|
|
| Line 1: |
Line 1: |
| In [[mathematics]], the point <math>\tilde{\mathbf{x}}\in \mathbb{R}^n</math> is an '''equilibrium point''' for the [[differential equation]]
| | Alyson is the name people use to contact me and I think it sounds quite great when you say it. I've always loved living in Kentucky but now I'm contemplating other options. To perform lacross is some thing he would by no means give up. Credit authorising is how she tends to make a residing.<br><br>Feel free to visit my page ... psychic chat online ([http://www.dabestrapperalive.com/users/HWechsler read full report]) |
| | |
| :<math>\frac{d\mathbf{x}}{dt} = \mathbf{f}(t,\mathbf{x})</math>
| |
| | |
| if <math>\mathbf{f}(t,\tilde{\mathbf{x}})=0</math> for all <math>t\,\!</math>.
| |
| | |
| Similarly, the point <math>\tilde{\mathbf{x}}\in \mathbb{R}^n</math> is an '''equilibrium point''' (or [[Fixed point (mathematics)|fixed point]]) for the [[difference equation]]
| |
| | |
| :<math>\mathbf{x}_{k+1} = \mathbf{f}(k,\mathbf{x}_k)</math>
| |
| | |
| if <math>\mathbf{f}(k,\tilde{\mathbf{x}})= \tilde{\mathbf{x}}</math> for <math>k=0,1,2,\ldots</math>.
| |
| | |
| Equilibria can be classified by looking at the signs of the eigenvalues of the linearization of the equations about the equilibria. That is to say, by evaluating the [[Jacobian matrix]] at each of the equilibrium points of the system, and then finding the resulting eigenvalues, the equilibria can be categorized. Then the behavior of the system in the neighborhood of each equilibrium point can be qualitatively determined, (or even quantitatively determined, in some instances), by finding the eigenvector(s) associated with each eigenvalue.
| |
| | |
| An equilibrium point is ''hyperbolic'' if none of the eigenvalues have zero real part. If all eigenvalues have negative real part, the equilibrium is a stable equation. If at least one has a positive real part, the equilibrium is an unstable node. If at least one eigenvalue has negative real part and at least one has positive real part, the equilibrium is a [[saddle point]].
| |
| | |
| [[Category:Stability theory]] | |
| [[Category:Dynamical systems]]
| |
| | |
| {{Mathanalysis-stub}}
| |
Alyson is the name people use to contact me and I think it sounds quite great when you say it. I've always loved living in Kentucky but now I'm contemplating other options. To perform lacross is some thing he would by no means give up. Credit authorising is how she tends to make a residing.
Feel free to visit my page ... psychic chat online (read full report)