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
or, as a system of equations,
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 x3 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 (δ = 0) and unforced (γ = 0) Duffing equation, an exact solution can be obtained using Jacobi's elliptic functions.
