Duffing JavaScript integrator

This webpage uses the Runge-Kutta-Fehlberg fourth-order method with fifth-order error checking (RKF45) to approximate the solution to the problem of the Duffing oscillator:

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

Below you can specify these various parameters, as well as the initial conditions and starting and end times. The default values give chaotic behaviour.

Simulation parameter form.
Parameter	Value	Explanation
$\alpha$ :		Linear stiffness parameter.
$\beta$ :		Nonlinearity in restoring force parameter.
$\gamma$ :		Amplitude of periodic driving force.
$\delta$ :		Damping parameter.
$\omega$ :		Angular frequency of periodic driving force.
$t_0$ :		Starting time for the simulation in seconds (s).
$t_f$ :		End time for the simulation in seconds..
$x_0$ :		x coordinate in metres (m) at time $t_0$ .
$\dot{x}_0$ :		First derivative of $x$ with respect to $t$ at $t_0$ (in $\mathrm{m}\cdot \mathrm{s}^{-1}$ ).
$\epsilon$ :		Absolute error tolerance.
$h_{\mathrm{Initial}}$ :		Initial step size.
$h_{\mathrm{Min}}$ :		Minimum allowed step size.
$\Delta t$ :		Time increment for skipping ahead in animation.
$t_1$ :		Time you want to skip ahead to in animation when you press the skip button.
$\mathrm{Width}$ :		Width (in px) of Plotly windows used for plotting and animation below.
$\mathrm{Height}$ :		Height (in px) of Plotly windows used for plotting and animation below.
$\mathrm{Delay}$ :		Proportion of animation time passed per real time. Delay=1.0 means animation and real time match. Delay<1 means the animation is going more slowly than real time. Delay>1.0 means it is going more rapidly.

© Brenton Horne. I do not claim to be infallible, so feel free to open issues or pull requests on GitHub if you notice flaws or things that can be improved.
I favour Oxford spelling, so if you notice I generally use different spelling to Australian spelling, that is probably why.
Homepage. Source code: https://github.com/fusion809/fusion809.github.io.
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Last modified: July 13, 2025.