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# Duffing equation

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The **Duffing equation** (or **Duffing oscillator**), named after Georg Duffing, is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by

- <math>\ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 = \gamma \cos (\omega t)\,</math>

where the (unknown) function *x*=*x*(*t*) is the displacement at time *t*, <math>\dot{x}</math> is the first derivative of *x* with respect to time, i.e. velocity, and <math>\ddot{x}</math> is the second time-derivative of *x*, i.e. acceleration. The numbers <math>\delta</math>, <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math> and <math>\omega</math> 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.

## Contents

## Parameters

- <math>\delta</math> controls the size of the damping.
- <math>\alpha</math> controls the size of the stiffness.
- <math>\beta</math> controls the amount of non-linearity in the restoring force. If <math>\beta=0</math>, the Duffing equation describes a damped and driven simple harmonic oscillator.
- <math>\gamma</math> controls the amplitude of the periodic driving force. If <math>\gamma=0</math> we have a system without driving force.
- <math>\omega</math> 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 <math>x^3</math> 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 (<math>\delta = 0</math>) and undriven (<math>\gamma = 0</math>) Duffing equation, an exact solution can be obtained using Jacobi's elliptic functions.

## Boundedness of the solution for the undamped and unforced oscillator

Multiplication of the undamped and unforced Duffing equation, <math>\gamma=\delta=0,</math> with <math>\dot{x}</math> gives:^{[1]}

- <math>

\begin{align}

& \dot{x} \left( \ddot{x} + \alpha x + \beta x^3 \right) = 0 \\ &\Rightarrow \frac{\text{d}}{\text{d}t} \left[ \tfrac12 \left( \dot{x} \right)^2 + \tfrac12 \alpha x^2 + \tfrac14 \beta x^4 \right] = 0 \\ & \Rightarrow \tfrac12 \left( \dot{x} \right)^2 + \tfrac12 \alpha x^2 + \tfrac14 \beta x^4 = H,

\end{align} </math>

with *H* a constant. The value of *H* is determined by the initial conditions <math>x(0)</math> and <math>\dot{x}(0).</math>

The substitution <math>y=\dot{x}</math> in *H* shows that the system is Hamiltonian:

- <math> \dot{x} = + \frac{\partial H}{\partial y}, </math> <math> \dot{y} = - \frac{\partial H}{\partial x} </math> with <math> \quad H = \tfrac12 y^2 + \tfrac12 \alpha x^2 + \tfrac14 \beta x^4.

</math>

When both <math>\alpha</math> and <math>\beta</math> are positive, the solution is bounded:^{[1]}

- <math> |x| \leq \sqrt{2H/\alpha}</math> and <math> |\dot{x}| \leq \sqrt{2H},</math>

with the Hamiltonian *H* being positive.

## References

### Inline

- ^
^{a}^{b}Bender & Orszag (1999, p. 546)

### Other

- Bender, C.M.; Orszag, S.A. (1999),
*Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory*, Springer, pp. 545–551, ISBN 9780387989310 - Addison, P.S. (1997),
*Fractals and Chaos: An illustrated course*, CRC Press, pp. 147–148, ISBN 9780849384431

## External links

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