Bifurcations and Center Manifold

[Rubik]

Homework Statement

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

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

Homework Equations

The Attempt at a Solution

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

\dot{u} = \bar{u}
\dot{v} = \bar{v}² + \bar{u}

Is that all I have to do?

 

[jackmell]

Homework Statement

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

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

Homework Equations

The Attempt at a Solution

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

\dot{u} = \bar{u}
\dot{v} = \bar{v}² + \bar{u}

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 w=v+1/2, then w'=v' and v=w-1/2. Now, in terms of u and w, the first one is unchanged but what is w' in terms of u and w:

u'= u
w'=