Duffing equation

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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

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

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

  • \delta controls the size of the damping.
  • \alpha controls the size of the stiffness.
  • \beta controls the amount of non-linearity in the restoring force. If \beta=0, the Duffing equation describes a damped and driven simple harmonic oscillator.
  • \gamma controls the amplitude of the periodic driving force. If \gamma=0 we have a system without driving force.
  • \omega controls the frequency of the periodic driving force.

Methods of solution[edit]

Duffing oscillator limit cycle γ>0
Duffing oscillator limit cycle phase animation γ>0
Duffing oscillator chaos oscillation γ<0
Duffing oscillator attractors animation γ<0

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 undriven (\gamma = 0) Duffing equation, an exact solution can be obtained using Jacobi's elliptic functions.

Boundedness of the solution for the undamped and unforced oscillator[edit]

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


\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}

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

The substitution y=\dot{x} in H shows that the system is Hamiltonian:

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

When both \alpha and \beta are positive, the solution is bounded:[1]

 |x| \leq \sqrt{2H/\alpha}   and    |\dot{x}| \leq \sqrt{2H},

with the Hamiltonian H being positive.

References[edit]

Inline[edit]

  1. ^ a b Bender & Orszag (1999, p. 546)

Other[edit]

  • 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[edit]