Coupled FitzHugh - Nagumo system - equilibrium position

The coupled FitzHugh-Nagumo oscillators produce an attractor (previous post) which has an obvious resting / equilibrium point. The trajectory spends a long time spiralling about this point before making one or more fast excursions, before returning to the equilibrium point, and so on.

The following shows the location of the centre of the equilibrium region for Sprott's figure 6.7 [1]. I've added my estimates as to the coordinates of the equilibrium centre:

Given that the trajectory moves slowly around the equilibrium point, it seems reasonable that one could work out an analytic solution as to the whereabouts of this point. In pursuit of this, I took Sprott's coupled FitzHugh-Nagumo system (equation (6.7)), and set the right hand sides to zero (i.e. the game being to find where the speed of the trajectory is zero):

After a fair bit of algebra, I worked out the following rather compact equations for the coordinates of the equilibrium position (see box below). These equations are valid for small values of c - consistent with Sprott's comment that 'chaotic solutions require that c be rather small' [1]. These equations were derived from a power series development of an exact solution for the coordinates, which was worked out, the derivation of which is given also, below.

It is evident that the y-coordinate does not depend on parameter c, when c is small.

The following tables gives the calculated coordinates of the equilibrium point, for k = -1 and k = 0 (no coupling). All other values are as per Sprott's figure 6.7 (i.e. a = 0.4, b = 0.5, c = 0.01). The first table gives the results obtained using the exact solution, the second gives results obtained using the above, boxed, equations.

Coordinates of equilibrium point calculated using
exact derivation.

Coordinates of equilibrium point calculated using
approximate derivation.

These agree rather nicely with the values I measured directly off Sprott's figure 6.7 - namely -0.78 for x₁ and +0.51 for y₁.

What's interesting (to me) is that the equilibrium point actually depends on the value of the coupling parameter k. I (perhaps naively) assumed that the equilibrium point for a single decoupled oscillator would be the same as that for the coupled case - but experiments with the analog computer indicated otherwise:

Analog computer results for coupled (k = -1) and uncoupled (k = 0) cases. It is evident
that the equilibrium points have very different y-values.

In particular the y-coordinate of the equilibrium point varies strongly with k.

In contrast, the x-coordinate of the equilibrium position does not depend strongly on the coupling parameter k. To first order, the x-coordinate is dominated by the -a/b term which, in this case is equal to -0.8.

For reasons which remain unexplored (offset voltages?), the analog computer's results are slightly different to Sprott's. My equilibrium positions are shifted with respect to the origin of the plot.

Coordinates of equilibrium point measured from

analog computer output plot.

Derivation of Exact Coordinates of Equilibrium Point

 

Note

The ninth order (degree) polynomial appears to have just one real root. This is equation (7) above.

Reference

[1] Sprott J.C. Elegant Chaos: Algebraically Simple Chaotic Flows, pp. 123 - 125.