Bifurcations and Center Manifold

  • Thread starter Rubik
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  • #1
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Homework Statement



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])

Homework Equations





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?
 

Answers and Replies

  • #2
1,800
53

Homework Statement



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])

Homework Equations





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?

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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