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Bifurcations and Center Manifold

  1. Apr 27, 2012 #1
    1. The problem statement, all variables and given/known data

    If β=0 the neurone model is [itex]\dot{u}[/itex]= -u
    [itex]\dot{v}[/itex]= v2 + v - u + [itex]\delta[/itex]
    If [itex]\delta[/itex] = 1/4 it has critical point (0,-1/2)

    Transform the system so that the critical point is at the origin so let [itex]\bar{v}[/itex] = v +1/2 and find the equations of motion for (u,[itex]\bar{v}[/itex])
    2. Relevant equations



    3. The attempt at a solution

    Does this mean for a change of variables I let [itex]\bar{u}[/itex] =u and [itex]\bar{v}[/itex]= v + 1/2 subs in gives

    [itex]\dot{u}[/itex] = [itex]\bar{u}[/itex]
    [itex]\dot{v}[/itex] = [itex]\bar{v}[/itex]2 + [itex]\bar{u}[/itex]

    Is that all I have to do?
     
  2. jcsd
  3. Apr 27, 2012 #2
    No. You have a system in terms of v and u and you want to represent that system in terms of new variables which have a critical point at the origin so you can study the system for small changes of t near the origin. If I let say [itex]w=v+1/2[/itex], then [itex]w'=v'[/itex] and [itex]v=w-1/2[/itex]. Now, in terms of u and w, the first one is unchanged but what is w' in terms of u and w:

    [tex] u'= u[/tex]
    [tex]w'=[/tex]
     
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