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

  1. Jun 2, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the following set of differential equations:

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

    The parameters [itex]b \gg 1[/itex] and [itex]\alpha \ll 1[/itex] are fixed, with [itex]8\alpha b < 1[/itex]. Show that the system exhibits relaxation oscillations for [itex]c_1 < c < c_2[/itex] where [itex]c_1,c_2[/itex] 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

    [tex]\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)[/tex]

    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!
     
  2. jcsd
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