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+
+Bistability in 2 variable systems
+
+review: mutual activation (positive feedback loop) 
+        and mutual inhibition (negative feedback loop)
+
+[review of simple mutual activitation and inhibition physical systems]
+
+when doing analysis, often want to plot nullclines in phase (aka, variable)
+space. the nullcline of a variable is a curve in phasespace where the time derivative of the given variable is zero.
+
+find these by expressing the differential of the variable (w/r/t time) as a
+symbolic expression (probably involving both variables) and solving for equals
+0. then we will analyse the intersecting points (which are equilibria, though
+not necessarily stable). might need to plot nullcline for varying "other"/free
+variables to find a state where there are 3x (or more) intersections, which are
+bistable systems.
+
+overall a bit confused; shouldn't this lecture have come earlier, before the
+stability analysis? oh, no, that was a single variable system.
diff --git a/lectures/lec14_bistability5 b/lectures/lec14_bistability5
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+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