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diff --git a/lectures/lec14_bistability5 b/lectures/lec14_bistability5 new file mode 100644 index 0000000..e53785f --- /dev/null +++ b/lectures/lec14_bistability5 @@ -0,0 +1,25 @@ +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 |