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

[tjackson3]

Homework Statement

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.

Homework Equations

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!

 

[mighty2000]

Thank you for sharing your thoughts on this set of differential equations. It is always exciting to see someone interested in exploring the dynamics of a system.

Firstly, let us try to understand what relaxation oscillations and excitability mean in the context of this system. Relaxation oscillations refer to a type of nonlinear oscillations where the system exhibits a slow and fast response, leading to a relaxation-like behavior. Excitability, on the other hand, refers to the ability of the system to respond to small perturbations, resulting in a transient change in its behavior.

Now, to show that the system exhibits relaxation oscillations, we need to determine the conditions for which the system shows a slow and fast response. Looking at the equations, we see that the first equation contains a term (b(v-u)(\alpha+u^2)) that is much larger than the other terms (since b \gg 1 and \alpha \ll 1). This suggests that the system will have a fast response when this term dominates, i.e., when v \approx u. On the other hand, the second equation contains a term (c-u) that will be dominant when u \approx c. This suggests that the system will have a slow response when u \approx c.

Therefore, we can conclude that the system will exhibit relaxation oscillations when v \approx u and u \approx c. Now, to determine the values of c_1 and c_2, we need to look at the behavior of the system when v \approx u and u \approx c. For v \approx u, we can rewrite the first equation as:

\dot{u} \approx b(u-u)(\alpha+u^2)-u = -u(1+bu^2)

This equation shows that the system will have a fast response when u is small (since the term -u dominates). On the other hand, for u \approx c, we can rewrite the second equation as:

\dot{v} \approx c-u \approx c-c = 0

This equation shows that the system will have a slow response when u is close to c (since the term c dominates). Therefore, we can say that the system will exhibit relaxation oscillations for c_1 < c < c_2, where c_1 is the value of c for which the fast response dominates and c_2 is the value of c for which the slow