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author | bnewbold <bnewbold@robocracy.org> | 2014-05-29 23:17:46 -0400 |
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committer | bnewbold <bnewbold@robocracy.org> | 2014-05-29 23:17:46 -0400 |

commit | d59f471befbcc62aa178b46e2aaad99d58e350f1 (patch) | |

tree | e2853f034a48881a86ffed4692ba8e8243e6cb86 | |

parent | 392a611d547287920c46edbdaa8c2025596c8edb (diff) | |

download | dmmsb2014-d59f471befbcc62aa178b46e2aaad99d58e350f1.tar.gz dmmsb2014-d59f471befbcc62aa178b46e2aaad99d58e350f1.zip |

tonight's lectures

-rw-r--r-- | lectures/lec13_bistability4 | 20 | ||||

-rw-r--r-- | lectures/lec14_bistability5 | 25 |

2 files changed, 45 insertions, 0 deletions

diff --git a/lectures/lec13_bistability4 b/lectures/lec13_bistability4 new file mode 100644 index 0000000..f0c6363 --- /dev/null +++ b/lectures/lec13_bistability4 @@ -0,0 +1,20 @@ + +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 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 |