Local stability and equilibria

In summary, the conversation discusses the existence of a finite number of isolated equilibria for a dynamical system, each of which is locally stable. The question is whether this can happen for more than one equilibrium point, to which the answer is yes, as shown by the example of a pitchfork bifurcation system. However, the system does not satisfy the requirement of all equilibria being stable.
  • #1
Ali 2
22
1
Hi all
Suppose for a dynamical system [itex]\dot x=f(x) , x \in \mathbb R^n [/itex] 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
 
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  • #2
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
example, pitchfork bifurcation system: x' = rx - x^3 has 2 stable equilibria for r > 0
 
  • #4
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
Oh, sorry I read it wrong. I did not see it saying ALL equilbira are stable
 

1. What is local stability?

Local stability is a property of a system where small perturbations or changes in the initial conditions of the system do not significantly alter its long-term behavior. In other words, the system is able to return to its previous state after experiencing small changes.

2. What is an equilibrium point?

An equilibrium point is a state in which the system remains unchanged over time. It is a point where the forces within the system are balanced and no net change occurs.

3. How is local stability determined?

Local stability is determined by analyzing the behavior of a system around its equilibrium point. This is done by linearizing the system's equations of motion and examining the eigenvalues of the resulting matrix. If all eigenvalues have negative real parts, the system is locally stable.

4. What is the difference between local and global stability?

Local stability only refers to the behavior of a system around its equilibrium point. Global stability, on the other hand, takes into account the behavior of the system for all possible initial conditions. A system can be locally stable but not globally stable if there are regions where the system's behavior is unstable.

5. How is local stability important in real-world applications?

Local stability is important in real-world applications because it allows us to predict the long-term behavior of a system and identify potential instabilities. This knowledge is crucial in fields such as engineering, economics, and ecology, where small changes can have significant impacts on the overall system.

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