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# Duffing equation

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

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

or, as a system of equations,

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

where u is the displacement of x, v is the velocity of x, and $omega$, $beta$, $delta$, $gamma$ and $phi$ 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.

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

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