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Duffing equation

File:Forced Duffing equation Poincaré section.png
A Poincaré section of the forced Duffing equation suggesting chaotic behaviour

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.

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:

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

  1. ^ 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 
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External links

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