Can a Dynamical System Have Multiple Locally Stable Equilibria?

[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

 

[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

 

[wisvuze]

example, pitchfork bifurcation system: x' = rx - x^3 has 2 stable equilibria for r > 0

 

[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.

 

[wisvuze]

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