Coupled duffing oscillator solver

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 coupled Duffing oscillator

\begin{aligned} \ddot{x} + \delta \dot{x} + \alpha x + \beta x^3 + \kappa (x-y) &= \gamma \cos{(\omega t)} \\ \ddot{y} + \delta \dot{y} + \alpha y + \beta y^3 + \kappa (y-x) &= 0. \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.
$\kappa$ :		Coupling term coefficient.
$t_f$ :		End time (in seconds or s) for the simulation.
$x_0$ :		Initial $x$ coordinate in metres (m) .
$\dot{x}_0$ :		Initial value of $\dot{x}$ ( $\mathrm{m}\cdot \mathrm{s}^{-1}$ ).
$y_0$ :		Initial $y$ coordinate in metres (m) .
$\dot{y}_0$ :		Initial value of $\dot{y}$ ( $\mathrm{m}\cdot \mathrm{s}^{-1}$ ).
$\epsilon$ :		Error tolerance.
$\mathrm{Tol}_{\mathrm{type}}$ :		Tolerance type, can be either absolute (0) or relative (1).
$h_{\mathrm{Initial}}$ :		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.
$t_\mathrm{Scale}$ :		Proportion of animation time passed per real time. $t_{\mathrm{Scale}}=1.0$ means animation and real time match. $t_{\mathrm{Scale}}<1.0$ means the animation is going more slowly than real time. $t_{\mathrm{Scale}}>1.0$ 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.
Website built with Franklin.jl.

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Last modified: August 13, 2025.