## What is the difference between an excitable system and a relaxation oscillator?

1. The problem statement, all variables and given/known data

Consider the following set of differential equations:

$$\begin{eqnarray*} \dot{u} & = & b(v-u)(\alpha+u^2)-u \\ \dot{v} & = & c-u \end{eqnarray*}$$

The parameters $b \gg 1$ and $\alpha \ll 1$ are fixed, with $8\alpha b < 1$. Show that the system exhibits relaxation oscillations for $c_1 < c < c_2$ where $c_1,c_2$ are to be determined, and is excitable for c slightly less than c1.

2. Relevant equations

3. The attempt at a solution

There aren't a lot of good resources online for this sort of thing. If it helps, the linear stability matrix for this is

$$\left(\begin{array}{cc}-1-b(a+c^2)+2bc(-c+\frac{c+abc+bc^3}{b(a+c^2)}) & b(a+c^2) \\ -1 & 0\end{array}\right)$$

You can get the eigenvalues and such from that, but I'm not sure how they help. Can anyone explain what sort of difference I should be looking for? Thank you so much for your help!

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