The next step is combining the local quasi-potentials into a single surface. If the system only has a single equilibrium point or basin of attraction (e.g., a limit cycle) then the local quasi-potential is the global quasi-potential. If the system has two or more basins of attraction, then the local surfaces need to be aligned, adjusted, and combined. Details can be found in Moore et al. (2016) and references therein, but we briefly describe what's happening in the function that creates the global quasi-potential,
Briefly, the most probable way for a trajectory to transition from one basin to another involves passing through the lowest point on the surface specified by each local quasi-potential surface along the separatrix (the line between two basins of attraction). The two local quasi-potentials should be translated so that the minimum heights along the separatrix are the same. In this example, the minima of both local quasi-potentials occurred at the same point---the saddle at $(4.201, 4.004)$---so the algorithm amounts to matching at that point. This is a fairly straight-forward example, but see Moore et al. (2016), Example 3, for a more complicated example.
The functions that calculates the global quasi-potential is
QPGlobal. Its arguments minimally include:
For our example, we have:
unstable.x <- c(eqs[2, 1], eqs[4, 1]) # values are c(0, 4.2008) unstable.y <- c(eqs[2, 2], eqs[4, 2]) # values are c(0, 4.0039) ex1.global <- QPGlobal(local.surfaces = list(eq1.local, eq2.local), unstable.eq.x = unstable.x, unstable.eq.y = unstable.y, x.bound = bounds.x, y.bound = bounds.y)
And that's all she wrote---you've created your first global quasi-potential!
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.