# Local stability and equilibria

1. Apr 14, 2012

### Ali 2

Hi all
Suppose for a dynamical system $\dot x=f(x) , x \in \mathbb R^n$ there exists finite number of isolated equilibria, each of which is locally stable (i.e eigenvalues of the associated Jacobian have negative real parts).
My question is: Can this happen for more than one equilibrium? (sorry if it is a trivial question

Regards

Last edited: Apr 14, 2012
2. Apr 14, 2012

### wisvuze

what do you mean can it happen for more than 1 equilibrium point? didn't you just hypothesize a finite number of equilibrium points such that blah blah blah? or do you mean if the finite number in the statement can exceed 1. If so, then the answer is yes

3. Apr 14, 2012

### wisvuze

example, pitchfork bifurcation system:

x' = rx - x^3 has 2 stable equilibria for r > 0

4. Apr 14, 2012

### Ali 2

hi
yes, my question means: can the number of equilibria in the statement exceed one?
The example you provided don't satisfy the requirement. The equilibrium 0 is unstable
thanks.

5. Apr 14, 2012

### wisvuze

Oh, sorry I read it wrong. I did not see it saying ALL equilbira are stable