. It is an example of a dynamical system that exhibits chaotic behavior. The equation is given by

$\ddot{x} + \delta \dot{x} + \omega_0^2 x + \beta x^3 = \gamma \cos (\omega t + \phi)\,$

or, as a system of equations,

$\dot{u} = v\,$

$\dot{v} = -\omega_0^2 u -\beta u^3 - \delta v + \gamma \cos (\omega t + \phi)\,$

where u is the displacement of x, v is the velocity of x, and $ω$ , $β$ , $δ$ , $γ$ and $φ$ are constants.

The equation describes the motion of a damped oscillator with a more complicated potential than in simple harmonic motion; in physical terms, it models, for example, a spring pendulum whose spring's stiffness does not exactly obey Hooke's law.

[edit] Methods of solution

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 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 Newton's method and Runge-Kutta can be used.

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