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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## Methods of solution

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

- $ddot\{x\}\; +\; delta\; dot\{x\}\; +\; omega\_0^2\; x\; +\; beta\; x^3\; =\; gamma\; cos\; (omega\; t\; +\; phi),$

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 $omega$, $beta$, $delta$, $gamma$ 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.

- 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.

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Last updated on Wednesday June 18, 2008 at 20:00:42 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday June 18, 2008 at 20:00:42 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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