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where the (unknown) function x=x(t) is the displacement at time t, is the first derivative of x with respect to time, i.e. velocity, and is the second time-derivative of x, i.e. acceleration. The numbers , , , and are given constants.
The equation describes the motion of a damped oscillator with a more complicated potential than in simple harmonic motion (which corresponds to the case β=δ=0); in physical terms, it models, for example, a spring pendulum whose spring's stiffness does not exactly obey Hooke's law.
The Duffing equation is an example of a dynamical system that exhibits chaotic behavior. Moreover the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour.
- controls the size of the damping.
- controls the size of the stiffness.
- controls the amount of non-linearity in the restoring force. If , the Duffing equation describes a damped and driven simple harmonic oscillator.
- controls the amplitude of the periodic driving force. If we have a system without driving force.
- controls the frequency of the periodic driving force.
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 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 Euler's method and Runge-Kutta can be used.
In the special case of the undamped () and undriven () Duffing equation, an exact solution can be obtained using Jacobi's elliptic functions.
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