Duffing oscillator

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{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.

Methods of solution

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

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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