Diffusively coupled oscillators

Below are some examples of nonlinear oscillators where we think of each spatial point as an ODE and couple these by diffusion. Warning: As these are oscillators, do be mindful that some parameters may have rapid flashing.

Coupled Van der Pol oscillators

The (unforced) Van der Pol oscillator is a well-known example of a nonlinear oscillator of the form

\[\diff{^2X}{t^2} = \mu(1-X^2)\diff{X}{t}-X,\\\]

where $\mu$ is a coefficient of nonlinear damping. We now consider adding linear diffusive coupling to a spatial variant of it. Writing this as a system of first-order equations in time, we then have

\[\begin{aligned} \pd{X}{t} &= Y ,\\ \pd{Y}{t} &= D (\nabla^2 X + \varepsilon \nabla^2 Y) + \mu(1-X^2)\pd{X}{t}-X, \end{aligned}\]

where $D$ is a diffusion coefficient, and the term involving $\varepsilon$ is a kind of artificial diffusion to dampen numerical instabilities.

The interactive Van der Pol simulation shows how this system behaves with random initial data. Changing $\mu$ gives different structures acting on different time and length scales. You can also set the initial condition to be a constant (e.g. 0) and click to initiate a wave.

Coupled Duffing equations

The Duffing equation is another example of a nonlinear oscillator, where again we can add linear diffusive coupling to a spatial variant of it to get:

\[\begin{aligned} \pd{X}{t} &= \varepsilon D \nabla^2 X + Y,\\ \pd{Y}{t} &= D \nabla^2 X -\delta Y-\alpha X-\beta {X}^{3}+\gamma \cos{\left(\omega t\right)}, \end{aligned}\]

where $D$, $\alpha$, $\beta$ are parameters of the model, $\varepsilon$ is again an artificial diffusion term, and we have included a temporal forcing term (with parameters $\gamma$ and $\omega$) to increase the variety of observed dynamics.

The interactive Duffing equation simulation simulates this model with random initial conditions. Changing the value of $\alpha$ can lead to a variety of interesting dynamics.

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