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

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

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

• $\delta$ 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

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 $x^3$ 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 ($\delta = 0$) 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

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

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

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:

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

with the Hamiltonian H being positive.

External links

• Duffing oscillator on Scholarpedia
• MathWorld page