blob: e53785fb8b7d9e298ef3d28f4c4c9c63e6993b81 (
plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
|
Stability analysis for multi-variate systems
could do a crude thing: chose points (on nullclines) near an equilibrium and
integrate to see if there is convergence or divergence.
jacobians! horray. the eigenvalues determine stability. stable if both real
parts of eigenvalues are negative; otherwise unstable.
claims that analytical evaluation of jacobians is hard but numerical is easy.
hmm.
the above is the algebaic way.
geometric way is to calculate gradient w/r/t to time in the areas around and
between nullclines (by calculating at zero/zero and/or very far away at
inf/inf, then "flipping" every time a null cline is crossed (which makes sense
because these are points were derivative is zero). the arrows/gradient "points
to" the stable intersections.
aka, vector looks like: | d[A]/dt |
| |
| d[B]/dt |
can also do one-dimensional analysis on individual nullcline lines, again using
"flip if crossing (other) nullclines" trick
|
