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:

$$u'= u$$
$$w'=$$