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<- [[:chaosbook]]

====== Local stability ======

(ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#stability|Local stability]])  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-11 12:55//

Enter latest posts at the bottom - flows better. 

:?: Is that Schwarzenegger quote at the beginning of section 4.2 something he actually said?  When did he say that?  //[[jshyatt@gatech.edu|John Hyatt]] 2009-02-10// 

:!: Actually, it was a skit on [[http://prairiehome.publicradio.org/programs/20031129/scripts/guy_noir.shtml|Prairie Home Companion]]  --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-10//

:?:  In the tutorial we went through on 1/20/2008, we only generated one eigenfunction for time considerations.  When I ran it on my own, I let the calculation run longer and it generated about 30 eigenfunctions before I wanted to move on.  How many of these eigenfunctions are worth generating for a particular solution and why? //Dustin 2009-01-25//

:!: For an equilibrium, you need to get all eigenvalues with positive real parts, and all negative eigenvalues whose magnitude is less or comparable to the largest expanding eigenvalue. More precise statement is the Kaplan-Yorke criterion (keep adding negative eigenvalues as long as the sum of leading eigenvalues is positive; for much more detail, see the [[:gtspring2009:predrag:blog#blog|2009-10-29 discussion of Lyapunov vectors]]). You do not need to worry about the remaining (60 thousand!) eigendirections for which the negative eigenvalues are of larger magnitude, as they always contract: nonlinear terms cannot mix them up in such a way that expansion in some directions overwhelms the strongly contracting ones. In plane Couette 20 or so eigenvalues have so far sufficed.   --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-10//



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