Here we look at the FitzHugh–Nagumo model, given by

\[\begin{aligned}\pd{u}{t}&=\nabla^2 u +u-u^3-v,\\ \pd{v}{t}&=D\nabla^2v+ \varepsilon_v(u-a_v v-a_z),\end{aligned}\]

where we take $D>1$.

  • Load the interactive simulation

  • Click in the domain to initiate a pattern-forming instability, which will form roughly concentric rings as it expands.

  • This system has many different kinds of solutions which are stable over long time periods. To see this, change the initial condition, under  Initial conditions so that $u|_{t=0}$ has the value ‘RAND’. Then press to restart the simulation. It should now exhibit patterns which are much more spot-like.

Turing–Hopf bifurcations

We now vary the parameters from the previous simulation so that it supports both pattern formation, but also oscillations. These oscillations come from steady states undergoing Hopf bifurcations. In such regimes, one can often find a range of complex spatial, temporal, and spatiotemporal behaviours, many of which can be simultaneously stable for different initial conditions.

To illustrate this, we consider the initial conditions

\[u(x,y,0) = \cos\left(\frac{m \pi x}{L}\right)\cos\left(\frac{m \pi y}{L}\right), \quad v(x,y,0)=0,\]

for some integer $m$ and domain length $L=280$.

  • Load the interactive simulation

  • This simulation can display long-time solutions that exhibit all three kinds of behaviour, depending on the values of $m$, $D$, and the other parameters. Try $m=4$, $m=3$, and $m=6$ for example.

Three-species variant

A three-species variant of the FitzHugh–Nagumo model is

\[\begin{aligned}\pd{u}{t}&=\nabla^2 u +u-u^3-v,\\ \pd{v}{t}&=D_v\nabla^2v+ \varepsilon_v(u-a_v v-a_w w-a_z)\\ \pd{w}{t}&=D_w\nabla^2w+ \varepsilon_w(u-w).\end{aligned}\]
  • Load the interactive simulation

  • The simulation demonstrates the dynamics of this system in a regime which has both homogeneous limit cycles and pattern formation competing against one another.

  • The initial pattern formed in this simulation will eventually be destroyed by the oscillations. You can increase the value of $a_v$ to stabilise the pattern for longer, and if $a_v=0.2$, the pattern will eventually overtake the oscillations and fill the entire domain.